1 64644 1 g = CD OSMANIA UNIVERSITY LIBRARY Call No. 2X3 -7/^ ^(Accession No. /9* Author ^cie*^/^^/-- t&u&uv . r S/fa Vi: This book should be returned on or before the date last marked b$loiv. CONTINUOUS GROUPS OF TRANSFORMATIONS CONTINUOUS GROUPS OF TRANSFORMATIONS By LUTHER PFAHLER EISENHART Dod Professor of in Princeton University PRINCETON PRINCETON UNIVERSITY PRESS LONDON: HUMPHREY MILFORD OXFORD UNIVERSITY PRESS 1933 COPYRIGHT, 1933, BY PRINCETON UNIVERSITY PRESS PRINTED IN THE U. 8. A. BY THE MAPLE PRESS COMPANY, YORK, PA. PREFACE The study of continuous groups of transformations inaugurated by Lie resulted in the development of an extensive theory by him and under his inspiration by Engel, Killing, Scheffers and Schur. Cartan in his thesis placed many of these results on a more rigorous basis. Bianchi and Fubini developed geometrical applications of the theory. This chapter of the history closed about thirty years ago. A new chapter began about ten years ago with the extended studies of tensor analysis, Riemannian geometry and its generalizations, and the application of the theory of continuous groups to the new phys- ical theories. This book sets forth the general theory of Lie and hisjr contempo- raries and the results of recent investigations with thfejtid of the methods of the tensor calculus and concepts of the new 'differential geometry. The first three chapters contain in the main the results of the first period. Chapter Four is devoted to the theory of the adjoint group and the sequence of theorems basic to the character- ization of semi-simple groups, as developed by Cartan and recently by Weyl and Schouten. Although geometrical ideas are used throughout the book, Chapter Five contains the geometrical applications of the theory both in the space of the transformations and in the group-space, and here are to be found particularly the concepts of the new differential geometry. The theory of contact transformations with applications to geometry and mechanics is set forth in the closing chapter. Of the many exercises in the book some involve merely direct applications of the formulas of the text, but many of them constitute extensions of the theory which would properly be included as por- tions of a more extensive treatise. References to the sources of these exercises are given for the benefit of the reader. All references are to the papers listed in the Bibliography. In the writing of this book I have had invaluable assistance and criticism by my colleagues Professors Bohnenblust, Knebelman and Robertson, and by my former student Dr. T. S. Graham particularly in connection with Chapter Four. The galley proof has been read by the students in my course this year and revised on the basis of their suggestions; Mr. F. D. Cubello has aided me considerably while seeing the book through the press. LUTHER PFAHLER EISENHART. May, 1933. CONTENTS CHAPTER I THE FUNDAMENTAL THEOREMS SECTION PAGE 1. Certain systems of partial differential equations. Mixed systems . , 1 2. Linear operators. Complete systems of linear partial differential equations 6 3. Essential parameters in a set of functions 9 4. Groups and groups of transformations 13 5. Fundamental differential equations of a group 18 G. First fundamental theorem 22 7. Conditions upon the constants of structure. Transformations of coordinates and parameters 25 8. Semi-group of inverse transformations 28 9. The parameter-groups of a group G r 31 10. One-parameter groups 32 11. Sub-groups G\ of a group G r 36 12. Canonical parameters 44 13. Abelian groups 48 14. The vectors U a a 51 15. The second and third fundamental theorems 53 CHAPTER II PROPERTIES OF GROUPS. DIFFERENTIAL EQUATIONS 16. Sub-groups of a G r 59 17. Absolute and relative invariants of a group 61 18. Ordinary and singular points. Order of a transformation at a point . . 64 19. Invariant varieties 67 20. The group induced in an invariant variety 70 21. Transitive and intransitive groups 71 22. Equivalent groups 76 23. Imprimitive and primitive groups 80 24. Systatic and asystatic groups 83 25. Differential equations admitting linear operators 85 26. Extended groups. Ordinary differential equations of the first order . . 92 27. Extensions of the second and higher orders. Ordinary differential equations of the second order 97 28. Differential invariants 101 CHAPTER III INVARIANT SUB-GROUPS 29. Groups invariant under transformations 109 30. Commutative groups. Reciprocal simply transitive groups. ... 113 vii viii CONTENTS SECTION PAGE 31. Invariant sub-groups 118 32. Isomorphisms of groups 124 33. Series of composition of a group. Theorem of Lie- Jordan 127 34. Factor groups 129 35. Derived groups 131 36. Integrable groups 134 CHAPTER IV THE ADJOINT GROUP 37. Linear homogeneous groups and vector-spaces 140 38. Canonical form of linear homogeneous transformations. Fundamental vectors 145 39. Adjoint group of a given group 150 40. The characteristic equation of a group. The rank of a group 155 41. Determination of invariant sub-groups 160 42. Integrable groups 162 43. The root-spaces of the matrix ij (U Q ) for a regular vector u 165 44. Canonical form of the matrix i)(u). Cartan's criterion for integrable groups 169 45. Semi-simple groups 173 46. Classification of the structure of simple and of semi-simple groups. . 180 CHAPTER V GEOMETRICAL PROPERTIES 47. Riemannian spaces 186 48. Linearly connected manifolds. Linear connection determined by a simply transitive group 192 49. Simply transitive groups determined by linear connections of zero curvature 195 50. Geometry of the group-space 198 51. Groups of motions. Equations of Killing 208 52. Translations 211 53. The determination of groups of motions 213 54. Equations of Killing in another form 217 55. Intransitive groups of motions 221 56. Spaces V* which admit a group of motions 226 57. Spaces Vt admitting a G* of motions 228 58. Motions in a linearly connected manifold 229 CHAPTER VI CONTACT TRANSFORMATIONS 59. Definition of homogeneous contact transformations 238 60. Geometrical properties of homogeneous contact transformations. . . 242 61. Definition of homogeneous contact transformations by means of funda- mental varieties 245 62. Infinitesimal homogeneous contact transformations 251 63. Non-homogeneous contact transformations 255 64. Restricted non-homogeneous contact transformations 260 CONTENTS ix SECTION PAGE 65. Homogeneous contact transformations of maximum rank 263 66. Geometrical properties of continuous groups of maximum rank. Waves 267 67. Application to geodesies of a Riemannian space 274 68. Application to dynamics 277 69. Function groups 281 70. Homogeneous function groups 287 Bibliography 293 Index 297 CHAPTER I THE FUNDAMENTAL THEOREMS 1. Certain systems of partial differential equations. Mixed systems. In this section and the next we establish certain theorems concerning differential equations which are used later. Consider a system of partial differential equations c)Q a (1.1) - ^(0>, - - , 0"; xS , *) B t/^(0; x) (a = 1, - , m; i = 1, , n), where the ^'s are functions of the 0's and the z's; it is understood that the following treatment applies to a domain in which the ^'s are analytic in the 0's and z's. These equations are equivalent to the system of total differential equations (1.2) dff* = tfdx* (a 1, , m] i = 1, , n). In these equations we have made use of the convention, to be used throughout, that when the same index appears twice in a term this .term stands for the sum of the terms obtained by giving the index each of its values; thus the right-hand member of (1.2) stands for the sum of n terms as i takes the values 1 to n. The conditions of integrability of (1.1) are >j = *'' ' ' ' n ' we remark that in these equations and 7 are summed from 1 to m. If these equations are satisfied identically, the system (1.1) is said to be completely integrable. In this case the solution is expressible in the form (1.4) /. THE FUNDAMENTAL THEOREMS (dtf*\ ) = d/J(c; XQ) and the other coefficients are obtained dxyo by differentiating (1.1) and evaluating by equating the x's to the xo's and the 0's to the c's. Thus for the domain of the x's for which the series (1.4) are convergent we have a solution of (1.1) determined by m constants. We denote such a solution by (1.5) r = +l , ,, x) = ( r, we repeat this process and obtain a system of t( ^ s) equations. If t > s, we continue the process and so on. Finally we have either n independent equations in which case the only solution of (2.7) is / = const., or we get a number u less than n for which a, 6, c in (2.10) take the values 1, . . . , u. In this case we say that the system is complete of order u. Hence the solution of a system of the form (2.7) is that of the solution of complete systems in which (2.10) hold. If (2.7) is a complete system, so also is (2.11) Xif - \ a b X a f = 0, where the X's are any functions of the X'B such that the determinant 8 7. THE FUNDAMENTAL THEOREMS |\S| is not zero. For, from (2.10) it follows that the right-hand member of (2.6) is linear in X\f, . . . , X r f, and since the latter are expressible from (2.11) linearly in the X'f's, we have from (2.6) expressions of the form (2.10), which proves the statement. Since the rank of M (2.8) is r, there is no loss in generality in assuming that |{J|, for a, b = 1, , r is different from zero. Hence we may solve equations (2.7) for p t| . . . , p r and write the result in the form (2.12) Xaf = Po + tip* = (a = 1, - - t r;t = r + 1, - , n). Evidently these are of the form (2.11) and hence form a system equivalent to (2.7); when a complete system is expressed in this form, it is called Jacobian. If we put X a f = oV> we have that (2.13) ft = S 6 , {'* = ft (a, 6 = 1, , r\ p = r + 1, , n), where 60, called the Kronecker deltas, are defined by (2.14) dl = 1 or 0, as a = b or a 7* 6; the Kronecker deltas are used frequently throughout this book with different indices but they always have the values (2.14) for the indices involved. Analogously to (2.10) we have (X a , X b )f = yabXcf. From (2.2) and (2.13) we have (X. 9 X' b )f = (Xtfj - Xfflp. + (XM - XiMpt = (Xa^b ~ Xb^Pt (c = 1, ,r;t = r + 1, , n). Since pi, , . . , p r do not appear in this last expression, it follows that 706 = 0, that is, for a complete system in the Jacobian form (x' a , X'M = o. Hence X'ati - Xiti = 0, that is, ^ P + ^ = ^ + ^ ( M-i.'">'A dx" * **dx 9 dx b + * b dx \p, q = r + 1, , n) Comparing these equations with (1.3), we see that the equations 3. ESSENTIAL PARAMETERS 9 dx p (2.15) - ft (a = 1, - , r; p - r + 1, , n) are completely integrable. Moreover equations (2.12) are the associated system of (2.15). Hence from the results of 1 we have that equations (2.12), and consequently (2.7), admit n r inde- pendent solutions, and that they do not admit more than n r independent solutions. Hence we have: [2.1] A complete system of r homogeneous linear partial differential equations of the first order in n(>r) variables admits exactly n r independent solutions. 3. Essential parameters in a set of functions. Consider a set of n functions /* of n variables a; 1 , . . . , x n and r parameters a 1 , . . . , a r , say /O i \ /j/ i n I r\ fif \ ( ' 1 T^T =0 (a = 1, - , r; 3. ESSENTIAL PARAMETERS 11 and we denote by MI the rank of the matrix (3 .6) v ' da a da" da" We get thus a sequence of ranks (3.7) MO ^ MI ^ M2 ' ^ M ' ^r. If any /i is equal to r, the ^>'s are zero and the r parameters are essen- tial. Suppose that M-I < r and that M* = M-i> then we shall show that M+i = M- Since /* = M-i> it follows that the sum on the right involving terms of the matrix Af.-i. Differ- entiating with respect to x k , we have that the terms ' * l * g ' ** are expressible linearly in terms of the members of the matrix M,_i, which proves the statement. Hence the terms of the sequence (3.7) continually increase and attain the maximum r, in which case the r parameters are essential, or they cease increasing after a certain one M(O), in which case they are not essential. We shall show that in the latter case M of the parameters are essential. If fjL 8 < r, the equations (3.4) and the sequence up to and including (3.5) being of rank M r M of the r ~ /l * be these functions and take them as functions of the a's but not the x'a. Then /O ON V \ each a is replaced by -> and consequently we have OX dx" p dx k from which it follows that

r -"' = and so on. These independent equations form a complete system. For, the commutator (2) of any two equated to zero admits /* as solutions and hence it must be a linear combination of the given equations. Hence the functions are expressible in terms of Ma essential parameters, and we have: [3.2] The number of essential parameters in terms of which the functions /*(x; a) are expressible is equal to the maximum number attained in the sequence (3.7). Exercises 1. Find the mixed system of equations (1.1) for which u = ax + by, v - axy, w &x* + <*2/ J is the general solution, a and 6 being arbitrary con- stants, and determine the system F for this case. 2. When the functions \l>" in (1.1) are linear and homogeneous in the 0's, the equations of the sets Fi, . . . , FN have this property, and in order that (1.1) admit a solution it is necessary and sufficient that there exist a positive integer N(m) such that the rank of the matrix of the sets FI, . . . , FN is m q(q ^ 1) and that this be the rank also of the sets Fi, . . . , FN+I. 3. A complete system admitting p independent functions p 1 , ...,?* as solutions is obtained by choosing for y for a = 1, , n p any n p inde- pendent solutions of the equations - (a - 1, -.., p); all complete systems admitting the same solution are in the relation (2.11). 4. Find a complete system admitting the solution x*x* + z, x* -f x 4 . 5. Find the most general complete system in 3 variables admitting the solu- tion (* l ) + (*)' + (*). 6. If for a complete system of r equations in n variables a number ( x > a i t a ) J \ x ) a ) \ l ~" " t n ) for each set of values of the a's define a transformation of a point P(x) in F n into a point P'(x'). If the values of the x's and o's are such that the jacobian of the fa with respect to the x's is not zero, that is, () | * 0, |dX in accordance with the theory of implicit functions* equations C4.1) may be solved uniquely for the x's, at least in the neighborhood of a point for which (4.2) holds, so that we have (4.3) x<=7'*(x';a) (i = 1, , n). These equations define the inverse transformation of P' into P. * Fine, 1927, 4, p. 253. 14 /. THE FUNDAMENTAL THEOREMS *""** It is understood that the functions /* are continuous in the x'a and in the a's, and that the same is true of their derivatives of such orders as arise in the subsequent discussion. From this it follows that if P'(z') is the transform of a point P(x) for particular values of the a's, then for small changes in the values of the a's we get points in the neighborhood of P'. Accordingly we say that the trans- formations are continuous. We indicate the transformations (4.1) and (4.3), respectively for a set of values a? of the a's in the symbolic form (4.4) T ai x = x' f 27>' = x, that is, T ai is the operation which sends a point P(x) into a point P'(x f ), the latter depending upon P and the values a?. If we apply a transformation T 0l and then another T ai we obtain a transforma- tion which we indicate by T a ,T ai , which is called the product of the two.* It does not follow necessarily that the product of two transformations is a member of the set (4.1), that is, that there are values of the a's, say a?, such that T a ^T ai and T a > have the same effect. From the definition of T^ 1 it follows that (4.5) 77X* = , Tattxf = x' 9 that is, either product T^lT ai or T ai T^* leaves each point unaltered, which transformation is called the identity. However, it does not follow that there are values of the a's in (4.1) yielding either the inverse transformation or the identity. Before proceeding further with a discussion of the particular operators T a defined by equations such as (4.1), we consider a set of operators T a abstractly. They constitute the elements of a class of objects. What in the above discussion we called the product of two operators constitutes an operation between two members of the class; we indicate the result of the operation on two elements T a and T* in this order by T*T a . Such a class, however its elements and the operation are defined, are said to form a group with respect to the operation under the following conditions: 1 If T a and T& are elements of the class, then TbT a is uniquely defined and is an element of the class; 2 The associative law holds for the operation, thus * Earlier writers indicated this product by T Cl T a ^ but many recent writers use the notation proposed. 4. GROUPS OF TRANSFORMATIONS 15 T c (T b T a ) = (T e T b )T a , that is, the element given by the operation upon the element TbT a and T c is the same as when the operation is applied to T a and T c T b ; 3 There is in the class an element / called the identity such that TJ = IT a = T a for every element T a ] 4 For each element T a of the class there is an element 77 l such that T~ l T a = T a T7 l - /. The question to what extent these conditions are independent has been investigated by various writers,* and for a discussion of this question the reader is referred to the articles indicated. If only the conditions 1 and 2 of the above set are satisfied, the elements are said to form a semi-group for the operation. As an example of a group we consider the set of all non-singular analytic transformations that is, those for which the jacobian is of rank n for the domain dx of the x's considered. Then from the theory of implicit functions! it follows that the above equations admit an inverse x< = /<(s'S , *"). If Ti and T 2 are given by x' = /i(z), x' = / 2 (z), then T 2 Ti is given by x' = /2(/i(z)), and thus the conditions 1 is satisfied for the domain in which the functions so defined are regular; moreover, the jacobian of this function, being the product of the jacobians of /i and / 2 , is not zero for those values of the x's for which these two functions satisfy this condition. Whenever the elements of a class are defined by equations, as in this case, the condition 2 is satisfied, because the process of substitution is associative. Since X H = x { are equations of the set, all the conditions are satisfied, and thus we have the group of analytic transformations. Since the functions f* may be thought of as power series in the x'& (including poly- nomials), the transformations involve an infinite number of arbitrary constants and consequently it is an infinite continuous group. * Cf. Dickson, 1905, 2; Huntington, 1905, 3; Moore, 1905, 4. t Fine, 1927, 4, p. 253. 16 /. THE FUNDAMENTAL THEOREMS Equations such as (4.1) involve a finite number of arbitrary constants, and when they define a group for continuous values of the a's, the group is said to be finite and continuous. Before we proceed to the determination of the conditions which (4.1) must satisfy in order that they define a group, we wish to point out that, when these conditions are realized, they may obtain only for a limited domain about the point to which any transformation of the group is applied. For such a point the functions /* and their derivatives of required order must be continuous in the x's and a's. Furthermore, the functions must be such that for a value of the a's, say aj, the equations (4.1) are x fi = x*, so that (4.2) must hold for these values of the a's, and consequently for values of the a's sufficiently near aj, since we are dealing with continuous functions. Accordingly when we speak of a group of transformations in what follows it is understood that the statements hold for those values of the x'a and the a's for which the conditions 1, 2, 3 and 4 hold. We return to the consideration of equations (4.1) and observe that in order that these equations satisfy the condition 1 of a group it is necessary that for each set of values a? and a? there exist a set of values a?, functions of the former, say (4.6) a? = *(ai\ a 2 ) (a = 1, , r), such that the equations (4.7) /*[/(* ;ai); a,] /*(*; a,) are identities in the re's, ai's and a 2 's, and thus for arbitrary values of the ai's and a a 's we have (4.8) T a ,T ai = T a , When these conditions are satisfied, the inverse of any trans- formation is not necessarily a member of the set, that is, it is not necessarily true that to each set of values a a there corresponds a set d a , such that (4.9) a-7*(*';a) =/<(*'; a), where the functions f* are defined by (4.3), or symbolically (4.10) 77V TaX f = x. If this condition is satisfied, we have (4.11) T.T&' = x', TaT a x = x, 4. GROUPS OF TRANSFORMATIONS 17 that is, when the inverse of each transformation is a member of the set, so also is the identity. Hence there is a set of values a?, such that a? = >(<*; a), and (4.12) /''(*; a ) - *'. Since the parameters in (4.1) are essential, there is only one set of values a? yielding the identity. Thus T aQ denotes the identity /, and from (4.11) we have so that in addition to (4.12) we have (4.13) aS = ^(a; a) = /'(a; a) and also (4.14) a a = . If only (4.6) are satisfied, the transformations form a finite, continuous semi-group. If T ai and T at are two transformations of a semi-group, and 77, 1 and 77, 1 are their respective inverses, then T^^T^^T ai T ai is the identity. From this result and (4.8) we have that (4.15) 77K - TV/, and consequently [4.1] The inverses of the transformations of a semi-group constitute a semi-group. These two semi-groups are the same, when and only when they are groups. If we denote by S the transformation defined by (4.16) < = V(x\ , a), \d\l/\ for which the jacobian is not identically zero, and we denote by |ox| S~ l its inverse, then the transformation (4.17) f a = ST a S~ l 18 /. THE FUNDAMENTAL THEOREMS is called the transform of T a by S. It is readily shown that if a set of transformations satisfy (4.8) then T a ,T 0l = T 0j . Hence we have : [4.2] // a set of transformations form a group, the transforms of these transformations form a group. (S-G) * From (4.17) we have (4.18) T a = S- l TaS and consequently T a is the transform of T a by S" 1 . If the inverse of (4.16) is written (4.19) a?' - JV, , y), the equations of T a are (4.20) 0" = *(/(*; a)) = i If instead of looking upon (4.16) as defining a point trans- formation in F n we consider it to be a non-singular transformation of coordinates, then equations (4.20) are the equations of the trans- formations T a in the coordinates y\ Accordingly the foregoing theorems show that: [4.3] The group property of a set of transformations is invariant under a change of coordinates. (S-G) This result is evident from the abstract definition of a group as given, since it does not depend upon the particular coordinates used. 6. Fundamental differential equations of a group. In this section we determine the conditions which the functions/* in (4.1) must satisfy so that the transformations form a group (? r , that is, so that there exist functions (ai; a 2 )). In the first place we shall show that (5.2) * * In what follows we shall be concerned primarily with groups as distin- guished from semi-groups, but some of the theorems hold equally well for the latter more general class; we indicate this by the notation (S-G) at the end of a theorem. 5. DIFFERENTIAL EQUATIONS OF A GROUP 19 From (5.1) we have da? I = 0, there exist a set of functions A? of the ai's and a 2 's such do that Then from the preceding set of equations we have, on replacing /<(*;,) by*", that is, the a's are not essential parameters If we write (5.1) in the form (5.3) /(* / ;a,)y(*;^(ai;a we have If dx'*daR " = 0, there exist functions A" of the ai's and o 2 's, such that A? - = 0, and consequently da i W Aa dx'* a/(x';a 2 ) Since the jacobian is not identically zero (4), we have da? and consequently the a's are not essential parameters. This result follows indirectly from the second of (5.2), when we observe that the roles of the parameters a? and aS are interchanged in the set of inverse transformations (cf. 4.15). Consequently (5.2) is established. 20 /. THE FUNDAMENTAL THEOREMS Because of the second of (5.2) equations (4.6) can be solved for a? as functions of a? and a". When these expressions are sub- stituted in (4.7), the latter become identities in the z's, ai's and a 3 's. Consequently we have from (4.7) = dz'* da? dal da? Because of (4.2) there exist functions $(x f i a 2 ) defined by where fly are Kronecker deltas (2.14), and consequently the above equations are equivalent to The quantities are functions of the a^s and a 3 's, but by means da\ of (4.6) are expressible in terms of the GI'S and a 2 's. The other quantities in the right-hand member of each of the above equations are functions of the x"s and a 2 's. But the left-hand member of each of these equations does not involve the a 2 's and consequently the combinations on the right are independent of the a 2 's. Conse- quently if (a?) are particular values of the a 2 's and we put (5.4) * d ^ a (6 ' 2) **&* ~~ *M* where Since (6.2) are identities in the z"s and the a's, and the latter do not enter in the 's, we have V - da *'" ' From (6.3) it follows that the c's do not involve the a:"s. Conse- dcji quently we have T-^r 0; otherwise we have relations of the form da (5.10). Hence we have: [6.1] The quantities c a l in equations (6.2) and (6.3) are constants. They are called the constants of structure of the group; some writers call them the constants of composition. From (6.2) we observe that Cab = Cfca, that is c<& is skew-symmetric in the indices a and 6. In consequence of (5.8) the equations (6.3) can be written in the form dA dA <* d(f - ^ and also in the form, because of (5.7), 6. FIRST FUNDAMENTAL THEOREM 23 Equations (6.4) are due to Maurer.* If we write (4.7) in the form (6.6) *'"*=/<(*'; a 2 ) -/(*; ai), we have nruiloicoiisly to (5.5), according as we use the first or second expression for re"*', dx"* dx"* _ , da? ~ ** (X ^ If we substitute the second expression for the x'"a in the first set of these equations and note that the a 3 's are functions of the ai's and o 2 's, we have ft MS From these two sets of equations it follows that (6.7) % = At(a 3 )A t M, otherwise we should have relations of the form (5.10). Suppose conversely that we have a completely integrable set of equations (5.5) with (5.6) satisfied; then (6.2) and (6.5) hold. Since the equations (6.5) are of the same form as (6.2), we see on comparing (6.7) with (5.5) that the former is necessarily com- pletely integrable, in consequence of (6.5). These equations admit solutions a? such that a? = a? when a? have the initial values a?, say (6.8) a? = a (o>i'> 02; 0o)> 0? ^ a (0i> 0o > 0o)> Suppose then that a set of solutions of (5.5) is/'(x; a) such that (4.2) is satisfied, and we put (6.9) 4 = /*(*; as), * 1888, 2, p. 117. 24 7. THE FUNDAMENTAL THEOREMS where a" have the values (6.8). We have accordingly in consequence of (5.5), (6.7) and (5.8). Since these equations are of the form (5.5), they admit solutions /'(; a 2 ), where the x's, being independent of the a 2 's, are functions of the z's, a/s and a a (ai;a 2 ;a ) = a?, is the inverse of the given transformation. The foregoing results may be stated as the following first funda- mental theorem (cf. 11) of continuous groups: [6.2] A necessary condition that a set of n functions f*(x; a) of n variables x i and r essential parameters a a define a group G r of continu- ous transformations is that these functions satisfy a system of differ- ential equations of the form (5.5), where the determinant of the functions A(a) is of rank r and the functions & are independent (constant coefficients). (S-G) Conversely, if /*(#; a) are solutions of a completely integrable system of equations of the form (5.5), where the A's and %'s satisfy the conditions stated above, such that for values a? of the a* 8 the determinant of the A's is not zero and (6.12) /<(*; ao) - *', 7. CONSTANTS OF STRUCTURE 25 these functions define a continuous group of transformations, thai is, the inverse of each transformation is a member of the group.* We shall show that the conditions of the second part of this theorem can be met when we have a completely integrable system (5.5). In fact, if /*'(#; a) are n independent solutions of such a system, that is, such that (4.2) is satisfied and a? are values of the a's for which |4(a )| 7* and we write the equations of the trans- formation T a , and T~ a \ in the forms xl = f { (x; a ), x* = /'(a**; ao), then the equations of a transformation T a T^ are These expressions evidently are a solution of (5.5) in which x* are replaced by /'(zo; o). Moreover T a *T^l = / and consequently these transformation satisfy the conditions of the second part of the above theorem ; in fact / f [7(z ;a );a ] = xl Hence we have the theorem: [6.3] // the functions a and A a a satisfy the conditions (6.2) and (6.4), equations (5.5) admit one independent set of solutions defining a group G r of transformations, for which any given values of the a j s such that \Aa\ 7* determine the identity transformation.^ 7. Conditions upon the constants of structure. Transforma- tions of coordinates and parameters. The functions a which enter in equations (5.5) serve to determine r linear operators (2) (7.1) X a f = (x)jt = &*. From (6.2) it follows that (7.2) (X a , X b )f = cJiXJ (a, 6, e = 1, - , r). Applying the Jacobi identity (2.4) to (7.2), we obtain 0. * The proof of this theorem is an adaptation of the proof given by /Sc/iwr, 1891, 1, pp, 264-268, t Cf. Bianchi, 1918, 1, p. 85. 26 /. THE FUNDAMENTAL THEOREMS Since the 's are linearly independent (constant coefficients), we have: [7.1] The constants of structure of a group satisfy the conditions (7.3) c a l = -ol, (7.4) cj>c/ c + c b lc/ a + c c ' a c/ b = 0. (S-G) We call these the Jacobi relations. If we subject the x's to a non-singular transformation, such as (4.16) and (4.19), it follows from (5.5) that where (7.5) rf a (y) = &(*)g. Two sets of quantities and rja related as in (7.5) are said to be the components in the respective coordinate systems of a contra variant vector in F n .* Also we have (7.6) that is, X a f is a scalarf under transformation of the coordinates. The functions & and A a a have been defined by (5.4). Suppose, however, that we define a new set of functions 'I by (7.7) (*) = (*), where the c's are constants, such that the determinant \c b a \ 5^ 0. Then the "s are linearly independent (constant coefficients). A set of constants cl is uniquely defined by (7.8) c b a c b = :, c b a c a e = d b . If then we put (7.9) A1(fl) = ciA b a (a\ A' a (a) = cU & a (a), we have that (5.5) may be written * Cf. 1926, 3, p. 4. t Cf. 1926, 3. p. 6. 7. CONSTANTS OF STRUCTURE 27 so that we have the generality in the choice of the f f s and A 9 a set forth in (7.7) and (7.9). If we effect a change in the parameters a a by putting (7.10) a = ?(a'S , a''), where dtp 0, it follows from (5.4) that Thus a change in the a's may be said to induce a linear trans- formation on the functions &(%') However, in view of the gener- ality referred to above we may choose the 's for the parameters a' a so that for the 's in the a"s we take &[/(;

') = A b a (a)jj~ (a, ft 6 = 1, , r). Thus the A'a obey the law of transformation of co variant vectors* in the space S of coordinates a a , the index b indicating the vector and a the components; they constitute an ennuple of independent covariant vectors in this space S, which is called the group-space of G>. From (5.8) it follows that the quantities AS are the components of the ennuple of contravariant vectors in S, conjugate to the ennuple A b aJ where the index 6 indicates the vector and a. the component, f We return to the consideration of (7.7) and put X' a f = 'a(z)p'- In place of (7.2) we have C7 1**>) \-A-ay A-bJj * where (n i o\ '' <*i &!=* V / . Lfj) C a b == C a C& C e * Cf. 1926, 3, 7. t Cf. 1926, 3, p. 15. 28 /. THE FUNDAMENTAL THEOREMS Lie calls X a f for a = 1, , r the symbols of the group; we shall refer to them as the basis of the group; the significance of these terms appears more clearly in 10, 11. When the basis is changed in accordance with (7.7), the new constants of structure are given by (7.13). Two r-parameter groups whose constants are in the relation (7.13) are said to have the same structure. 8. Semi-group of inverse transformations. In 4 we saw that the inverses of the transformations of a semi-group G> form a semi-group 5 r , and that if they coincide G> is a group. For the present we treat the general case. For the semi-group U r we have analogously to (5.5) the com- pletely integrable system of equations <9r* (8.1) = l(x)Al(a), oa of which the f unctions 7 l (x'; a) indicated in (4.3) are solutions. In this case the equations analogous to (5.1) are, in consequence of (4.15), (8.2) /'[/(a'; a 2 ); a,] = /'(a:'; *(cn; a,)). Consequently in place of (6.7) we have, since the roles of ai and a 2 are interchanged, (8.3) ^ = AKaJA'M. da" For any values a? for which the determinant |A*(ao)| is different from zero a set of constants c? can be found such that and in consequence of (7.7) and (7.9) we have that the functions & and A* a can be chosen so that (8.4) J(a ) = A(a ), ASM = 4?(a ). When we express the conditions of consistency of (6.7) and (8.3), we obtain Since these must hold for all values of the ai& and c^'s, the quantity in parenthesis must be zero, so that on changing indices we have 8. SEMI-GROUP OF INVERSE TRANSFORMATIONS 29 9Al, f dA a a , _^ _ __ A? = o. In order to consider these equations further, we define functions of the a's by the equations (8.6) which because of (5.8) are equivalent to /o 7 N Ja _ A^Al _ A (8.7) I*- -A, - From (8.7) and (6.5) we have (8.8) LI, - Ift = c a U a ft If we put, analogously to (8.6), (8-9) + AiL^ = 0, we find from (8.5) and (8.7) that (8.10) Analogously to (6.5) we have ' -a da and similarly to (8.8) (8.12) Z 7 - L^ From (8.10), (8.8) and (8.12) we have, on putting a" = a" and making use of (8.4), (8.13) c a i = -cj. Comparing these equations with (7.13), we see that the latter arise from the former, if we put c = d b a . Consequently the semi- group of inverse transformations of a semi-group has the same structure as the latter. In fact, the minus sign is a consequence of the choice (8.4), 30 /. THE FUNDAMENTAL THEOREMS Since the determinant \A\ is of rank r (5), a set of functions p b a (a) are defined by the equations (8.14) Al = p b a A b , which transform as scalars, when the parameters a a undergo a general analytic transformation; in particular, (8.15) P 6 a (a ) - J, as follows from (8.4). When the above expressions for A% are substituted in (8.9) and LJj 7 are replaced by L"^, we obtain, in con- sequence of (8.6), (8.16) 2* = P :AlAl(Lf y - Ly>-, these are reducible by (8.8) and (5.8) to A b /Q t^ dpq 6 A f (8.17) = c. f p a A J a . The conditions of integrability of these equations are reducible by means of (8.6) and (8.8) to ct + c&i + c c ' h c e b d )p h a A c a A d p = 0, which are satisfied identically because of the Jacobi relations (7.4). Hence equations (8.17) are completely integrable, and p are the solutions satisfying the initial conditions (8.15). That (8.17) are completely integrable is seen also from the fact that equations (8.9) are completely integrable. Equations (8.14) may be written in the form (8.18) P l = AAl Another set of scalars p are defined by (8.19) A = A b a A, which are such that (8.20) pip*, - dl From (8.20) and (8.17) we have (8.21) 4 - 9. THE PARAMETER-GROUPS 31 Moreover, from (8.8), (8.10), (8.12) and (8.13) we have (8.22) c = c&'M. 9. The parameter-groups of a group G r . In 4 we indicated by (9.1) r, = T a% T ai the first condition that the transformations T a form a group, the parameters being in the relation (9.2) a? - *> a (ai; a 2 ). We may look upon (9.1) as defining a transformation of the point of coordinates a? in the group-space S into the point of coordinates a", the transformation being induced by T at . Then (9.2) are the equations of this transformation, a? being the parameters. We shall show that these transformations form a semi-group. In fact, if we put a" = ? a (a 3 ; a 2 ), we have 1 a (a 2 ; ^2). Thus we have shown that equations (9.2) in which the a 2 's are parameters define a semi-group of transformations, and furthermore that the functions a connecting its parameters are the same as for a semi-group G>. If G r is a group, as distinguished from a semi- group, a? being the values of the parameters for the identity, then for these values of a? equations (9.2) define the identity of the latter, as follows from (4.14). Conversely, if equations (9.2) define a group, then G r is a group. We remark further that equations (6.7) are the equations for (9.2) analogous to (5.5) for G>. In similar manner the symbolical equations m 1 mmm mlml L 0| * 01 * at may be looked upon as defining a transformation of the point of coordinates a? into that of coordinates a? in the group-space /S, the transformation being induced by 77, 1 ; and the equations of the transformation are (9.2) with a" as parameters. If now we put 03* = ? a (al; a 3 ), 32 /. THE FUNDAMENTAL THEOREMS we have ml /TT 1/jT 1 /m lml\ frr1 m 1m 1 L oi oi * a, = V-* oi -* 01 ; * a, ~ J- ;' 1 a , , where Consequently equations (9.2), in which a? are the parameters, define a semi-group or a group, according as G> is a semi-group or group. In this case equations (8.3) are the equations for this group (or semi-group) analogous to (8.1) for G r . When G r is a group, the groups defined by (9.2) as a? and a? are considered as the parameters are called respectively the first and second parameter-groups of G r . Their respective symbols are (9.4) A j= A *lL, Aj = AljL From 8 it follows that these two groups have the same structure as G r (for a semi-group G> the same structure as G> and its inverse U r respectively). In consequence of (4.14) and (6.7) we have /H K\ At/ \ A a/ x (9.5) Af (01) = A?(a Q ) and from (8.3) (9.6) 10. One-parameter groups. Suppose that (10.1) *"-/*(*!, - ,*;a) (t = 1, ,n) are the equations of a one-parameter group Gi and a is the value of a for the identity, that is, (10.2) /*(*; a ) = x*. The f unctions /* are the solution of a system of ordinary differential equations (cf. (5.5)) (10.3) aa satisfying the initial conditions (10.2). If we define the parameter *by (10.4) t = f*A(a)da, Jo 10. ONE PARAMETER GROUPS 33 then t = yields the identity, and equations (10.3) become (10.5) ^ = t-<(x'). at Corresponding to (4.6) and (4.14) we have (10.6) a 3 = P(OI, a 2 ), a = 's are independent and the c's are constants. Without loss of generality we may assume them solved for x' l t . . . , x' n ~ l as functions of x' n and the c's. When these are substituted in t n (x f ) yielding n (z' n , c 1 , . . . , c"" 1 ), and we put Jdx' n -p = ?(*', c\ -p = ?(*', c\ - , c"- 1 ) = v(x' 1 , , x' n ),

~dT~ 1 ' of which the solution is (10.14). Incidentally we have established the theorem : [10.2] // *(#) are the components of a contravariant vector, there exists a coordinate system in which the components of the vector are d* n . Equations (10.10) define the congruence of integral curves of equations (10.5), one through each point. Equations (10.12) give the curve through the point P(x) in terms of the parameter t. They are called the trajectories of the group, each of which is described by a point as the latter undergoes a continuous transformation of the group. We return to the consideration of equations (10.5) and seek their solution as power series in t. Evidently we have 42 d^(x] (10.17) x = x i + t?(x) + 2^^-^T + * ' ' Making use of the notation in (10. 16) and indicating by X m f the result of operating with X m times on /, these expressions may be written (10.18) x'* = x i + tXx i + ^XV + . + .X"x* + . 2! ml Moreover any function f(x' 1 , . . . , x' n ) regular in the domain of x* is expressible in the form (10.19) /(*') = /(*) + tXf + - + -.X"f + . ml For, we have **>-** and the result follows from (10.5). 36 7. THE FUNDAMENTAL THEOREMS If the series (10.18) are convergent f or ^ t ^ t\, the equations so defined, say (10.20) x" = F'(; 0, are equivalent to (10.1) for the range of values of a as given by (10.8) for these values of t. This may mean that only a portion of the trajectory through P(x), as defined by (10.1), is given by (10.18). By a transformation of coordinates as (10.13) equations (10.20) take the form (10.14) for the limited values of t. From this result it is seen that if P(x') is a point on the segment of the trajec- tory referred to above, then on applying (10.20) to P' we get another segment. Hence by sufficient repetition of (10.20) we obtain the trajectory through P(x) in so far as it is defined by (10.1). For the rotation in the plane with the symbol (10.15) we have that (10.18) become t 2 t s x' 1 = x l - xH - x l + x*- + = x 1 cos t - x* sin t, 2i\ o! t 2 t 3 x' 2 = x 2 + x l t x 2 - x 1 - + = x 1 sin t + x 2 cos t. 2! o. Consequently (10.18) holds for the whole trajectory in this case. If in (10.17) we replace t by an infinitesimal dt and neglect powers of dt higher than the first, we obtain (10.21) x'* = z* + ?(x)tt, which is called the infinitesimal transformation of the group G\. By the integration of the infinitesimal transformation in the sense of the preceding paragraph the group Gi is obtained. Lie was the first to undertake a systematic study of the construction of con- tinuous groups by means of infinitesimal transformations in this sense. 11. Sub-groups Gi of a group G r . If in equations (4.1) of a group G r the a's are expressed in terms of a single parameter t, say (11.1) a" = a"(0, we obtain the equations, (11.2) x" = f*( X ; a'(0, - - , a'(0) s F<(x; 0, of oo l transformations. The question arises under what conditions these transformations form a group G\. For this to be the case there must be a relation *3 = 0(ti, /f) 11. SUB-GROUPS Gi OF A GROUP G r % 37 such that from (4.6) we have a"(ts) = ^(a^O, - ,a'(*i);ai(, , a'(*,)) m *> a (a(*i);a(* 2 )). From these equations we have by differentiation with respect to i* which in consequence of (6.7) become where A b ft (a(t)) = A b ft (a*(t), - , a'(0). If we put (11.4) P(t) m A b a (a(t))^-, at equations (11.3) are reducible in consequence of (5.7) and (5.8) to V(t*)~ alt From these equations it follows that Since these relations must hold for all values of J 2 , we have that each term is a constant, and hence we have where the e's are constants, and equations (11.4) become at If we define a new parameter ' by t' = J^^(0*> where * is such that a a (< ) = o, the latter giving the identity, then the above equations in terms of the new parameter which we now call t assume the form da? (11.5) 38 /. THE FUNDAMENTAL THEOREMS From the choice of t it follows that, when i = 0, the transformation is the identity. Hence the desired functions a a (t) are the solutions of (11.5) such that a a (0) = a?. By means of (11.5) equations (11.3) reduce to (11.6) = 1. 0(2 From (4.14) it follows that 0(0, U) = / 2 and 6(ti } 0) = ti. Hence the solution of (11.6) is (11.7) d - h + h. Conversely, if & in (11.5) are any set of constants and the functions a a (t) are the solutions such that a a (0) = a?, then (11.3) are satisfied, if we make use of (11.7). Consequently is at most a function of t\. But this expression vanishes when <2 = 0, whatever be h, and consequently it vanishes identically. Hence if the functions a a (f) are substituted in (11.2), the resulting equations define a one-parameter group for each set of values of the e's and in each case f or t = we have the identity. Suppose then that we have the solution of (11.5) involving the e's as parameters and t, and satisfying the initial condition a(0) = a?. Then using (5.5) we have flj.fi Jfj" (11.8) = ^(*'M = e-W) - $V). The integrals, as power series in t, of these equations such that x'< = x i when t = 0, are (cf. 10) (11.9) x'< = a< + tXx i + + ,X~x< + - , ml where (11.10) Xf = ?(x)p< = rtfajH. Because of the second expressions for Xf in (11.10) equations (11.9) may be written 11. SUB-GROUPS Gi OF A GROUP G r 39 (11.11) x fi = x* + te a X a x i + i m ml - e am X ai From the form of these expressions it follows that the a's as functions of the e's and t t to which we have referred, are in fact functions of the form (11.12) a" = t"(u l , , W), u a = e?t\ the expressions for the \l/'s as infinite series are obtained in 12. The values of t for which the series in (11.11) converge depends evidently upon the values of the e's and the z's. In each case and for the appropriate domain of t equations (11.11) may be written in the form (11.13) xf* = F'(x-, u). Evidently when the expressions (11.12) are substituted in (11.2), we get (11.13). Hence for the values of t for which the series in (11.11) converge, for given values of the e'a and z's, equations (11.11) define transformations of the given group Gi, and these transformations, in the terminology of Lie, are generated by the corresponding infinitesimal transformation (11.14) x fi = x* + e*(x)dt in the sense of 10. For given values of the x's in (11.11) there are values of u a = e?t, in general limited, for which the series can be inverted, and we have u a as functions of x i and x'\ Consequently for any point P'(x') sufficiently near P(x) there exists a transformation which sends P into P'. If in the equations x fi = /*(#; a), we give the a's a set of values for which these equations have a meaning, we get the point P'(x') into which P(x) is transformed by the corresponding transformation. If these values of the a's are such that the u's are defined by (11.12) (that is, if it is possible to solve these equations for the u'& as functions of the a's with these values), then a sub-group G\ is defined and the trajectory of this group through P(x) passes through P f (x r ). In this case, as seen in 10, P'(x') can be obtained from P(x) by the transformations (11.11), for the corresponding values of the e's or by a product of such transformations. 40 /. THE FUNDAMENTAL THEOREMS There are cases, however, for which the values of the a's giving P' are such that it is not possible to find a group G\ whose trajectory through P passes through P 1 . The question has been raised in these cases as to whether the passage from P to P 1 can be accomplished by a set of trajectories, so that P is transformed into a point PI by means of one set of e's, PI into P 2 by another set and so on. It has been shown that, if P(x) and P'(x') are points of a connected manifold, then P can be transformed into P' in this manner, that is, by the product of transformations of a number of groups of order one.* It is stated sometimes that a group G> consists of the totality of transformations of the groups G\ generated by the infinitesimal transformations with the symbols e a X a f, where the e'a are arbitrary constants, X a f being called the generators of G>. We have seen that the above results should be stated as follows: [11.1] The transformations of a continuous group G r , whose symbols are X a f, consist of the transformations of the groups G\ generated by the infinitesimal transformations with the symbols e a X a f, where the e's are arbitrary constants, or of the products of such transformations. A continuous group satisfying these conditions is called a Lie group. In view of the preceding discussion we have : [11.2] A necessary and sufficient condition that the group G\ generated by the infinitesimal transformation with the symbol Xf be a sub-group of a G r with the symbols X\f, . . . , X r f in the same coordinate system is that Xf be a linear homogeneous combination (constant coefficients) of the latter. And as a corollary of this we have : [11.3] A necessary and sufficient condition that two sets of operators X a f and X a f, expressed in the same coordinate system, be the symbols of the same group is that (11.15) X'j - clXJ, where the c's are constants such that the determinant \c*\ is not zero. If we denote by e' a the constants for the symbols X^f yielding the same infinitesimal transformation as e a X a f, then from (11.15) and (7.8) we have (11.16) e' = * Schreier, 1925, 2, p. 18, 19. 11. SUB-GROUPS Gi OF A GROUP G r 41 In terms of the parameters a" appearing in (11.2) an infinitesimal transformation is defined by (11.17) x'< = /<(*; a in consequence of (5.5). From this result and (11.5) we have: [11.4] When the finite equations of a group are given, the vectors * of a general infinitesimal transformation of the group are obtained from the equations x i + 5t = f l (x\ a\ + e^t, - , a r + e'St), where the e j s are constants, on neglecting second and higher powers of8t. Consider now the transformation determined by the values a a + 5a a , when (11.2) gives the transformation for a a ; the equations are x "i = f*( xl a + da) = x fi + &(x')A a a (a)da a . Comparing this result with (11.17), we see that if we determine quantities 8ia" by the equations the above equations become and consequently (11.18) We obtain the same result from (4.6), (4.14) and (9.5); thus a a + da a =

a (fl fl o) = a a -\- A"(a)Ap(aQ)8iaQ. In similar manner we have from (9.6) a? + 8a a =

. Exercises 1. The symbol of the Gi of dilatations, or homothetic transformations, whose equations are x'* = ax* is Xf - x' Pi . 2. Show that each of the following sets of equations defines a one-parameter group, and find the symbol and the trajectories in each case: 0) (ii) (iii) x = ax, *-- = a n y (m, n integers); ~ ay 3. Find the finite equations of the one parameter groups whose symbols are the following: (i) (ii) (iii) **P + xyq (x - y)p -f (x + y)q ; a/ Of df a;- -- y -- h ^ dy dx dz / I p = Y dx 4. For the group Gi of motions in the plane defined by the equations the equations (5.5) are satisfied by the functions 1 x 1 - x' l \ 1 1 a 8 -a 1 -1 A\A\A\ Al At At The corresponding symbols of the group are 11. SUB-GROUPS Ot OF A GROUP G, 43 and the constants of structure are cii - & = 1 2, 3; a not summed). 6. Find the symbols of the group (7j of rotations of Ex. 5 in terms of polar coordinates. 7. The vectors * of the general infinitesimal transformation of the linear, or affine, group x'* = a\x* -f- o* are (cf. theorem [11.4]) ' a* -f- aty, where the a's are arbitrary constants; also find the constants of structure. 8. Derive the 's for the group of motions of euclidean 3-space as a special case of Ex. 7. 9. The vectors * of the general infinitesimal transformation of the group G s of projective transformations of the plane > - i g/ + 0> ** "" bj& + I are where the a's are arbitrary constants; also find the constants of structure. 10. Show that the equations of the infinitesimal transformations and T^Ta+ta are respectively 44 I. THE FUNDAMENTAL THEOREMS Bianchi, 1918, 1, pp. 71, 72. 11. Show that the symbol of the transform ( 4) of the infinitesimal trans- formation with the symbol Xf by the infinitesimal transformation with the symbol F/is Xf + lt(X, Y)f. 12. Canonical parameters. If we apply to equations (6.7) considerations similar to those which led from (5.5) to (11.8) with the aid of (11.5), we see that in fact equations (11.5) are the differ- ential equations whose integral curves are the trajectories in the group-space S of the first parameter group. We can therefore write down the integrals of (11.5) in a form analogous to (11.11), namely (12.1) a' a = a a + te a A a a a + - - - im a ml l where we have put A a (a) = Aa(a) - In particular, the solution of (11.5) such that a a = a" when t = 0, which was used in 11, is given by (12.2) a" = a? + te a A a a^ + ml ai We shall write the latter in another form by making use of the functions rj T defined by /1O o\ TV* /T I 7\ v A ^'W 10-y oV^-y i" L/-Y0)) the functions L% y being defined by (8.7) ; thus T 7 is the symmetric part of LJ r In fact, if equations (11.5) are differentiated with respect to t, the resulting equations are reducible by (11.5) and (8.6) to (12.4) + r 12. CANONICAL PARAMETERS 45 Thus the trajectories of the first parameter group are paths of S determined by the linear connection rj 7 .* Differentiating (12.4) and making use of these equations in the reduction, we obtain dV a da^da^da^ dt dt dt dt where !- _lpte _!-!-!- r .\ 3 \ da* ty * y j> where P indicates the sum of terms obtained by permuting the subscripts cyclically. Continuing this process, we have where r- lrJdr/ 0-* V* r" T^ ' ftm = ml - ~dc^ -- ^ * l ' ' ' *'- t Moreover, the expressions (12.5) become (12.9) a" = o? + y a - r^oJ/V If in place of the a's we had started with another set of coordinates a' a , corresponding to (12.8) we should have ' = (da' a \ t = ( da ' a \ J> \ dt /o \ da / Q Because of this result and (7.11) we have and consequently, as follows from (12.7), the quantities u a are unchanged when the coordinates a" undergo a general analytic transformation. If we take linear combinations with constant coefficients of the vectors A and of A" as in (7.9), it follows from (8.7) that the coefficients L% y are not changed. Accordingly the coordinates y a , defined by (12.9), are not affected by such changes of the vectors A*, but the quantities u a are changed. In fact, from (7.9) and (12.7) we have, on denoting the new coordinates by u', (12.10) u' = clu b . We call the u a canonical parameters. Equations (12.10) are in agreement with (11.16). If we denote by U the components in the u'a of the vector of components A% in the a's, we have (12.11) V a = 45. Then from (12.5), on observing that u" = when a" = a?, we have (12.12) (0) - C W) - 12. CANONICAL PARAMETERS 47 Analogously to the equations (11.5) we have From these equations and (12.6) it follows that e a = e*U, and consequently we have (12.13) u*U = u a , u a U a a = u\ We observe that, if a? are the coordinates of any point in the group-space S, then in terms of the coordinates y a defined by (12.9) the equations of the paths through the point are given by (12.8). Evidently the coordinates y a are determined for the domain about the point for which (12.9) are convergent and such that no two paths through the point meet again in the domain. They are in fact the normal coordinates with this point as origin of the space with the linear connection determined by the quantities r^ 7 .* Although the results of the preceding development are of use in the theory of continuous groups, the ideas of the latter have not been used in their development and on reviewing these results we have : [12.1] Given an ennuple of vectors A^(a) in a space of coordinates a a and define as in (8.7) a linear connection; any point for which the determinant \A a a \ is not zero may be taken as the origin of a set of canonical parameters u a , defined by (12.5), where F 7 are the symmetric parts of the coefficients of the linear connection; in this system of coordinates the components U" of the vectors satisfy the equations (12.13); the coordinates u a are unchanged by a general analytic trans- formation of the a's, but undergo a linear homogeneous transformation with constant coefficients, when the vectors are subjected to such a transformation. These coordinates u a have been used in other theories recently, f For the values of t for which the right-hand members of (12.5) converge these equations may be written in the form (11.12), and * 1927, 1, p. 59. t Cf. Thomas, 1930, 4; Michal, 1929, 3. 48 /. THE FUNDAMENTAL THEOREMS these in turn lead to (11.13). If equations (11.11), written in the form (12.14) X'* = X* + U'XaX* + + -u" - - u-X ai - X am x* + m! converge for these values of t } then (11.13) are equivalent to (11.11). 13. Abelian groups. If all the constants of composition of a group are zero for a given basis of the group, the same is true for every basis, as follows from (7.13). In this case we have from (7.2) that (X a , X b )f = 0. In 29 it is shown that, when and only when this condition is satisfied, any two transformations, T a and Tb, are commutative, that is, that T& T a = T a 7V If all the trans- formations of a group possess this property, the group is said to be Abelian. Before proceeding to the discussion of certain canonical forms for non- Abelian groups, we shall develop canonical expressions for the equations of Abelian groups. For an Abelian group equations (6.4) become ds _ *d? - n da" da" ~ ' so that there exist functions A" (a 1 , . . . , a r ) such that Ml and the jacobian ^ 0. I da I If we introduce a new set of parameters u a defined by (13.2) u" = A a (a\ , a'), and denote by U a a (u) and V a a (u) the functions of the components in the w's of the vectors of components A a a and A" in the a's, we have in consequence of (13.1) and (7.11) (13.3) Ul(u) = C Vl(u) = K. We observe that these results are in keeping with (12.13), but this is the only case in which the U's have the particular values (13.3), as follows from the equations in the u'a analogous to (6.4). The fundamental equations (5.5) are in this case 13. ABELIAN GROUPS 49 (13.4) The corresponding equations (6.2) are in this case, on dropping primes for the present, (13.5) g - $ = 0. dx' dx* We choose the coordinate system in accordance with theorem [10.2] so as to have for the vector I of an Abelian group (13.6) { = Si. If in (13.5) we put a = 1 and 6 = 2, . . . , r, it follows that the vectors & are independent of x l . The rank q of the matrix (13.7) M = ||g|| (i = 1, , n; a = 1, , r) for general values of the coordinates x i is called its generic rank; evidently q ^ r and also q ^ n. We consider first the case when q = r. If 2 ^ 0, the equation 2 p,- = admits n 1 independent solutions ^i = 31 + ^(x\ . - , *), ^( X 2, .-.,) (or = 3, - , n); if j = 0, we may take ^ 1 = 0. Denote by ?? 2 (z 2 , . . . , x n ) a solution of jp = 1. Then the equations x fi = ^ define a non- singular transformation of coordinates. In this coordinate system, which we now call x\ we have from (7.5) that (13.6) hold and 2 = 5 2 , and then from (13.5) that the other 's are independent of x l and x z . Then we take the equation jp = and get solutions 9 i = a-i + ^(x\ . . , x n ), *(x\ , x), ^(x 3 , - - , Z") (cr = 4, , n) and

, p = q + 1, , r). Proceeding with this case in a manner similar to the preceding, we have in consequence (13.10) (13.11) = Si, g = **(x+i, - - , *), = /t - 1, , n; A = 1, , q; \ \p = q + 1, , r; s = ? + 1, , nj- In this case equations (13.4) are du A, a = 1, , q; a = 1, , r; \ \s = g + 1, - , n; is a group for which the identity is determined by a% and the vectors A% are chosen so that (8.4) are satisfied, it follows from (12.5) that the canonical parameters with respect to the ennuple AS are the same as with respect to the ennuple A". This is a conse- quence also of the fact that the symmetric part of the coefficients Ifpy is the same as of L% y in consequence of (8.10). Accordingly, if Ua and (7* are the components in the u's of the respective vectors A" and A, we have (14.7) u a TJ" a = w, u"U = u a , U a a (Q) = & 17^(0) = C Since the functions I7(u) are defined by (14.6) with the c's replaced by the c's, in consequence of (8.13) we have (14.8) WM = U'(-u), and because of (5.7) (14.9) T72(u) = ITS(-u). If analogously to (8.14) we put then the cr's are equal to the p's in (8.14) under the transformation of the a's into the u's. In place of (8.17) we have (i4.il) ^ = cb:u' M and in consequence of (12.13) jb (14.12) a = cfr&. at From (12.12) and (14.7) we have <7a(0) = 6. The solutions of (14.12) satisfying this condition are of the form * Schur, 1891, 1, p. 270. 15. SECOND AND THIRD FUNDAMENTAL THEOREMS 53 (14.13) r ~ l infinitesimal transformations e?Xaf or of products of such transformations. In the foregoing proof no use has been made of the functions & except as giving the constants c & and thence the conditions (7.3) and (7.4). What has been shown in fact is that, if a set of constants satisfy these conditions, the functions C7 defined by (14.6) satisfy equations (14.1), that uU* a = U-, and consequently the determinant |t/(w)| is of rank r. Further- more the functions UZ(u), satisfy the equations rj _ , .r* ~ Ub ~ Cal> " Consequently the equations are completely integrable, and the solutions such that y' a = y" when u = 0, define a group G r of transformations. This is the same as the first parameter group when the c's arise from a given group (? r . Hence we have the following third funda- mental theorem of Lie : [15.2] Any set of constants c a l t which satisfy the conditions (7.3) and (7.4) are the constants of structure of a group. It should be remarked that the preceding argument applies only to the domain about the origin for which the determinant of [/ is not zero * Exercises 1. Show that the inverse of T a9 +&> 9 is T ao -&* where a? determine the identity and that the inverses of T a + aT^ 1 and 27 1 2 T .+ao are T^ oa T^ 1 and T?T*-to respectively. 2. Show that T ai+fasTai+ftai =* T oj-/ on * Cf. Carton, 1930, 1, p. 17. 56 7. THE FUNDAMENTAL THEOREMS if 6a? and flo? are chosen so that i\ a a ) aa t + and that in this case T^^T at +sai and !*!+$, T"* are inverses of one another, and consequently (Ex. 1) *ai+6aiT a ^ = 7 aj 7^ai-aa a . 3. Given two sets of transformations in which the a's are essential, and denote by TV and T at transformations of the two sets; in order that T at T ai be of the form where the o/s are essential, in which case a? - #(oi; a,), it is necessary that the functions /} satisfy a set of equations of the form Then show that a necessary and sufficient condition that the above conditions be met is that where &, and Sb t are transformations of a group G r , and 7Vo and 7 T ( a2 ) are particular transformations of the two sets. Bianchi, 1918, 1, p. 118. 4. If the r parameters in the equations y i =/(aj;a 1 , , a r ) are essential, the maximum number in the sequence (3.7), say /i, is equal to r and consequently the + 1 system of equations y* "/*! y'f/i -A /!>> y'f/ !/. =/*,/!,/. can be solved for the a's; when the a's are eliminated from these equations, q equations (1) ^afo 2/, y, /i, , y, /, .' . /.) - ( = !,,$) result, where g = + i + + c r, e< being the number of equations involving the t th derivatives and o = w. Then y*, /, . . . /. +1 are expressible in the form (2) y\h /.+i - *}, /.+ifo y, y, / y, /t /.) Thus (1) and (2) constitute the differential equations arising from y i /*'(*> a ) by the elimination of the a's, and the latter is the general integral of the system (1). 5. If the equations j/ /*(; a) in Ex. 4 define a group, and if v*(x) and \I>*(x) are solutions of the system (1), so also are ^(. In this case equations (1) are linear and homogeneous in (* and their derivatives of order 1 to s, the coefficients being functions of the X'B. These equations admit as solutions each set Q and also * J r The corresponding equations (1) are called the equations dx* dx* of definition of the infinitesimal transformations of the group. Lie-Engel, 1888, 1, vol. 1, p. 185. 8. Show that if a system of differential equations linear and homogeneous in * and its derivatives to a finite order possesses the property that when j and 2 are any two sets of solutions so also is \ -. j -. and the general solution dx l dx l , df . * involves a finite number, r, of arbitrary constants, then ' . is the general dx v symbol of a group G T . Lie-Engel, 1888, 1, vol. 1, p. 188. 9. The equations of definition of the infinitesimal transformation of a group follow from the equations of definition (Ex. 4 (1)) of the group, if y* are replaced by x* + tfdt and the coefficient of 5t is equated to zero. Bianchi, 1918, 1, p. 128. 10. Find the equations of definition of the infinitesimal transformation of the affine group (Ex. 6) and show that ' == G { +e}x* (ij = 1, , n), where the e's are constants. 11. The group of motions in euclidcan n-space in cartesian coordinates is defined by x fi = a* H- a)V, where the a's are constants subject to the conditions Show that the equations of definition (Ex. 4) are 12. For the group of motions in euclidean w-space (Ex. 11) the equations of definition of the infinitesimal transformations are 58 7. THE FUNDAMENTAL THEOREMS of which the integrals are - 6* + 6'*', 6} = -6J- Consequently the symbols are Pi, x*pi - x*pi (t, j = 1, , n). 13. When a group of transformations is defined not by a single set of equa- tions as (4.1), but by m(>l) sets of equations, say *'* =/(!)(*; a) (* = 1, - , w), in each of which there enter r essential parameters a", the equations are said to define a mixed group. Show that in this case each set of functions / ( J } satisfy a set of equations where the f* are the same functions for each value of s, but not the A (t) 'a. Lie-Engel, 1888, 1, vol. 1, p. 314. 14. If the functions ? a (x) in Ex. 13 are such that X a f = */>, are the symbols of a group by theorem [6.31, and if a," are values for which the determinant |4 (t) 2(a, )| is not zero, then T^ 09 T ( , )a are transformations of the mixed group. Since the identity is in one set say T^*, this set is the group of symbols X f, and 15. Show that x' = x cos a y sin a, y' x sin a + y cos a, x' = x cos a + y sin a, y f = a; sin a y cos a define a mixed group; that the symbol is y x ; and that if T^o denotes the dx By second transformation when a = 0, then Tta =* TwTia. 16. In order that m sets of equations define a mixed group it is necessary that the same number of essential parameters enter into each set of equations. Lie-Engel, 1888, 1, vol. 1, p. 319. CHAPTER II PROPERTIES OF GROUPS. DIFFERENTIAL EQUATIONS 16. Sub-groups of a G r . We have seen that for each set of values of the constants e? in e a X a f, the latter is the symbol of a sub-group Gi of the given G>. The question arises whether m(, and find under what conditions v a X a f is the symbol of a G\, so that the two form a sub- group Cr 2 of G r . For this to be the case it is necessary and sufficient that (16.7) (u*X a , v b X b )f = *u*X f + P v b X b f. In consequence of (7.2) we have (16.8) ri(u)v b - ffu 9 - pv< = 0, where (16.9) i,J(ii) = c a lu*. Since vjl(u)u b = in consequence of (7.3), we have that the rank of the matrix \\rjl\\ is less than r. Hence in order that (16.7) with cr = p = be satisfied by v a different from u a , it is necessary that the rank of this matrix be less than r 1, and we have: [16.2] A necessary and sufficient condition that u a X a f be the symbol of a GI contained in an Abelian sub-group G* of a G r is that the rank of the matrix of c a lu a be less than r 1. 17. ABSOLUTE AND RELATIVE INVARIANTS OF A GROUP 61 In order that p = 0, cr 5^ in (16.7), in which case we may take or = 1, as in theorem [16.1], it follows from (16.8) that the aug- mented matrix ||77&, u e \\ must be of the same rank as the matrix \\fjl\\f and that if the rank is r p, then certain p of the t/s may be .chosen arbitrarily and the others are determined. In order that p 9* in (16.7), we may take 0- = as in the case of theorem [16.1] and equations (16.8) become (16.10) fa - P dl)v 0. From theorem [16.2] it follows that, if zero is a multiple root of the determinant equation (16.11) \il - tlp\ = 0, there are values of v a other than u a satisfying (16.10) and these lead to Abelian sub-groups G z . If p is a non-zero root of (16.11), v a determined by (16.10) are different from u a t and we have a G* such that P V*X a f. As a result of the preceding investigation we have : [16.3] Any sub-group G\ of a G r is contained in at least one sub- group G% of G r . 17. Absolute and relative invariants of a group. A function F(x) which is unaltered by all the transformations of a group G>, that is, such that F(x f ) ss F (x) is called an absolute invariant of the group. For a sub-group Gi of transformations the trajectories are given by (11.8), that is, where the e's are definite constants. A necessary and sufficient condition that F be an absolute invariant of this G\ is that F(x) be an identity when the x n s are replaced by the solutions of the above equations determined by the z'S as the values when t = 0. Hence we must have 62 77. PROPERTIES OF GROUPS. DIFFERENTIAL EQUATIONS If F is to be an absolute invariant for G>, this condition must hold for all values of the e's and we have : [17.1] A necessary and sufficient condition that a function F(x 1 f . . . , x n ) be an absolute invariant of a G r is that (17.1) X a F = (a = 1, , r). If the rank of the matrix (17.2) M = ||il|| is g, then q of the equations (17.1) are independent, say (17.3) XJ = (o- = 1, , g). In consequence of (7.2) and the fact that the other symbols are expressible in terms of those in (17.3) the system (17.1) is complete. Hence if q < n, there are n q independent absolute invariants, and none when q = n. Suppose that a function F(x) is not an invariant of the group with symbols X a f, but that it is invariant for the transformations with the p linearly independent (constant coefficients) symbols Y l f = KXJ (I = 1, ' ' ' , p), where the X's are constants, and only for these symbols. If we take as basis these p symbols and r p of the X a fso that the whole set be linearly independent (constant coefficients), we have fv v \f *, k v f _i_ *, * v / i > *, wi = 1, , p',\ (Yi, Y m )f = yimYkf + yimXtf I . _ , 1 ] V p ~r A> ; ' / When / = F , we obtain from these identities the equations 0, from which it follows that yim = 0, otherwise we should have further symbols for which F is an invariant. Hence we have : (F,, Y m )f = TiilV, and consequently by the second fundamental theorem the trans- formations with the symbols Y ff f form a group, which is a sub-group of G>. This result may be obtained also by observing that the product of any two transformations applied to F(x) leaves it invar- iant and consequently is a member of the same set. We call this sub-group the sub-group of the function F(x). From the results 17. ABSOLUTE AND RELATIVE INVARIANTS OF A GROUP 63 at the beginning of this section it follows that the number of inde- pendent functions invariant for a sub-group is equal to n minus the rank of the matrix of the symbols of the sub-group. Suppose that we have p independent functions F\, . . . , F p such that not identically, but for the values of the x's for which (17.5) F a = 0. If we change the coordinates so that equations (17.5) are in the new coordinate system (17.6) x a = (a = 1, , p), then from (17.4) we have & equal to zero when x l , . . . , x p are zero; we indicate this by (17.7) (g) = 0. If now we take the sub-set of equations (11.8) and treat the quantities x p+l , . . . , x n entering therein as parameters, we have that the solutions of these equations which have the initial values (17.6) are in fact (17.6), in consequence of (17.7) and the theory of systems of ordinary differential equations. Hence we have: [17.2] // X a f are the symbols of a G r and there exist a set of inde- pendent functions FI, . . . , F p such that XaF a = for points of the variety defined by FI = = F p = 0, this variety is transformed into itself by each transformation of G r . In this case the functions F a are said to define a relative invariant of the group. Theorems [17.1] and [17.2] are established immediately, if we consider only infinitesimal transformations of the group, so that we have proved that invariance under the infinitesimal transforma- tions of a group implies invariance under the finite transformations generated by them. We show in 51 that the same principle holds in the case of another type of invariant. 64 //. PROPERTIES OF GROUPS. DIFFERENTIAL EQUATIONS 18. Ordinary and singular points. Order of a transformation at a point. Consider the matrix (18.1) M - |||| of the components of the vectors of a group G>. Let q be the rank of M for general values of the z's, that is, q is the generic rank (13). A point for whose coordinates the rank is q we call an ordinary point of the transformation, and a point for which the rank is less than q a singular point. In order to distinguish between types of singular points, we pro- ceed as follows. If the equations obtained by equating to zero all the elements of (18.1) are consistent, we denote by L the locus of points whose coordinates satisfy these equations and say that these points are singular of order zero. If there is such a locus it consists of the points satisfying a set of independent and consistent equations, say F l = 0, , F. = 0. If s = n, I/o consists of one or more isolated points. If s < n, then Lo is a sub-variety of n s dimensions. For example, in the case of the group of rotations in euclidean 3-space (Ex. 5, p. 43) the origin is the locus L . If the equations obtained by equating to zero all the minors of order two of M are consistent, we denote by LI the locus of points, whose coordinates satisfy these equations and do not belong to Lo, if any, and call these points of order one. Proceeding in this manner we have the possible loci L , . . . , Lg_i of singular points. The above remarks concerning the nature of L apply to each of the loci LI, . . . , L fl -i. Thus for any point in L p the rank of the matrix is p. In this sense the locus of the ordinary points is denoted by L q . Thus L q consists of all the points of V n other than those in Lo, . . . , L a _i, if any. If the components * of a transformation are regular in the neighborhood of a point PO of coordinates XQ, and they are expressed in the form (18.2) $<(*) - $<(*) we say that the transformation is of order zero at PO, when not all 18. ORDINARY AND SINGULAR POINTS 65 of the ^(XQ) are zero; that it is of order one when all of the are zero but not all of the ( . ) , and so on. \axvo If the rank of M at P is g , then in the equations (18.3) &#(x ) = 0, go of the b's are expressible linearly and homogeneously in terms of the remaining r g . Hence there are r g linearly independent transformations (18.4) b*X a f of order greater than zero at PO and go of order zero. Each of the infinitesimal transformations of order greater than zero leaves the point P invariant, as follows from (10.21). If two such transformations are applied successively, the point P is unaltered and consequently the ."':: transformation is of the same kind. Consequently all the infinitesimal transformations of order greater than zero generate a group of order r go, which is a subgroup of G>; it is called the sub-group of stability of PO. In order to find expressions for the generators of this sub-group, we assume that the indices in (18.1) have been arranged so that the matrix ||{pCco)|| for p = 1, , g is of rank g . Then we have where the a's are constants. Hence the symbols (18.6) X.f - af-Yp/, being linearly independent and of order greater than zero at P , generate the sub-group of stability of PO of order r go. From the foregoing considerations it follows that, when the rank of M is r, an ordinary point does not admit a subgroup of stability other than the identity. If the rank g of M is less than r, then there is a sub-group of stability for each point; it is of order r g for an ordinary point and r p for a singular point of order p. Exercises 1. The symbols of the general projective group Gg in x and y, if we put p = , q = , are (cf. Ex. 9, p. 43) * dx * dy p, q, xp y yp, xq, yq, x*p + xyq, xyp -f y*q; 66 II. PROPERTIES OF GROUPS. DIFFERENTIAL EQUATIONS show that the following are the symbols of sub-groups: (0 p> Q> X P> VP> xq, yq; (ii) xp, yp, xq, yq, x*p + xyq, xyp + y*q; (iii) p, q, xq, xp - yq, yp; (iv) xq, xp - yq, yp, x*p -f xyq, xyp + y*q\ (v) p, q, xp, xq, yq. Lie-Scheffers, 1893, 1, p. 272. 2. Show that the general projective group in the plane does not have a sub-group of order 7. Lie-Scheffers, 1893, 1, p. 267. 3. The (7 8 with the symbols p + xq, xp + 2yq, (x 3 - y)p + xyq leaves the parabola x 2 2y = invariant. 4. Show that p, g, yq, xp are the symbols of a G*, that p -\- yq,q and p + 2/ j/5 are the symbols of sub-groups Gt, and these are the only sub-groups G* of which p -f 2/9 is one of the symbols. Lie-Scheffers, 1893, 1, p. 554. 5. If the ranks of M (18.1) and for an ordinary point P(z ) are g , ^o + ?i, tfo + 9i + ^2, , there are g independent transformations of order zero at P, q\ of order one, g a of order two and so on. There is an M, such that (i) tfo -f gi -f -f q. = r. 6. If Xif and JiTj/ are of orders hi and /i2 at P(XO), (JTi, -X^a)/ is of order hi + h, - 1. 7. For a G> in one variable the sequence (i) of Ex. 5 is 0, 1, , r 1, from which and Ex. 6 it follows that r is at most equal to three. 8. If Xif, Xrf, Xtf are symbols of transformations of a Gi on one variable of order zero, one and two at an ordinary point, by a suitable change of base one has (i) (Xi, Xt)f = Xif, (Xi, Xt)f = 2Xrf, (Xi, Xt)f = XJ. If the coordinate x is chosen so that Xif = p, the symbols of G\ are (ii) p, xp, x*p, and the finite equation of the group is , ax +b x' *= > > that is, the projective group on the line; show also that the first two symbols of (ii) define the subgroup of affine transformations on the line. Bianchi, 1918, 1, p. 374. 19. INVARIANT VARIETIES 67 9. Find the varieties L , Li, . . . , (18) for the group in x, y, z whose matrix is y x X y z X* xy xz xy y 2 yz xz yz z* and interpret the result geometrically. 10. For the group with the symbols , XJ - p 2 the function Sx 1 ^ 2 a; 3 2CX 1 ) 8 is a relative invariant, and an absolute invari- ant of the sub-group with the symbols Xif and Xif. 19. Invariant varieties. Two points are said to be equivalent under a given group, if they are transformable into one another by one or more transformations of the group. A sub-space V m of V n is called an invariant variety for the group, if all the points equivalent to each point of V m lie in F m . Thus if equations (17.1) admit s independent solutions, say Fi, . . . , F 9 , then the equations where the a's are constants, define an invariant V m , where 771 = n S, and any one of these equations defines an invariant variety of order n 1. Consider a group G, (r > 1) and let the rank of the matrix M (18.1) at a given point PoOro) be q Q < n. Let the indices of the symbols of G> be so arranged that the rank of the matrix of the symbols X e f(e !,- , q Q ) is q Q at P . Then the transformations with the symbols a'X e f, where the a's are arbitrary constants, are of order zero at PO. The points equivalent to JPo under G> lie on the trajectories through P of the Gi's determined by these symbols. The locus of the points is a variety of dimension q Q and is the variety of least dimension containing points equivalent to PO under the group; it is called the minimum invariant variety for PO. Making use of the fact that the sub-group of stability of P is of order r which leaves V m point-wise invariant. Since the jacobian matrix is of rank w, from (20.4) and (20.2) we have dy (20.5) c a ri = 0. Hence the symbols F /are not independent (constant coefficients), and in fact there are only r p of them thus independent. These define a group r r _j> which is said to be induced in V m by G>. Conversely, if p relations (20.5) hold, then (20.4) follow from (20.2) and there is a sub-group G p of G r which leaves V m point- wise invariant. Hence we have: [20.1] // V m is an invariant variety for a G r and there is a sub-group G p which leaves V m point-wise invariant, the group induced in V m is a r r _ P ; and conversely. * Cf. 1926, 3, p, 46. 21. TRANSITIVE AND INTRANSITIVE GROUPS 71 If r p /for p = 1, , r p, are independent (constant coeffi- cients) and we have (20.6) Y T f = then from (7.2) and (20.3) we have (20.7) (Y Pi , Y p )f = (c, + c^d^YJ ( Pl , Pv p 3 = 1, , r - p) as the equations of composition of the induced group. If V m is the minimum invariant variety for each of its points, then the sub-group of stability of each point is of order r m, but only in case there is a sub-group of stability which holds for every point is the induced group of order less than r. Consider, for example, the group of rotations in 3-space about the origin. The spheres with centers at the origin are the minimum invariant varieties, but there is no sub-group of stability holding for all the points of any one of these spheres. When in particular the equations (20.1) are taken in the form (20.8) x r = V(x\ , x m ) (r = m + 1, , n) 9 we have from (20.2) & = ??a5* (a = 1, , m). Hence we have : [20.2] When the equations of an invariant variety Vm. are given in the form (20.8), the symbols Y a f of the induced group are of the form (20.9) Y a f = &(x l , , x m , \f/ m+l , , \l/ n } j /a = 1, , r\ * \c7 = 1, , m) 21. Transitive and intransitive groups. A group is said to be transitive when every two ordinary points of the space are equiva- lent under the group (19); otherwise the group is intransitive. For example, the group of translations in euclidean 3-space is transitive and the group of rotations is intransitive. From the consideration of the finite equations of a group G> it is seen that * Cf. Bianchi, 1918, 1, pp. 164, 165. 72 77. PROPERTIES OF GROUPS. DIFFERENTIAL EQUATIONS for a group to be transitive we must have r ^ n and the rank of the matrix (19.3) must be n\ when r = n, the group is called simply transitive, otherwise multiply transitive.* For an intransitive group there are invariant varieties of order less than n, otherwise by a combination of transformations any ordinary point could be transformed into any other ordinary point. If q is the generic rank of the matrix M (18.1) and q = n, there are no absolute invariants of the group (17) and consequently the invariant variety of an ordinary point is the space itself, so that the group is transitive. But if q < n there are absolute invariants of the group, and the group is intransitive, since there are invariant varieties for ordinary points. Hence we have : [21.1] A necessary and sufficient condition that a group G> be transitive is that r ^ n and that the generic rank of the matrix M be n. For example, the first and second parameter groups of a G r are simply transitive on r variables (the a's) (cf. 9). Also from 19 it follows that the group induced on an invariant variety is transitive, when it is the minimum invariant variety for its points, but otherwise it is intransitive. A transitive group may have invariant varieties, the points of which necessarily are singular. For example, the group G 3 of Ex. 10, p. 67 is transitive and the equation 33.13.2 _ x s _ 2 (x 1 ) 3 = defines an invariant variety. We shall develop in the remainder of this section certain equa- tions which will be used in the subsequent development. We recall from 6 the fundamental equations If the generic rank q of the matrix M (18.1) is less than r, we assign the subscripts of ! so that the matrix ||g|| for h = 1, , q is of rank q and put (2L2) _ % = h p $ (h - 1, , g; p = q + 1, - , r), *Some writers use the term multiply transitive in the sense in which we use the term k-fold transitive in Ex. 14, p. 108. 21. TRANSITIVE AND INTRANSITIVE GROUPS 73 where the & If we take b = p in (21.3) and replace & by its expression (21.2), the result is equivalent, in consequence of equations of the form (21.3) when 6 is replaced by h y to (21.4) Xad = $ap, where (21.5) $ a = C a K p + Catf. - /a=Y- ' ' ,r; \ (A, g = 1, ,q; I \p, s = q + I, , r] When a in (21.4) takes the values 1, . . . , q, we have In consequence of (21.2) and (21.6) when a in (21.4) takes the values q + 1, , r, the latter are reducible to which are fundamental identities connecting the <$>'s. Hence equations (21.4) are equivalent to (21.6) and (21.7). When q < n, that is, when G T is intransitive, the equations X a f = form a complete system of q independent equations, and consequently admit n q independent solutions, say \l/ ff . Without loss of generality we may assume that the determinant ox for t, u = q + 1, , n. If we effect the transformation z' x - x\ x = V (X - 1, , q; - ** d d+ t i* / = !, i n;p = g + l, , r;\ (21.18) A aM , = c w ,({/ I ) M, ", ft* *i J = 1, ' ' ' , ? / When g = r, there are no functions v4 and consequently (21.19) < = (" = lj ' ; ' ' n] \ ^ ' M \X, M, " = 1, ' ' ' i r; r = ?/ 22. Equivalent groups. Two groups G r and // r , whose finite equations are (22.1) f-n* 1 , ,*;aS , a'), a'S f a") (a = 1, , n), f are said to be equivalent when there exists a set of r independent functions

a (d) for a = 1, , r, there exists a non-singular trans- formation of coordinates which transforms either set of equations (22.1) into the other; some writers call two such groups similar. From the results of 7 we have that by a suitable choice of bases of the two groups the constants of structure are the same, that is, the two groups have the same structure. Hence we have : [22.1] A necessary condition that two r-parameter groups in the same number of variables be equivalent is that they have the same structure. Suppose that a (x) and a(z') are the vectors of the bases of two groups G r and H r respectively, such that the constants of structure are the same. In order that the two groups be equiva- lent it is necessary and sufficient that there exist a non-singular transformation (22.2) x' - f (*), such that Ar ta (22.3) = i~ A necessary condition is that the two matrices ||f|| and ||{'|| be of the same rank. 22. EQUIVALENT GROUPS 77 We consider first the case where the generic rank q of the matrix ||{|| is equal to r. If r < n, the coordinates x a and x fa can be chosen (cf. 21) so that (22.4) g = ' = (a = 1, - , r; a = q + 1, , n). From (22.3) for a = r + 1, , n it follows that dx la (22.5) = (cr = q + 1, - - - , n; A = 1, - , q). Hence x 19 are functions 9) c U = 1, - , q Since q = r, equations (21.15) are in this case Because of these relations it follows from (22.9) that the conditions of integrability of (22.7) are satisfied identically. Hence a solution is determined by taking for initial values of x' 1 , . . . , x' r arbitrary functions of x r+l 9 . . . , # n If these are chosen so that the jacobian 78 II. PROPERTIES OF GROUPS. DIFFERENTIAL EQUATIONS of the solution with respect to x l t . . . , x r is different from zero, these functions and (22.6) define a non-singular transformation of G r into H r . Hence we have: [22.2] Two intransitive groups G r and H r in the same number of variables, with the same constants of structure and whose matrices M are of rank r, are equivalent; the equations of transformation involve r arbitrary functions. If r = n, that is, if the group is simply transitive, we have in place of (22.4), (22.6) and (22.7) only equation (22.7) in which X, ju> a = 1, . . . , n. As before these equations are completely integrable, and their solution involves n arbitrary constants, the initial values of the x"s. Hence we have: [22.3] Two simply transitive groups in the same number of variables and with the same constants of structure are equivalent, and the equations of transformation of one into the other involve n arbitrary constants. We consider next the case when q < r. For the group G> we choose the 's so that ||f|| for h = 1, , q is of rank g, then the matrix ||& a || for h = 1, , q must be of rank q, and analogously to (21.2) we put (22.10) fr =

*'*) (7 1 v 22 - 11 ) ' / There are q(r q) of these equations. Evidently in order that the two groups be equivalent, it is necessary that these equations be consistent, and that it be impossible to eliminate the x"& and get a relation between the x's, and vice-versa. We shall show that this condition is also sufficient. If q < n, we assume that the coordinates x r and x are chosen so that (22.4) and (22.6) hold. As in the case when q = r, we get the system of equations (22.7) in which now X, ju, a = 1, , g. When in the "s in these equations and in (22.11) we replace x" for a = q + 1, , n by x*, we have a mixed system (1) of differ- ential equations in a;' 1 , . . . , x' q in the independent variables x l 9 . . . , x q which system involves x q + l , . . . , x n as parameters. As in the preceding case we have (22.9), and in consequence of 22. EQUIVALENT GROUPS 79 (21.15) the conditions of integrability of (22.7) are satisfied because of (22.11). From (21.11) we have / 99 19 \ d, , *'; *S , ) /p = S + 1, , q\ \X = 1, ' ' * , q ) 80 //. PROPERTIES OF GROUPS. DIFFERENTIAL EQUATIONS obtained from (22.7) on replacing x' 1 , . . . , x'* by the expressions from (22.13) and x' 9 by x 9 for h P (x Q )X h f (h = 1, , q; p = q + 1, , r). If n p of the functions ^ are independent, the equations (24.2)

not in the sub-group, so that S(Po) = PI, where PI is a point not in V p defined by (24.2). Then STS' 1 is a transformation of stability of PI, and of all the points S(P), where P is any point in V p , that is, of all points in the variety V' p into which V p is transformed by S. Conversely if T' is a trans- formation of stability of the points in V' p then S~ 1 T'S is a trans- formation of stability of the points in V p . Thus V p is a systatic variety for G>. Hence we have: [24.1] When a group G r is systatic and the generic rank of the matrix || (|| is less than r, the systatic varieties form a system of imprimitivity. Exercises 1. Show that a: 2 -f- y 2 = c 2 z 2 , where c is an arbitrary constant, are invariant varieties for the G* of Ex. 9, p. 67 and find the induced group on each by putting x = cu cos v, y = cu sin v, z cu. 2. If XJ and Y, x 3 = 3ut> 2w 3 , and the system of imprimitivity of the induced group. 6. Show that the group of rotations in euclidean 3-space is systatic and that the straight lines through the origin are the systatic varieties, and systems of imprimitivity. 7. Show that the concentric spheres with the origin for center are systems of imprimitivity of the G* x ip. _ x ip. t x , so that we have the theorem : [25.1] // Xif, . . . , X r f are linear operators, a necessary and sufficient condition that X a O be solutions of a complete system (25.1) AJ = <&>< = (a = 1, , p), when is a solution of this system other than a constant, is that (25.2) (^j/ where the \'s are at most functions of the x's. When these conditions are satisfied, we say that the complete system admits the set of operators, or that it admits the one- parameter groups of continuous transformations generated by these operators in the sense of 10. 86 II. PROPERTIES OF GROUPS. DIFFERENTIAL EQUATIONS From the Jacob! identity (2.4) applied to the operators X a f, X b f and A a f, we obtain in consequence of (25.2) (25.3) ((X a , X b ), A a }f = (Z a X 6 * - Z 6 X 6 f + XJ\4 - XoXM*/- Since this is of the form (25.2) we have: [25.2] // a complete system of linear homogeneous partial differential equations admits the operators X a f and Xi>f, it admits also the commuta- tor (X a , X b )f. If the operators X a f are the symbols of a group G>, no new operators are obtained as a result of this theorem. We shall consider the more general case of theorem [25.1]. Since the system (25.1) is complete by hypothesis, we have (25.4) (A a , AJf = in whose equations x h enters as a parameter. Accordingly we have : [25.7] // an equation Af = admits r independent operators X a f, either it admits a group G r of infinitesimal transformations, or solu- tions of the equation are obtained by direct processes. Consider the case when n = 2 and there is one operator Xf. If we eliminate X from the two equations da> d& (25-14) e- - -< - X> a = 1, 2), which are the equations (25.10) in this case, we obtain (25.15) \ dx 1 dx*/ \ dx 1 dx 1 This may be written in the form (25.16) ^M) + ~(a^M) = 0, dx 1 dx z where Consequently Af is an integrating factor of the equation (25.18) a*dx l - a l dx* = 0, and a solution of it and consequently of (25.9) is given by a quadrature. Conversely, if an integrating factor M of (25.18) is known and any functions r and 2 are chosen to satisfy (25.17), equation (25.16) becomes (25.15), and from the latter (25.14) follows. Evidently if f 1 and 2 are functions satisfying (25.17), so also are S 1 + a 2 , where

i(x), 172 = *(x) + y, where 2 = e j * l > y' = y ^ I -dx, J 2(z)<7, where ?/' = T/ - I dx, J v\ J , namely (26.1) *"=/<(*;<*), that (26.2) rf , grf. Equations (26.1) and (26.2) define a transformation in the 2n variables x i and x\ involving r parameters; the values a" for which (26.1) becomes the identity yield the identity in (26.2) also. It is readily shown that equations (26.1) and (26.2) define a group G> in these 2n variables which is called the extended group of G>. In fact, from the equations (4.7), namely (26.3) it follows that #*(*'; 01) dx k dx k Consequently we have , df*(x'; a 2 ) df*(x; a 3 ) ; - Xl = &T- Xl = ~toT-^ which establishes the group property. From (26.2) and (5.5) we have For the extended group equations (5.5) and (26.5) together in 2n variables play the role of (5.5) for the given group. Since the 26. EXTENDED GROUPS 93 vectors A a a (a) are the same for the extended group, we have from (6.3): [26.1] The constants of structure of the extended group are the same as for the given group. Since the parameter groups are determined by the vectors A t we have also : [26.2] The first and second parameter groups of the extended group are the same as for the given group. Analogously to (6.2) we have (6.2) and fort a\ frJ/ (26.6) (z ak dx> dx 1 dxi dx\ The symbols of the extended group are (26.7) Ox 1 dxi In consequence of (6.2) and (26.6) we have (26;8) which result follows also from theorem [26.1], In accordance with the theory of 17 any function of the z's and Xi& which satisfies the equations (26.9) Xcoa/ - is called an absolute invariant of the extended group. Furthermore, if a set of p independent functions F\, . . . , F p of the z's and Si's are such that X(i>aF a = (a - 1, , p) not identically but for values of the x's and xi& for which F a (x;xi) ~ 0, we say that these functions define a relative invariant of the extended group. Consider, for example, the p independent Pfaffian equations (26.10) \?x\ = (a = 1, - , p; i = 1, - , n) f 94 II. PROPERTIES OF GROUPS. DIFFERENTIAL EQUATIONS where the X's are functions of the x's. From (26.7) and (26.4) we have Since the right-hand member is linear in the #i's, in order that X?xl be a relative invariant, it is necessary and sufficient that foA i-n fcj^" i ^\ _ \0 (26.11) fc, + X, - r oS X,, where the ' ' > n > \ dx l \l, m = 1, , q/ Expressing the condition of integrability of these equations, we obtain (26.13) _ \l where A?j m is defined by (21.17). Moreover, A?/ m are zero, when the rank of |||| is r (cf. (21.19)). Hence equations (26.12) are completely integrable. If r < n, each set of solutions is determined by n arbitrary functions of or r+1 , . . . , x n , taken as initial values; if r = n, each set of solutions is determined by n arbitrary constants taken as initial values. Hence we have : [26.3] If the generic rank of the matrix ||(|| of a group G r is r(^n) there exist Pfaffian forms \x\ which are absolute invariants of the extended group; the \'s involve n arbitrary functions when r < n, and n arbitrary constants when r = n. We consider next the case where the generic rank q of the matrix 1 1 fill is less than r, and assume that the coordinates are chosen and the 's are designated so that the results of 21 obtain. From (26.11) for a = 1, , q we have a set of equations of the form 26. EXTENDED GROUPS 95 (26.12). When a in (26.11) takes the values q + 1, - - , r, the resulting equations are reducible by means of (21.2) and (26.12) to (26.14) X-rfi^j- = 0. When these equations are satisfied, the conditions (26.13) are satisfied in consequence of (21.18). Hence the problem of deter- mining Pfaffian forms when q < r, reduces to the solution of (26.12) with the finite conditions (26.14); that is, these are the equations Fo of 1. Moreover, the set FI are those arising from (26.14) by differentiation and reduction by means of (26.12) and (21.13). By the processes of 1 we determine in any case the generality of such absolute invariants of the extended group. We return to the consideration of equations (26.11) for the case of the single equation (25.18) and one operator Xf. Then we have P from these two equations, we obtain equation (25.15). In consequence of the preceding results concerning this equation, we have : [26.4] // a Pfaffian form \tx\ is a relative invariant of the extended group of a Gi in two variables x 1 and x 2 , the equation Xx} = can be integrated by a quadrature. Hence we say that the equation (25.18) admits the group Gi, if the left-hand member of the former is a relative invariant of the extended group of G\. In like manner we say that any differential equation F(x, y, xi, yi) = 0, homogeneous in xi and yi admits the group Gi, if F is an invariant, absolute or relative, of the extended group. In order to give this problem another form we write the equation thus (26.15) /(*, y, y') = 0, y' = % = ^- ax x\ Since JL = ^ JL _^' A A - 1 jL dxi dxi dy' ~ xi dy'' dyi ~ xi By'* 96 II. PROPERTIES OF GROUPS. DIFFERENTIAL EQUATIONS we have from (26.7) and (26.4) in the case of two variables x and y If the group is a O rt we have r symbols Xu) a /of the form (26.16), for a = 1, , r. When the finite equations of G> are written in the form (26.17) x = /(x, y] a), y = , (26.18) ' = - Q- s *(x, a curve of the plane goes into a curve and the extended group gives also the relation between the slopes of the tangent at corresponding points. If a differential equation (26.15) admits the G>, under a transformation of the group an integral curve is transformed into another integral curve. If /(x, y, y'} is an absolute invariant of the extended group of a Git that is, if it is a solution of X(\)f = in which x, y and y' are treated as independent variables, we say that it is a differential invariant of the first order of the group G\. For a given Gi 9 that is, for and ij given, the problem of finding its differential invariants reduces to the integration of dx _ dy^ dy^ t *L + (*a - *H\ * - ( >)* d J- dx \dy dx) y (y } dy Let u(x, y) = c be a solution of t\dx dy = 0, or in other form y = (p(x, c). Substituting this value of y in the equation we have a Biccati equation for the determination of y f as a function of x and c. It is readily shown that y 1 = TJ/ is a particular solution of this equation, and in consequence of the known theory of Riccati 27. EXTENSIONS OF THE SECOND AND HIGHER ORDERS 97 equations we have that the complete solution of this equation reduces to quadratures. If we denote this solution by w(x, y', c) = d, where d is an arbitrary constant, then v(x, y, y r ) s w(x, y', u) is a solution of Xd)f = and the most general solution is any function of u(xj y) and v(x, y, y'). The function u(x, y) is an absolute invariant of d, that is, the curves u = const are the trajectories of the group. Hence we have : [26.5] // the trajectories of a group G\ in two variables are known, the differential invariants of the first order of Gi can be found by quadratures. If the finite equations of G\ are known, then the extended group is found by differentiation, and in accordance with 19 the deter- mination of the differential invariants of the first order results from the elimination of the parameter a from the equations of the extended group. 27. Extensions of the second and higher orders. Ordinary differential equations of the second order. If we consider the extended group, which we now call the first extension of the given G r , as a group in 2n variables and form its extended group, which we call the second extension of G>, we obtain a group in x\ x\ and the second differentials of the x's which we denote by x\. Con- tinuing this process we get extensions of all orders. Thus the p-th extension involves (p + l)n variables x\ x\ f . . . , x l p . From the manner in which (26.7) were obtained it follows that the symbols of the second extension are foj n V, f - t< & a. t * 9 f a. t d f (27.1) A( 2 )a/ = fo . + la-"i + ?2aT-l> dx l dx[ dx\ where in consequence of (26.4) (97 ^ t At < (27.2) &, = dfc. Generalizing this result, we have that the symbols of the p-th exten- sion may be written in the form (27.3) Xwaf = d?A (a = 0, 1, , p), dx a with the understanding that d = and XQ = x. 98 II. PROPERTIES OF GROUPS. DIFFERENTIAL EQUATIONS As in the case of the first extension we have : [27.1] The constants of structure and the first and second parameter groups of the p-th extension of a G r are the same as for G>. From the foregoing discussion it follows that the matrix of the first extension is obtained from ||(||, the index i indicating the row and a the column, by augmenting the former by n rows containing the differentials of the f 's. And in general the matrix of the p-th extension is that of the (p l)-th extension augmented by n rows, the elements being the differentials of the preceding n rows. Since the p-th extension involves (p + l)n variables, if (p + 1)^ > r this extension is intransitive, and consequently there are absolute invariants of the p-th extension. Consider for example the second extension of the group (? 3 of motions of euclidean 2-space. Its matrix is y 1 -x 2/1 -xi 2/2 -z 2 Its absolute invariants are solutions of (27.4) = 0, -i dx dy df df *~* dx 2 9f 0. From the first two of these equations it follows that these invariants are independent of x and y. By inspection we see that three independent solutions of the third equation are (27.5) (i) x\ + y\] (ii) (iii) + In accordance with the general theory of complete systems any absolute invariant of the second extension is a function of these three. The first and third of these are respectively the square of the linear element and one half of its differential. For a curve y = /(#), we have y\ = y'x\ t y* = y"x\ + y'x*, consequently (27.6) (x\ + y!)* ( which is the curvature of the curve. 27. EXTENSIONS OF THE SECOND AND HIGHER ORDERS 99 We consider now the general case in two variables, x and y. We write (26.16) in the form (27.7) Z(1)/ _+#+* dx dy dy' where (27.8) , t - - ,f , dx dx where is the total derivative, that is, dx drj drj drj = 7/ dx dx dy u Looking upon (27.7) as the symbol in three independent variables x, y and y', we have (27.9) dx = , Sy = rj8t, dy' = y> y'> y"} = admit the G\ with the symbol Xf is that F be a relative invariant of the second extension ofGi, that is, that XwF = when F = 0. If, however, F = is given in the form y" = co(a;, y, y'), then the condition that this equation admit the G\ is that w satisfy the equation do) do) do) (27.14) r; 2 - { - TJ - rj, = 0, dx dy dy' it being understood that y" in 772 is replaced by w. We denote by u(x, y) a solution of (27.15) Xf = fr + yq = 0. From 10 it follows that, when u(x, y) is known, a function v(x, y) can be formed by quadratures such that dv dv 28. DIFFERENTIAL INVARIANTS 101 If we put x = v(x, y), y = u(x, y), then in the coordinates x and y Xf=P and the finite equations of the group are x r = x + t, y' = y. Consequently y, ,..., , ... are differential invariants. ax ax p The symbol of the pth extension in these coordinates is = p. Hence any differential invariant is a function of ii fi' ?}(P) y * y > i y > . . . . Consequently we have : [27.3] The determination of the differential invariants of a G\ with the symbol Xf reduces to the determination of a solution of (27.15), a quadrature and differentiation. Since the property of invariance is independent of the coordi- nate system, it follows that the differential invariants of the pth order of Xf are of the form Flu. >> V In general the \ dv dv p / differential equations of order p which admit the G\ with the sym- bol Xf are obtained by equating such functions to zero. However, there are exceptional cases. For example, if in the coordinate system of (27.15) the coefficient of df/dy (p) in X( p )f involves y (p) as a factor, then y (p) = is invariant under (27.15) for this value of y (p} . 28. Differential invariants. In 26, 27 we have considered in particular the case where there are two variables and the effect of the transformations of a G r on a curve y = f(x) and the deriva- tives y', . . . , y (p \ .... This is a particular case of the general problem in a V n for a sub-space V m of m dimensions defined by (28.1) x p = &(x l , , x m ) (p = m + 1, , n). 102 II. PROPERTIES OF GROUPS. DIFFERENTIAL EQUATIONS If these values are substituted in the finite equations of the group (28.2) " = ?(x; a), we obtain for each set of values of the a's a V' m into which F m is transformed. In order that the equations of V' m may be expressed in the form (28.3) X 'P , x' m ] a) (p = m + 1, , n), it is necessary and sufficient that the first m of equations (28.2) after the substitution from (28.1) be solvable for x l , . . . , x m and the resulting expressions substituted in the last n m of (28.2). Hence the matrix (28.4) df a df* d dx b dx* dx b /a, b = 1, , w; \ \p = m + 1, , n/' must be of rank m. When the determinant of this matrix is d

it is seen that the coefficients dx are minors of order m of the matrix ; for a = 1, , rw; i = 1, , n, and every such minor is a coefficient. Since all of these minors cannot be zero in accordance with the requirement (4.2), it follows that the rank of (28.4) is m for general 's are such that the rank of (28.4) is m, we have that each V m is transformed into a V'm defined by (28.3), in which the a's have particular values for each trans- formation of G>. From (28.1) and (28.3) we have (28.5) dx* - ~dx a = 0, ox dx' a r' a = (a- 1, From the first of these sets and (28.2) we have (28 . 6) v J C= 1, - - , n; a = 1, - , rn;\ = m + 1, , n / 28. DIFFERENTIAL INVARIANTS 103 by means of which the second set of (28.5) are expressible in the form + ._ , d _ dx" dx* dx a dx' b \dx a dx* dx a /a, b = 1, , m; \ \P, q = w + 1, , n) Since the differentials dx 1 , . . . , d# m are arbitrary, the expressions in parenthesis are zero. The resulting equations can be solved for as functions of the z's, and the a's, because the rank of ox ox a (28.4) is w. If we denote these partial derivatives by x fp b and x p a , the solution may be written as (28.7) x'l=ft(x;xl\a). In order to find the second derivatives of x' p with respect to x' 1 , . . . , x' m , we make use of . dx' b dx' a By means of (28.7) and (28.5) the left-hand member is reducible to a linear function of dx 1 , . . . , dx m , the coefficients being func- tions of the x's, the first and second derivatives of x m+l , . . . , x n with respect to x l , . . . , x m and the a's. By means of (28.6) the right hand member is dV p /d/ df a d

because the rank of the matrix (28.4) is m. dx /b dx' a If we use the notation ^L^ (A , , + ... + .), 104 II. PROPERTIES OF GROUPS. DIFFERENTIAL EQUATIONS and similarly for z p , these may be written in the form (28.8) a... =/*... .(*;*...,.; a) p, g = m + 1, i + + Pi + ' + for A = 2; thus we have each second derivative of any #' p equal to a function of the z's, first and second derivatives of x m+l 9 . . . , x n with respect to x l 9 . . . , x m and the a's. If we apply to (28.8) for A = 2 the process which was applied to (28.7), we obtain a set of equations (28.8) for A = 3. Con- tinuing this process step by step, we obtain equations of this sort for any integer N. Consider now the set of equations (28.2) and (28.8) as A takes the values 1, . . . , N 9 where N is some positive integer. These equations define a group in the x's and the derivatives of x m+l 9 . . . , x n with respect to x l , . . . , x m of the first, second, . . . , Nth order. For, if we give the a's in these equations the values a?, and apply the above processes to (28.9) *"< =/*(*';!), we get in place of (28.8) (28.10) x '. . . am = J* ai . . . >'; x%... pj a,). Eliminating the z"s from (28.7) and (28.9), we have (cf. 4) x m =/'(z;a 3 ) a? =

. From the results of 19 it follows that when the finite equations of a group G> are known, the determination of the differ- ential invariants of any order N involves differentiation in order to obtain the Nth extension and then the elimination of the parameters a a from its equations. When the symbols of the group are given the problem of finding the differential invariants is that of finding solutions of the complete system obtained by equating to zero the symbols of the extension of given order. We proceed to the determination of these symbols. We have seen in 10 that dx i = { 8t and accordingly we put We must have (cf. 27) (1 -m < ...a.- 2 < - ...,..+1..,, *<& = 0. Interchanging the operations 6 and d, we have (28.11) d? ai ... am - I- m Since f J . m involves the x's and derivatives of x m+ 1 , . . . , x n to order JV(= i + + a m ), we have K... an d^... amdx ,^de at ... am dxi <... a .ax V ^.... g .<.....L + da; ^ + ^i^|,..., m ax- f x ' where S indicates the sum of all the terms when fti + ' + A. ^ N. 106 II. PROPERTIES OF GROUPS. DIFFERENTIAL EQUATIONS > If we denote the expression in the parenthesis by - - - and put dx a dp = dp d^dx^ dx a "~ dx a dx* dx on equating to zero the coefficients of each of dx l , . . . , dx m in (28.11), we have d^ a (28.12) ^ . . . ._><, + 1. 1 ,= By means of this formula we obtain the 's for any extension from those of the preceding one. Equation (27.11) is a particular case of (28.12). If we denote the symbol of the Nth extension by Xwf, we have (28.13) where S indicates the sum of terms when 01 + ' * + m takes the values 1 to N. If we have a group G r , for each symbol X a f we form a symbol X( N ) a f by the above process and thus obtain the r symbols of the Nth extension. Exercises 1. If an equation Af = admits two operators Xif and X 2 /, such that X z f = x l x*x* is a solution of the equation. Show that x l -\- x* -f- x 1 is a second solution. Lie-Scheffers, 1891, 3, p. 451. 3. Show that the equation Af m (*> + 2)pi - 2 -X^/ = (x* + 2x 3 + l)(pi + 2pj pi) and that and consequently x 2 + 2x 8 is a solution of Af = 0. Lie-Scheffers, 1891, 3, p. 447. 4. If an equation Af B a*p =0 (i = 1, 2, 3) admits two operators Xi/and X 2 /, and the rank of the matrix ||a, }, J|| is three, the integration of Af = reduces to at most two quadratures. Lie-Scheffers, 1891, 3, p. 444. 5. If F(x) is an absolute invariant of a group <7 r , -.x\ is an absolute invar- da;* iant of the extended group. 6. If Fi(x) = =s Fp(z) = define a relative invariant of a group G r , these equations and x{ = (a 1, , p) define a relative invariant of dx* the first extended group. 7. Show that y" = w(a;, y, t/') is invariant under a Gi, if and only if the equation df_ t df_ jtf_ _ aa; 3y W dy' admits the first extension of the symbol Xf of Gi. Lie-Scheffers, 1891, 3, p. 364. 8. Show that the following differential equations are invariant under the accompanying operator: (1) y" = F(ax + by, y'), bp - aq; / y y' \ (2) y" = & 'Ff -, ], xp + nyq; - n)x n+l y f = F\ xy'-nyY \*" / (3) 2/" 9. If u(x, y) is an absolute invariant of a G\ and t;(x, y, y') is a differential dv invariant of the first order, then - is a differential invariant of the second order. du Lie-Scheffers, 1891, 3, p. 377. 10. If u(xj y) and v(x } y, y') are invariants of a G\ t the most general differen- tial equation invariant under G\ is given by dv_ du where ^ is an arbitrary function of u and v. Lie-Scheffers, 1891, 3, p. 377. 108 II. PROPERTIES OF GROUPS. DIFFERENTIAL EQUATIONS 11. Show that because of Ex. 10 the solution of a differential equation of the second order admitting an operator requires at most the solution of two differ- ential equations of the first order and quadratures. Lie-Scheffers, 1891, 3, p. 378. 12. An ordinary differential equation of the second order in x and y is invariant under at most eight linearly independent (constant coefficients) operators. Lie-Scheffers, 1891, 3, p. 405. 13. A differential equation of the second order which admits two operators can be integrated by quadratures. Lie-Scheffers, 1891, 3, pp. 457-464. 14. When a group G> has the equations (4.1), the equations W~H A * a / I * = ' ' ' W I I _ j ... 1. / define a group G r in the nk variables x. If this group is transitive, any k ordi- nary points in F can be transformed into k other arbitrary ordinary points, and the given G> is said to be k-fold transitive, when k is the maximum number for which this is true. The symbols of the group (i) are XiJ + + where / \ a = 1, , 15. For k arbitrary points of coordinates z is fc-fold transitive. 16. If vk in Ex. 15 is less than nk, the equations XiJ + + XkJ = admit nk vk independent solutions, called the invariants of the k points x^ If k is such that vk = r for a G>, the system of ft -f 2 points, or of a larger number, does not have any invariants which are not invariants of a lesser number of points. Bianchi, 1918, 1, p. 336. 17. The projective group G 9 in one variable with the symbols p, xp } x*p is 3-fold transitive. The invariant of four points is their cross-ratio Xi Xt Xt X* Xi - Z 4 X t X t and is the solution of CHAPTER III INVARIANT SUB-GROUPS 29. Groups invariant under transformations. Consider a group G> with symbols X a f and also a transformation S, namely (29.1) x* = ?'(*). In order that the symbols (29.2) Xj= (*)| generate in the x's the same group as the given group, it is necessary and sufficient that each of the operators (29.2) be a linear combina- tion with constant coefficients (11) of the symbols arising from X a f by the transformation (29.1), that is, (29.3) &(*) = cl&(xf dx j must be satisfied identically in consequence of (29.1). From (29.3) we have X a f = clX b f, J(x) = f(x). If u a are canonical parameters for G> and we put (29.4) u a = ciu b , u b = c b a u a , where the constants cl and cl are in the relations (cf . 7) (29.5) cl c c 6 = 5?, then we have (29.6) uX a f = u*X b f, and consequently equations (11.11) are transformed into the same form in the x's by means of (29.1). Thus we have schematically (29.7) ST U S~ 1 = T, or (29.8) ST U = TS. ino 110 ///. INVARIANT SUB-OROUPS When the above conditions are satisfied, we say that the group G r admits the transformation S, or is invariant under it. When and only when we have cl = $?, equations (29.3) become (29.9) in which case (29.8) is (29.10) ST U = TJS, that is, S is commutative with each transformation of the group. Consider next the case when the transformation (29.1) is replaced by a one-parameter group FI with the symbol Xf = *??. In this case the c's in (29.3) are functions of the parameter r of the group such that, if T = defines the identity, we have (ca) T -o = d. In order that (29.3) hold for the infinitesimal transformation (29.11) & = V + we find on substitution that it is necessary that where gl are the constants defined by From (29.12) we have that (29.14) (X,X b )f is a necessary condition that the group admits the infinitesimal transformation. If the coordinates are chosen in accordance with theorem [10.2] so that (29.15) {' = ai, equations (29.12) become (29.16) ^ + 9l = 0. dx n In this coordinate system the finite equations of PI are (29.17) x* = x< + d ( n r. 29. GROUPS INVARIANT UNDER TRANSFORMATIONS 111 If in (29.6) we put / = x\ then in consequence of (29.17) we have (29.18) u"&(x) = u&(x). Differentiating with respect to T, we have - o. Since equations (29.16) are identities in the x's and consequently hold in the z's, the above equations are reducible to from which it follows that u" are the solutions of the equations (29.20) ^ = u"gl dr which reduce to u a when r = 0. Conversely, if equations (29.14) are satisfied, the coordinate system being any whatever, then in the particular coordinate system for which (29.17) are the finite equations of the group FI we have equations (29.15). If u a is any set of solutions of (29.20), then u a &(x) is independent of r as is shown by differentiation with respect to T, and we have (29.18) where u b = (u b ) T -Q, from which follow (29.6) in consequence of (29.17). We have thus established that equations (29.14) constitute a necessary and sufficient condi- tion. Hence we have: [29.1] A necessary and sufficient condition that a group G r generated by the infinitesimal transformations with the symbols X a f admit the transformations of a group FI with the symbol Xf is that equations (29.14) be satisfied, where the g'$ are constants. In order to obtain equations (29.3), we have made use of the fact that the transformations with the symbols X a f form a group. We shall show that the preceding theorem holds, even if the trans- formations u"X a f, where the u's are arbitrary parameters, do not form a group. In fact, if equations (29.6) are satisfied with the understanding that (w a )7-o = w, we have equations (29.18) for the special coordinate system in which the equations of the group FI are given by (29.17). If in (29.19) we put r = 0, we get 112 ///. INVARIANT SUB-GROUPS Since these equations must hold for arbitrary values of the w's, if we take r different sets of u a such that all but one is zero in a set and the other is unity, we obtain equations of the form (29.16) in which the g's are constants. Then the discussion proceeds as before. Hence we have: [29.2] // X a f are the symbols of r infinitesimal transformations, in order that the transformations of symbols u a X a f, where the u j s are arbitrary parameters, admit the group FI of transformations with the symbol Xf, it is necessary and sufficient that (29.14) be satisfied, the g's being constants. Then a given G\ defined by u a X a f is trans- formed by all the transformations of FI into the group G\ defined by u a X a f, where the u's are the solutions of (29.20) such that u a = u a when r = 0.* As a corollary to this theorem and because of the second funda- mental theorem we have: [29.3] // X a f are the symbols of r transformations and all of them are invariant under each of them, the given transformations form a group of order r; and G r is invariant under any sub-group G\. From considerations analogous to those applied to the trans- formations (29.1) leading to (29.9), we have that, when and only when u a = u a , each transformation of G> is commutative with each transformation in IY Then from (29.20) we have gl = 0. Hence we have: [29.4] A necessary and *n Writ hi condition that each transformation of the group with the symbol Xf be commutative with each transforma- tion of the group G r with the symbols X a f is that (X, X.)f = (a = 1, , T-). As a corollary we have : [29.5] A necessary and sufficient condition that every two transforma- tions of a group G r with the symbols X a f be commutative is that (X a , X b )f = 0, that is, the constants of composition are zero. *Cf. Bianchi, 1918, 1, p. 198. 30. COMMUTATIVE GROUPS 113 A group satisfying this condition is called Abelian (cf. 13). When a sub-group G\ of a group G> is commutative with all the transformations of G>, the G\ is called exceptional. If its symbol is e a X a f, then we must have (e a X a , Xb)f = and consequently (29.21) e?c& = 0. These r 2 equations, as b and d take the values 1 to r, must be con- sistent. Hence the rank of the matrix (29.22) HC.JH where a(= 1, , r) denotes the columns, and 6 and d the rows must be less than r. If the rank is s, there are r s independent sets of solutions of equations (29.21). If we put r s = m, we may change the symbols of the given group G> so that Xif, . . . , Xmf are the symbols of exceptional groups Oi. Hence we have that the corresponding constants of composition satisfy the conditions (29.23) c t h a = (a, 6 = 1, - , r; t = 1, , m), Since (X t , X u )f = (*, ii = 1, - - - , m), it follows that : [29.6] The exceptional groups G\ of a group G>, if any, form an Abelian sub-group. 30. Commutative groups. Reciprocal simply transitive groups. If X a f are the symbols of a group G> and Zf = * . is the symbol of c/x a group (?i, then, as follows from theorem [29.4], a necessary and sufficient condition that each transformation of G> be commutative with each transformation of G\ is that We consider first the case when the group is simply transitive, that is, r = n and the generic rank of the matrix |||| is n. By means- of (21.13), where in this case X, ju, 0, h = 1, , n, equa- tions (30.1) are equivalent to (30.2) ^ + f*A& = 0. Expressing the conditions of integrability of these equations, we find that they are satisfied identically because of (21.19). Hence 114 ///. INVARIANT SUB-GROUPS equations (30.2) admit n sets of solutions f &, for 6 = 1, , n, such that the generic rank of the f 's is n. Moreover, any solution of (30.2) is a linear combination (constant coefficients) of these n sets of solutions. If we put Zaf 55 f *p t , since (Z., X,)/ = 0, (Z b , Z c )/ = (a, 6, c = 1, , n), it follows from the Jacobi identity (2.4) that ((Z a> Z>), JQ/ = 0. lt.7 /)/' Hence fi : f*& -. is a solution of (30.2) and in consequence dx* dx 1 of the above observation we have (30.3) fig? - rtg = Mri, where the c's are constants. Consequently Z a f are the symbols of a continuous group G r > Clearly the relation between the groups G r and G r is reciprocal; they are said to be reciprocal groups. From (8.5) and (30.1) it is seen that the first and second parameter groups of any group G r are reciprocal groups. From the results of 29 it follows that if x'* = of the symbols (x') - Consequently dx 1 ox * G r and G> are equivalent groups, and from theorem [22.3] it follows that the equations of transformation, which are in fact the equations of the group G T) involve n parameters. From (22.7) it follows that they are the integrals of the completely integrable system of equations dr'* (30.4) -j - (*')(*) (a, f, J - 1, ' ' , n). dx' Thus we have : [30.1] // &(#) are the vectors of a simply transitive group, the equa- tions (30.4) are completely integrable, and their solutions, involving n 30. COMMUTATIVE GROUPS 115 arbitrary constants, are the transformations of another simply transi- tive group, the reciprocal of the given group. Returning to the consideration of the solutions of equations (30.2), we may write (30.5) rf = titl Substititing in (30.2) and making use of (21.13), we have = 0. For the present case equations (21.15) are (30.6) A/, - A?* = CkT&&& (h, i, j, k, I, m = 1, , n). From these two sets of equations it follows that (30.7) ^ + *jc& = 0. If we substitute for the derivatives of the f 's in (30.3) their expressions from (30.2), we obtain From these equations, (30.5) and (30.6) we get, since the deter- minant of the 's is not zero, From (30.7) it follows that the ^'s are not constants, and they are not all zero when the x's are zero, otherwise they would be zero identically, as follows from the theory of differential equations of the form (30.7). Since the above equations must hold identically, we have when the x's are zero where the c's are constants. Hence the groups G> and G> have the same structure and consequently we have (22) : [30.2] Two reciprocal simply transitive groups are equivalent. If r < n and the generic rank q of the matrix |||| is equal to r, we choose a coordinate system such that (21.8) is satisfied, and proceed as in the case of simply transitive groups. Then equations (30.2) hold with t, j, k = 1, , r, and in addition we have from (30.1) 116 ///. INVARIANT SUB-GROUPS - The system of equations (30.2) is completely integrable because of (21.19), and consequently a solution is determined by taking for initial values arbitrary functions of x r+1 , . . . , x n . Thus each solution of (30.2) and (30.8) involves n arbitrary functions of these coordinates. For any two of these solutions we have equations of the form (30.3), where now the c's are in general functions of the coordinates x r+l , . . . , x n . Hence although each solution yields a symbol Zf which generates a group Gi each of whose transforma- tions is commutative with every transformation of G r , these Gi's are not sub-groups of a group of higher order. However, if it is possible to choose the coordinates so that (21.8) hold and the other 's are independent of x r+l , . . . , z n , then solutions of (30.2) can be found not involving these variables, and the r operators Z a f are the symbols of a group G r reciprocal to Gr- in this case we have in fact two simply transitive groups in r variables and the above result is in keeping with theorem [30.1]. Hence we have : [30.3] // the rank of the matrix ||&|| of an intransitive group G r is r, there exist groups Gi, whose vectors involve arbitrary functions, such that each transformation of the latter is commutative with each trans- formation of G r , but in general these G\s are not sub-groups of a group of higher order. If the group G r is transitive but not simply transitive, we may in all generality assume that the matrix |||| for a = 1, , n is of rank n, and that for p = n + 1, , r we have (21.2). When a takes the values 1 to n in (30.1), we have the equivalent system (30.2) and for a = n + 1, , r equations (30.1) reduce by means of (30.1) for a = 1, , n to (30.9) r*U' = 0, in consequence of (21.11). Hence the problem reduces to the solu- tion of the mixed system (30.2) and (30.9). The conditions of integrability of (30.2) assume the form (cf. (21.17)) (30.10) r*Aj,< = 0. From (21.18) it follows that in this case equations (30.10) are not satisfied identically. 30. COMMUTATIVE GROUPS 117 Equations (30.9) are in this case the set F referred to in 1. Equations (30.10) and those resulting from (30.9) by differentiation and substitution from (30.2) are the system Fi. We observe that Fi is linear and homogeneous in the f 's and the same is true of the succeeding systems F 2 , . . . . The number of independent equa- tions, say s, in this set is at most n 1. From algebraic consider- ations it follows, in accordance with theorem [1.1], that the conditions of the problem are that there exist a positive integer N such that the rank of the matrix of the sets F Q , Fi, . . . , F N is n s (s ^ 1) and that this is also the rank of the matrix of FQ, . . . , FAT+I. If these conditions are satisfied, the solution of the problem reduces to the integration of a completely integrable set of equations of the form (1.11) in n s functions 6* and the ^'s are linear and homogeneous in these functions. Consequently any solution is expressible as a linear function with constant coefficients of n s particular solutions, and such an expression is a solution. If then there are n (>!) solutions, we have (30.3) holding for any two of them, where the c's are constants. Hence we have: [30.4] // a group G r is multiply transitive, there is not in general one, or more groups, (71 each of whose transformations is commutative with every transformation of G r ; if there are s(>l) of such groups Ui, they are sub-groups of a group (j 8 . If G r is intransitive and the generic rank q of the matrix ||&|| is less than r, we choose the coordinates so that (21.8) hold. We make use of (21.13) to obtain (30.2) for i, j, k = 1, , q. In this case we have (30.8) for i = 1, , q and . When a group G> does not have any invariant sub-groups other than the group itself and the identity, the group is said to be simple. From 29 it follows that, if a group G r admits a sub-group G m of exceptional Cri's, then G m is an invariant sub-group of G r . 31. INVARIANT SUB-GROUPS 119 Consequently, if the matrix \\CaZ\\ in which a indicates the columns and 6 and e the rows is of rank less than r, the group is not simple (cf. 35). Let V p be an invariant variety for an invariant sub-group G m of a G r and S any transformation of G m , and denote by V p the transform of V p by a transformation T of G r which is not of (? m , then But by (29.7) TST~ l is a transformation of G m , since the latter is an invariant sub-group of G r . Hence V p is an invariant variety of G m . If a sub-group of G m leaves V p point-wise invariant, the transform by T of the members of this sub-group leaves V p point- wise invariant. Hence the groups induced by G m in V p and V p are of the same order. If T is infinitesimal, V p is infinitely near V p , unless it coincides with it. The latter case arises, if V p is isolated in the sense that it is not one of a continuous array of invariant varieties in which the induced group is of the same order for all of these varieties. Hence we have: [31.1] // V p is an invariant variety for an invariant sub-group of a G r , the transform of V p by any transformation of G T , not of the sub-group, is an invariant variety of the sub-group and the group induced by the sub-group in the two varieties is of the same order; if V p is isolated, the transform coincides with V p . * If there are two invariant sub-groups G m and G m > of a (? r and there are p independent transformations common to the two sub- groups, these form an invariant sub-group of G r . For, the transform of any of the latter is contained in G m and Gw, and consequently is common to the two. Suppose that two invariant sub-groups G m and G m > of a given G r do not have a common sub-group, and we take as their symbols XJ, . . . , Xnjand X m +if, . . . , X m + m j. Then! (31.4) (X p , X u )f = CplXj = CplXJ /p, q = 1, , m; \ y, u = m + 1, , m + m'/' Since the operators of the two sub-groups are independent, we must have in this case * Lie-Scheffers, 1893, 1, pp. 529, 531. 120 ///. INVARIANT SUB-GROUPS (31.5) (^ = ^ = ff'*" 1 ' ' ' ' > m > , \ v ' \t, u = m + 1, , m + m '/' and consequently the transformations of the two sub-groups are commutative. It is evident that the symbols of the two groups generate a group (?m+m', which is G> itself when m + m' = r, and which is an invariant sub-group of G> when m + m' < r. In the latter case because of (31.5) the sub-groups G m and G m > are invari- ant sub-groups of (? m -f m'. We say that Gm+m> is the direct product of G m and G m '. Hence we have: [31.2J // a group G r has two invariant sub-groups not containing a common sub-group, the transformations of the two sub-groups are commutative, and the direct product of the two sub-groups is either G r itself, or an invariant sub-group of G r of which G m and G^ are inva- riant sub-groups. Suppose that G m and G m ' are two invariant sub-groups of a G> and they have a sub-group G p in common. In this case it follows that the second and third terms in (31.4) being respectively symbols of G m ' and G m must be a symbol of G> Consequently if the symbols are chosen, so that the symbols of G m and G m > are (31.6) G m : Xif, . . . , X p f, G m ''. Xif, . . . , X p f, Xm+lf) . . . , X m +m'pf, then (31.7) (X k , X u )f = c k h u X h f /k = 1, - , m; h = 1, - - , p; \ \u = 1, , p, m + 1, , m + m' - p) From this result and the fact that G m and G m f are sub-groups it follows that the m + m' p independent symbols in (31.6) gener- ate a group Gm+n'-p which is called the product of G m and GW; we use the term product in this case as distinguished from direct prod- uct when there is no common sub-group. In consequence of (31.7) Gm and G m > are invariant sub-groups of G m+m '-p and G p is an invari- ant sub-group of (?m, Gm', G m +m>-p and G>. Clearly m + m' p ^ r; if it is less than r, Gm+m'-p is an invariant sub-group of G>. Hence we have: [31.3] // two invariant sub-groups of G r , namely G m and G m ', have a sub-group G p in common, their product G m +m'-p is either G r or an 31. INVARIANT SUB-GROUPS 121 invariant sub-group of it; in the latter case G m , G m > and G p are invariant sub-groups of G m + m >- p and of G r . Suppose that we have two groups G m and G m , such that all the transformations of each group are invariant under all the trans- formations of the other but not commutative. Then we have equations of the form (31.4) in which the c's are constants (29), from which it follows that there are certain independent groups GI common to the two, say p. Combining the two sets of symbols we have m + m' p linearly independent ones (constant coefficients). Because of (31.4) and similar equations for each of the given groups it follows from the second fundamental theorem that these symbols generate a group G m +m'- P , and from these equations it follows that G m and G m > are invariant sub-groups of the enlarged group, as is also the sub-group of order p common to G m and G m >. Let T, Ti and T', 7\ be any two transformations each of G m and G m ' respectively; it follows that T'T is a transformation of the group (? m +m'-p. We have also that TiT' = TiTi, where T' 2 is another transformation of Cr m /; this is due to the hypothesis that the transformations of Gm> are invariant under those of G m . Con- sequently we have and thus the products T'T form a group T. As we take T' or T equal to the identity, these transformations are those of G m or G m >, and therefore F is not a sub-group of G m +m'- P , but is in fact this group, which we call the product of the two given groups. Hence we have: [31.4] // two groups Gm and G m > are invariant with respect to each other and are not commutative, they possess a common sub-group G p ; the two groups and also G p are invariant sub-groups of the group G m +m>- P generated by the independent (constant coefficients) symbols of the two groups; and each transformation of the enlarged group consists of the product of a transformation of G m and one of G m >.* By similar reasoning we have : [31.5] // two groups G m and G m > are commutative, their symbols generate a group of order m + m', the direct product of the two, of which each is an invariant sub-group. * Bianchi, 1918, 1, p. 204. 122 ///. INVARIANT SUB-GROUPS An invariant sub-group is said to be a maximum one, if it is not a sub-group of an invariant sub-group of higher order. We shall prove the theorem : [31.6] // G fi and G Pi are maximum invariant sub-groups of a G r and they have a sub-group G p of all common transformations, their product is G r , that is, n + pi p = r, and G p is a maximum invariant sub-group of G fl and G Pi .* The first part of this theorem follows from theorem [31.3]. For, if TI + pi p < r, the product of G fl and G Pi is an invariant sub- group of G r containing G Tl and G Pi , contrary to hypothesis. If G p is not a maximum invariant sub-group of G>,, there is an invariant sub-group Gp+k of it containing G> Consider the product of G p +k and G Pi . In consequence of (31.7) the commutator of any symbol of this product and a symbol of G Ti is a symbol of G p +k. The com- mutator of a symbol of this product and one of G Pi is a combination of symbols of G p and G Pi , that is, of G Pi itself. Hence the product is an invariant sub-group of G> and it contains (? Pi , contrary to the hypothesis that the latter is a maximum. Consequently G p is a maximum invariant sub-group of G ri and similarly of G PC If G>, and G Pi are maximum invariant sub-groups of G r and they have no sub-group in common, by the above reasoning it follows that ri + pi = r. Moreover, by theorem [31.2] it follows that the transformations of the two groups are commutative. Furthermore, each of these sub-groups is simple. For, if G>, contained an invari- ant sub-group Gk, then by reasoning similar to the above the symbols of Gk and G p generate an invariant sub-group of G> which contains G P9 contrary to hypothesis. Hence we have: [31.7] // G r , and G Pi are maximum invariant sub-groups of a G r and they do not contain a common sub-group, then ri + PI = r and G>, and G p are simple groups commutative with one another; consequently G r is the direct product of these sub-groups. Exercises 1. If a series of infinitesimal transformations u*X a f admit separately two Gi'a of symbols Yif and F 2 /, they admit the group Gi of symbol aYif + *Lie-Engel, 1893, 2, p. 706. 31. INVARIANT SUB-GROUPS 123 where a and b are any constants; also they admit the G\ whose symbol is (Yi, F 2 )/. Bianchi, 1918, 1, p. 197. 2. If a set of transformations x f< = f'(z; a) involving r essential parameters is such that a combination of any two, that is, T at T ai , involves only r essential parameters, then T a = SbT a9 , where S b are the transformations of a group G r and T a9 is a transformation of the set; moreover, G r is invariant under T a9 . Bianchi, 1918, 1, p. 118. 3. Show that, in consequence of theorems [24.1] and [30.5], a systatic group is imprimitive, and a primitive group is asystatic. Bianchi, 1918, 1, p. 190. 4. A Gz with the structure (Xi, X,)/ = XJ, (X lf X,)/ = 2X 2 /, (X 2 , X,)/ = XJ is simple (cf . 35) ; show also that the basis can be chosen so that (Xi, X 2 )/ - XJ, (X 2 , X,)/ - Xif, (X,, XO/ = XJ. 5. If a G> admits a non-invariant sub-group G>- 1, the basis can be chosen so that XJ, . . . , Xr-if are the symbols of G r -\ and ex., x r )f = c h r x>/, (x r _i, x r )/ = cr^jxrf + x f / Then show by means of the 'Jacobi identity (2.4) that Xi/, . . . , X r ^j/ are the symbols of a G r ~z, which is an invariant sub-group of G>_i. Lie-Scheffers, 1893, 1, p. 544. 6. If a G r contains a simple G>_i, the latter is an invariant sub-group of G r (cf. Ex. 5). Lie-Scheffers, 1893, 1, p. 572. 7. If a Gt has an invariant sub-group (TS with the structure (Xi, X 2 )/ - X,/, (Xi, X 8 )/ - 2X2/, (X 2 , X,)/ = X^, then (Xi, X 4 )/ = ^X// (i, j = 1, 2, 3). Because of the Jacobi identity the a's satisfy the conditions 2aJ = aj, aj - -a}, a? = 2aJ, a} - aj = aj = 0. If XJ is replaced by X'j - aJXi/ + a\X*J + aJX^, it follows that (X^ X' 4 )/ = (i - 1, 2, 3). 8. Establish theorem [29.6] by means of the Jacobi identity. 9. Show that the protective group (? with the symbols = p + 2/r, X,/ = sp + er yq + zr) - zg, XJ = q + xr (r ^^ \ ^ 2 leaves invariant the quadric z xy = 0, and that the induced group T on the quadric has the symbols p, xp, x*p, q, yq, y*q. Lie-Engel, 1893, 2, p. 199. 124 ///. INVARIANT SUB-GROUPS 10. Show for the induced group Y* in Ex. 9 the two sub-groups with the symbols p, xp, x*p and q, yq, y*q are invariant sub-groups of F; derive therefrom the two invariant sub-groups of G*. 32. Isomorphisms of groups. Let G> and H p be two continuous groups whose finite equations are (32.1) *'< = /'"Or; a), y" = h'(y, b) If p = r and ^"(a 1 , , a r ) for a = 1, , r are r functionally independent functions, the equations (32.2) b a =

, being any two pairs of corresponding transformations, the trans- formations Ta t T ai and /St,^ correspond. Later we shall have occasion to consider the case when p < r in which case the corre- spondence is said to be multiply isomorphic, and when p = r simply isomorphic. When the isomorphism is simple, if T ai is the identity, then T at corresponds to Sb t and also to $& 2 S &1 , and consequently &, is the identity, since the correspondence is one-to-one. Hence if T ai and S^ correspond so also do TV, 1 and Sij 1 . If the quantities b in (32.1) are replaced by their expressions (32.2), we obtain a new set of equations for H r in terms of the a's, say (32.3) ^ = to(y; a), and then corresponding transformations have the same values of the a's. If the correspondence is to be isomorphic, equations (4.6), namely a? = ^ a (aij 02), must apply to both groups and conversely. Consequently the two groups must have the same first and second parameter-groups. Since this condition is evi- dently sufficient, we have : [32.1] A necessary and sufficient condition that two groups be simply isomorphic is that they have the same parameter-groups. If this condition is satisfied it follows from (14.19) that the vectors Ua(u) are the same for the two groups, and then from the corresponding equations (6.3) that the constants of structure of 32. ISOMORPHISMS OF GROUPS 125 the two groups are the same. Conversely, when the latter condition is satisfied, it follows from (14.6) and (14.8) that the two groups have the same vectors U(u) and f7(tt), and from (14.18) and (14.17) that equations (14.16) are the same for both groups. Hence we have: [32.2] A necessary and sufficient condition that two groups be simply isomorphic is that their bases can be chosen so that the constants of structure of the two groups are the same for both. Thus from the results of 9 it follows that the parameter-groups of any group are simply isomorphic with it. Also from 26, 27 it follows that any extension of a group is isomorphic with it. We consider next the case of multiple isomorphism, that is, when p < r, and we understand that in (32.2) where now a = 1, ^ , p the jacobian matrix is of rank p; in accordance with the theory da\ of implicit functions this insures that there are values of the a's in (32.2) for any values of the 6's. Then there are oo r -p transfor- mations T of G r corresponding to a transformation S of H p . In particular, to the identity of H p corresponds a sub-group G r - P of G>. Since to any T there corresponds only one S, it follows by the argu- ment applied to the case p = r that the identity of G> is a member of this sub-group, and that if T ai corresponds to &, then 2T/ corre- sponds to Sb*. Since then T^T av T^ corresponds to S^fi^ST, 1 , if S bl is the identity, then T a ,T a ^ corresponds to the identity whatever be TV Since this is true for every T ai corresponding to the identity, we have : [32.3] // G r and H p (p < r) are two groups multiply isomorphic t the transformations of G r corresponding to the identity in H p form an invariant sub-group of G r of order r p. If we substitute from (32.2) in the second of (32.1) and obtain (32.3), these are finite equations of H p but all the parameters are not essential. If we take two transformations S^ and & and a pair T ai and T a , corresponding to them and apply (4.6) to get T at T 0l , then when (4.6) is applied to the transformations S in the form (32.3), we get the transformation <&,&,. In this sense the two groups have the same constants of structure, for suitable choice of the bases, following the procedure in the case of simple isomorphism. This means that, if we consider the infinitesimal 126 ///. INVARIANT SUB-GROUPS generators of the two croups and denote their respective bases by Xif, . . . , X r f and FI/, . . . , Y p f, we can put (32.4) where the a's are constants so chosen that (32.5) (F a , Y b )f = c a lY f f (a, 6, e = 1, - - , r), the constants c a e b being given by (32.6) (X a , X b )f = CatXJ. If now we put (32.7) (FI, FJ)/ = c%Ytf (a, ft 7 = 1, ' ' , p), it follows from (32.4) and (32.5) that (32.8) cCafctf = c.ta ("' ^ 6 = \> A 7 = 1, Conversely, if c & and c2 are the constants of structure of two groups G r and H p respectively, and equations (32.8) are consistent, then from (32.4) we have (32.5). If we change the basis of G r and put as in 7 / ^ then (32.8) becomes where fij t = aM,. Hence we have: [32.4] !// G r and H p are two groups with the respective constants of composition c a l and c^, a necessary and sufficient condition that they admit multiple isomorphism is that equations (32.8) be compatible. Thus the problem of determining whether two groups are isomorphic is an algebraic one. Suppose that the bases for H p and G> are chosen so that , , Y p f in (32.5) are independent, and that * = P + !> ' v f - tf v f * = > ' ' ' > YJ - b a Y t f ^ = ^ . . . ^ p where the 6's are constants. Then to each of the r p independent infinitesimal transformations 33. SERIES OF COMPOSITION OF A GROUP 127 (32.10) XJ - of G r corresponds the identity in H p . Evidently these transforma- tions determine a sub-group G r - p of G>. Conversely, if X a are con- stants such that Yf s xy a / = o, then V + X a &l = 0, and Xf = \X a f = \ a (X a - b' a X t )fis the sym- bol of a (?i of transformations of G>_ p . Since (F<, (F a - 6J.F,))/ = 0, we have = by means of which it can be shown that G r - P is an invariant sub- group, as previously established from the finite equations of the group. This result is a special case of the theorem: [32.5] // G r and H p (p ^ r) are isomorphic, to a sub-group of order m of G r corresponds a sub-group of order k( < m) of H p ; to a sub-group of order m of H p corresponds a sub-group of order m + r p of G r ; if either sub-group is invariant, so is the other. This theorem follows immediately from the definition of isomorphic correspondence at the beginning of this section. Also if infinitesimal transformations are used, we have at once that, if X^X\/ for , then X*F, and so on, the series (33.1) G r , (? ri , . . . , Or., 1 (where necessarily G fg is a simple group) is called a series of com- position of the group, and the differences (33.2) r - n, , r<_i - r q , r q 128 ///. INVARIANT SUB-GROUPS are called the indices of composition of the group; by some writers (33.1) is called a normal series. It may be that a group G> admits more than one series of composition. For example, if a group admits two maximum invariant sub-groups G>,, G p which do not contain a common sub-group, then as shown in 31 these sub-groups are simple, and we have the two series of composition Or, , and G Pi have a common sub-group T p , which by theorem [31.6] is a maximum invariant sub-group of both G ? r 1 and G p . We take a series of composition for T P) say TPI TPU Tp 2 > . . . , r p ., 1, and consider the series (33.1) and (33.4) G n G ri , r p , r Pll . . . , r p- , i. In 31 it was seen that r = ri + pi p. Assuming the theorem to hold for the two series for G ri , we have that (33.2) and r ri, ri p, p pi, , p,-i - p 8 , p* are the same numbers, possibly in different orders. Similarly the indices of (33.3) are the same numbers as r - PI, PI - p, p - pi, , Ps-i - p,, p. But r - ri = pi - p, r - pi = ri - p, and consequently the theorem is proved for G>. 34. FACTOR GROUPS 129 34. Factor groups. Suppose that a group G r admits an invariant sub-group G m and that the basis of G> is chosen so that X t f for t = 1, , m is the basis of the sub-group; then we have (34.1) d = (a = 1, , r; t = 1, , m; p = m + 1, , r). Consider the Jacobi relations c V 4. c V + c V* - o A>> ?> M = + 1, * ' ' , r \ kpgOta T^ t'flf' t p T^ CapC/tfl U I . ^ I V i, , r / In consequence of the preceding equations these relations become (34.2) cpfol + c^J + c.XJ = (p, ?, M w = w + 1, , r). By the third fundamental theorem there exists a group G r _ m whose constants of composition are c p u q . All such groups being simply isomorphic, we call anyone of them the factor group of G r determined by the invariant sub-group G m , and write symbolically a - = ' If we denote its basis by Y p f for p = m + 1, , r and adjoin the operators Yaf = for a = 1, , m (cf. 32), we have that G> and Gr- m are multiply isomorphic, and G m corresponds to the identity in G>_ m . Hence we have: [34.1] // a group G r admits an invariant sub-group G m , there exists a group G r -m multiply isomorphic to G>, and G m corresponds to the identity in Gr-m. Conversely, if a group Gr-m is multiply isomorphic to a group G r , the former is the factor group of G> determined by the invariant sub-group of G r corresponding to the identity in G r - m . If G 9 is a maximum invariant sub-group of Gr-m, then the corresponding invariant sub-group G 8+m of G r is maximum, and conversely. Consequently if G r -m is simple, then G m which deter- mines it as a factor group of G r is a maximum invariant sub-group, and conversely. Hence we have: [34.2] A necessary and sufficient condition that a factor group -- ur m be simple is that G m be a maximum invariant sub-group of G r . 130 ///. INVARIANT SUB-GROUPS If Gr-m is not simple, it admits a series of composition (33), say (rr m> Orr_mn . . , G>m tt , 1, each of the sub-groups being a maximum invariant sub-group of its predecessor; then the corresponding sub-groups of G>, namely G>_ mi + m , . . . , G r -m q +m, (?m, 1, form a series of composition. As a consequence we have: [34.3] Any invariant sub-group G m of a G r is a member of a series of composition of G>. Suppose that G> admits a series of composition (34.3) (? r , G ri , . . . , admits a second series of composition (34.5) G r , so that X a f for a = 1, , m is the basis of r m ; X a / and Xhf for A = m + 1, , ri the basis for (j ri ; and X/ and Zj/ for J = ri + 1, , r the basis for Pl . Then Z a /, Xft/ and Xi/ is the basis for G>. From the considerations at the beginning of this section and in particular equations (34.2), it C 1 C 1 follows that the factor groups - and have the constants of structure of G> for which the indices take the respective values TI + 1, , r and m + 1, , n, and that the factor groups 35. DERIVED GROUPS 131 G Pl G r and have these respective constants of structure. Hence F m (J PI we have the theorem : [34.4] // G>, and G Pi are maximum invariant sub-groups of G r with a G r G Pi common sub-group T m9 the factor groups and - are simply isomor- urri im , . i G r , Gr, phic, as are also - and - /nr In particular, if F m is the identity, G Pi is the factor group -~ On so that the factor groups of the series G>, G Tl , 1 are G Pi and G f| ; in like manner the factor groups of the series G r , G Pi , 1 are G fl and G Pi . We are now in position to establish the following theorem of Holder-Engel:* [34.5] // a group G r admits two series of composition, the simpk factor groups of the two series, to within their ordering, are simply isomorphic. We assume that this holds for order less than r and show that it holds for r. If T m admits the series T m , T m ,, . . . , 1, then the theorem holds for the series (34.3) and (34.6) G r , G ri , T m , T mi , . . . , 1 since by hypothesis it holds for G>,. Similarly the theorem holds for (34.5) and (34.7) G T) G, |f T m , T mi , . . . , 1. Then it holds for (34.3) and (34.5), because it holds for (34.6) and (34.7) in consequence of the above discussion. When r = 2, we have the particular case discussed above, and hence the theorem is proved by induction. 35. Derived groups. Consider the symbols X a i>f defined by (35.1) X ab f = (Xa, X b )f = CakXJ. Each of them determines a sub-group Gi of G>. Moreover, since (Xdb, Xcd)f = (CabX e , C c dXh)f = CabCcdXehf, *Lie-Engel, 1893, 2, p. 765. 132 ///. INVARIANT SUB-GROUPS we have that these symbols determine a group, which is either G> itself, or a sub-group of G r . Since also (Xd, X a b)f = C a b(Xd) Xe)f = we have that, if this group is a sub-group, it is an invariant sub- group. It is called the derived group of G>. Consider the matrix (35.2) C = ||c ffl Z||, where e indicates the columns, and a and 6 the rows. If the rank of the matrix is p, then any one of the symbols X ab f is expressible linearly (constant coefficients) in terms of p of them and the derived group is of order p. If the rank is zero, that is, if the group is Abelian, the derived group consists of the identity. If the rank is r, the derived group coincides with the given group. As a corollary we have: [35.1] // G r is a simple group, the rank of C is r; if the rank is less than r, G r is not simple. Suppose that the rank of C is r\ < r, then as seen above the derived group G fl is an invariant sub-group. If we denote by Xrf, . . . , X r J the symbols of this group, these and r r\ others X ri +if, . . . , X r f may be taken as the symbols of G>. From the definition of 6>, it follows that b ' ' ' ' ' (35.3) (X a , Xtf = _ . . \P Aj > / 1 If in these equations we let a and b take the values 1, . . . , n, r\ + 1, , ri + h (h < r n), we see that the symbols with these indices form a sub-group (?,-,+&, and then from (35.3), when a takes the values 1, . . . , r and b from 1 to r\ + h, it follows that G fl +h is an invariant sub-group of G>. Moreover, if a takes the values 1 to r\ + h and b the values 1 to r\ + h 1, we have from (35.3) that G>,+A-I is an invariant sub-group of G r ,^. Hence as h takes the values r r\ 1, r r\ 2, , 2, 1, we have a set of invariant sub-groups of G> (35.4) Or-l, Gr-2, . . . , a,+l, Gr lt each of which is an invariant sub-group of those of higher order. Hence if G' fl is a simple group, then G r and the set (35.4) form a series of composition of indices 1, . . . , 1, r\. 35. DERIVED GROUPS 133 As defined G>_i has the symbols Xtf, . . . , X r -if, but if any of the symbols X fl +if, . . . , X r f other than X r f had been omitted, we should have obtained another invariant sub-group of order r 1. Hence there are r r\ such sub-groups. Each of these in turn admits r r\ 1 invariant sub-groups of order r 2, and so on. Hence we have: [35.2] // the derived group of a G r is of order r\ < r t then G r admits r r\ invariant sub-groups of order r 1, each of the latter r r\ 1 invariant sub-groups of order r 2, and so on; and all these sub- groups contain the derived group. Conversely, we have:* [35.3] // a group G r admits an invariant sub-group of order r 1, the derived group of G r is this sub-group or is a sub-group of it; and consequently its order is less than r. For, if Xi/, . . . , X r _i/are the symbols of the sub-group G r -\, then and consequently the derived group either coincides with G r -\ or is contained in it. From theorem [35.3] it follows that, if the rank of the matrix C (35.2) is r, the group does not admit an invariant sub-group of order r 1. In particular, if r = 3 and the rank of C is 3, the group is simple. For, if it had an invariant sub-group, it would be of order one; assume Xif to be its symbol, then c a ? = for a = 1, 2, 3 and a = 2, 3, but in this case the rank of C is less than 3. From the definition of a derived group it is evident that : [35.4] The derived group of a sub-group G m of a G r is contained in the derived group of G r . If G m is an invariant sub-group of G r and the basis of the latter is chosen so that X a f for a = 1, , m is the basis of the sub- group, in the identities ((X a , X,,}, X.) + ((X fl XJ, X a ) + ((X a , XJ, Xj) = , ft = 1, , m; * = 1, ' ' ' , r a , / V ^ * Bianchi, 1918, 1, p. 212. 134 ///. INVARIANT SUB-GROUPS the second and third terms are symbols of transformations of the derived group of G mj and consequently it follows from these identities that the derived group of G m is an invariant sub-group of G>. The same reasoning may be applied to this derived group and we have : [35.5] The derived groups of an invariant sub-group of a group are invariant sub-groups of the group. If G ri in (35.4) is not a simple group, its derived group, called the second derived group of G r , may be of order less than n, say r 2 . Then as above we show the existence of sub-groups G fl _i, . . . , G rt , such that each of them is an invariant sub-group of the one of one higher order, and all of them are invariant sub-groups of G Tl . This process may be continued as long as each derived group is of lower order than the one preceding it. Consequently we get a sequence such as (35.5) or (35.6) G r , GU, . . . , Or., 1, the latter case arising when the (s + l)th derived group is the same as the sth derived group. In either case each group GK of the sequence for h = 1, , r 1 is an invariant sub-group of the group G*+i. 36. Integrable groups. A group is said to be integrable when there exists an array of sub-groups as (35.5) such that each sub- group Gh for h = 1, , r 1 is an invariant sub-group of the group Gh+i', the significance of the term integrable will appear later. If we denote by Xif the generator of Gi, by X\f and Xrf the generators of G 2 and so on, we have (X 3 , Xi)f = and in general (36.1) (X h + k , X h )f (h 1, ,r- l;t = 1, ,r- A;\ \l - 1, , h + k - 1 ) Hence each c a l for which I is equal to or greater than the larger of a and b is equal to zero. 36. INTEGRABLE GROUPS 135 From the method of 35 yielding (35.5) it follows that, if one of the successive derived groups is the identity, then the given G r is integrable. In particular, the first derived group of an Abelian group is the identity. Hence we have: [36.1] An Abelian group is integrable. Also from theorem [16.1] we have: [36.2] Any (? 2 is integrable. Conversely, if a group G> is integrable, its first derived group is contained in G r -i of a sequence (35.5) by theorem [35.3]. And by theorem [35.4] the second derived group is contained in the derived group of Gr-i, and consequently is contained in G>-2. Similarly the third derived group is in G>_ 3 , and so on. Consequently we have: [36.3] A necessary and sufficient condition that a G r be integrable is that there be an integer p (< r) such that the pth derived group be the identity. If G r is integrable and G p is a sub-group of (? r , then the first derived group of G p is contained in the first derived group of G r by theorem [35.4]. If these two derived groups are equivalent, then G p is integrable; if not, then by the same theorem the second derived group of G p is contained in the second of G>, and so on. Hence in consequence of the preceding theorem we have: [36.4] Any sub-group of an integrabk group is integrabk. The significance of the designation integrabk as applied to a group is set forth in the theorem: [36.5] // a linear homogeneous partial differential equation of the first order in n variables Af = admits an integrabk group G n -i for which the symbols and Af are independent, the integration of the equation reduces to quadratures. Let Xrf, Xrf, ... i X n -.if be the symbols of the group, so chosen that the first 6 symbols for any 6 from 1 to n 2 generate a sub- group which is invariant under the sub-group of the first 6 + 1 symbols. If Af = is the equation, then (36.2) Af = 0, Xif = 0, - , X-if = 136 ///. INVARIANT SUB-GROUPS form a complete system, and in consequence of the above statement this system admits the operator X n ~if. Accordingly, if u is a solution of (36.2) other than a constant, so also is X n -\u a solu- tion by theorem [25.1]. If X n -\u were zero, the system of equa- tions (36.2) and X n -if = would not be independent. Hence Xn-iM = ^>(M), since any two solutions of (36.2) are functions of J, -, we have that MI is a solution be con- tained in G m . Bianchi, 1918, 1, p. 289. 5. The first derived group is the invariant sub-group of lowest order deter- mining an Abelian factor group. Bianchi, 1918, 1, p. 289. 6. Show that the G* with the symbols Gti p, q, xq, x*q is integrable and that the sub-groups G 8 : q, xq, ap + bx*q\ G^: q, xq\ G\\ q are invariant sub-groups of G* and each one is an invariant sub-group of every one of higher order. Lie-Scheffers, 1893, 1, p. 538. 7. If p is a non-zero root of the determinant equation |c Jw p5J| =0, the quantities t> defined by (c a jw p8J)t; fr = are such that uXf and tPXbf determine a sub-group Gt of G>, and v*>Xbf is the symbol of the derived group of Gi (cf. 16). 8. If u a Xaf and v a X, its derived group has the symbol u a Xaf if c Jwt> 6 = u', and this is possible only if the matrix ||c Jw|| and the augumented matrix ||c a jw, u e \\ are of the same rank (cf. 16). 9. The basis of a Gt can be chosen so as to have one of the following structures : (1) (Xi, XM = XJ, (X lt X,)f = 2.XV, (X* XM = XJ; (2) (Xi, XM = 0, (X,, XM = X, f, (X,, Xia = '**f (c* 0,1)1 (3) (X,, X,)f = 0, (Xi, X,)f = XJ, (X,, X,)/ = XJ; (4) (X,, X,)/ = 0, (X 1( X,)/ = XJ, (X s , X,)/ = X,/ + XJ; (6) (Xi, X,)/ = 0, (X,, X,)/ - X ,/, (X 2> X,)/ = 0; (6) (Xi, X,)/ = 0, (X 1( X,)/ = 0, (X,, X 8 )/ = XJ; (7) (X,, X t )f = 0, (X,, X,)/ - 0, (X S) X,)/ = 0. All of these groups are integrable except the first. Lie-Scheffers, 1893, 1, p. 571. 138 ///. INVARIANT SUB-GROUPS 10. The basis of any non-integrable G s can be chosen so as to have the struc- ture (1) of Ex. 9; show that this G* is isomorphic to the protective group on the line (Ex. 8, p. 66). 11. In 4-space of coordinates x i the linear homogeneous transformations x fi = a)V leave invariant the hypersurface Sz 2 = > where K is a constant, if /C a\a{ = /* and thus form a G f . There are two sub-groups, each of the third order of this group for which x ri x* = where a is a constant. The values for K the a's in these cases are (i) a\ - a (i not summed), a\ = a{ (i, j = 1, , 4) and either (ii) oj = aj s ft a? = aj = 7, a| = a{ = 5, or (iii) aj = aj = ft a\ = aj = T, a* = i = 5, where in each case (iv) a a +0* +7 2 + 6 2 = 1. Bianchi, 1902, 1, p. 445. 12. Show that each of the two sub-groups of G* of Ex. 11 are invariant sub- groups of (?6, and the only ones. 13. If the symbols of the (? in Ex. 11 are denoted by (cf. Ex. 12, p. 57) the symbols of the two invariant sub-groups of G ? are 14. If we put x a = -^-=r, x' = -4= IT-, r = VV* ( - 1, 2, 3), 4 4 these equations define the hypersurface 2x* 2 = ~. The symbols of the K induced group r on this hypersurface (cf . 20) of Gt of Ex. 11 are given by X.J = IW = JT^ - V 4 ^, XJ = KJ (a, 6 = 1,2, 3), , (j f _i, . . . , GI, 1 such that for any k < r the sub-group Gk is an invariant sub- group of (?*+!. We choose the basis of G> so that Xif is the generator of Git Xif an d X, say E\. Then we project E n with respect to E\ and obtain an J? n -i- We can then carry on the same process for E n -\ as was done for E n and obtain in it a direction invariant for G>. Hence the J? 2 in E n represented by this direction is invariant with respect to G>. Then project E n with respect to 1? 2 , and apply the process to the resulting E n -2, and so on. Accordingly we have the theorem of Lie:* [37.5] // a group G r of linear homogeneous transformations is integrable t there is at least one sequence of invariant vector-spaces Ei t . . . , E n -i such that any Ek is contained in Ek+i. By choosing the coordinates as was done in the case of a one- parameter group, we have : [37.6] When a linear homogeneous group is integrable, the basis can be chosen so that (37.22) o} = (a = 1, - , r; \ V = 1, , n - l\t = j + 1, - ,n/ 38. Canonical form of linear homogeneous transformations. Fundamental vectors. The reduction by means of homogeneous linear transformations of the transformation (37.1) to the form for which (37.19) obtain is only a partial step in the reduction which can be made. Equation (37.12) sets forth the transformation of "1893, 1 p., 533. 146 IV. THE ADJOINT GROUP the matrix A under a linear homogeneous transformation. Hence we may take over the known results of the reduction of such forms. The reduction to normal form of a matrix A is based upon the elementary divisors of the matrix A (37.4), which are defined as follows: Let F+ be the highest common factor of the i-rowed minors of A and Z- the highest power of p p\ in F, where p\ is a root of (37.4). As i takes the values 1, . . . , n, we get numbers vi = li li-\ which are equal to or greater than zero; then for such values of vi which are not equal to zero the expressions (p p\) vi are the elementary divisors of A for the root pi. The fundamental theorem is as follows : If the elementary divisors of A are (38.1) (p - Pl )'', (p - p 2 )', - , (p - p.)"*, (the p a 's in these factors are not necessarily different, nor are the /s), there exists a real homogeneous transformation of the variables such that A assumes the form (38.2) A = M k where Af , represents the matrix of order Pi p 1 - P, - P 1 1 0. 0^ Pt p 1 0. Pt p and all the terms of A not in any Mi are zero.* We have remarked that several of the p 's in (38.1) may be the same. If we gather into one principal minor all terms involving the same p, and denote by v a the order of the root p a , we see that the minor corresponding to the root p a is of the form *B6cher, 1907, 1, pp. 270, 271, 288, 289. (38.3) M a = 38. CANONICAL FORM 147 Pa p e a i p a P 6 Pa ~ p 6 [,.-i] p a - where any e is one or zero, as the case may be, and that all the other terms in ||a)|| in the same rows or columns as terms of M a are zero. We denote by Xi a vector for the root pi satisfying (37.3), that is, (38.4) Axi = piXi. For the above normal form of A it follows that in this coordinate system one solution is (38.5) x ( = {, and also that = PI, 1, PI] (61 + 1)). Hence each of the vectors (38.6) xi, = < satisfies the corresponding equation (38.7) Ax^ = piXbi + eijbj-ijXbi- In particular, we have (ay - 5Jpi)X2 = 611X1, and consequently (a* - tpO(o/ - Jpi) = cu(o* ~ or, in matrix notation (A - 7pi) 2 Z2 = 0. And in general we have (38.8) (A - Ipi) bl x bl = (6, = 1, , vj. We say that the vectors (38.6) determine a vector-space E Pl of dimension v\ associated with the root pi, called the root-space 0, 148 IV. THE ADJOINT GROUP for the root pi. Any vector of this space being a linear combination (constant coefficients) of the vectors (38.6), it follows that any vector of the space E Pi , say xi, satisfies the condition (A - 7 Pl )% = 0, where b depends on x\ and is ^ v\. We shall refer to the vectors (38.6) as the fundamental vectors of E Pi corresponding to the first, . . . , ^th rows of the matrix Af i, and any two in the set for consecutive values of 61 we call consecutive. In like manner the vectors (38.9) xl t = 8$, (6 2 = vi + 1, , PI + v,) determine a space E Pt corresponding to the root p 2 . We have (38.10) Ax b , = P2 :c6 2 + 6 2(C2 _ 1] x 6a _ 1 , (A - I Pz ) Ct x bt = (c 2 = 6 2 v\), (A - 7p 2 ) c z 2 = (c^pi). In this way we proceed with the other roots p 3 , . . . , and we have : [38.1] // x a is any vector of the root-space for p a , then (38.11) (A - I Pa ) c x a = (c$ O- If we write equation (37.4) in the form ?(p) = (P - Pi)" 1 (p - PpY' = 0, there exist functions \f/ a (p) such that (P - Pa)"' a from which we obtain where /a(p) = ^a(p)n Ma (p - p 6 )"* (a, 6 = 1, , p). Hence we have the matrix equation (38.12) 7 - 2/ (4), f a (A) = 38. CANONICAL FORM 149 If x is a vector of the space, we have (38.13) x = xi + + x,, where x a is a certain vector of the root-space for p a . In consequence of the above theorem and the definition of f a (A) we have f a (A)x b = (6 ^ a), and then from the first of (38.12) (38.14) Xb = Ix b = f b (A)x b . Hence we have fa(A)x = fa(A)x a = X a . Now if (A p}I) h x = for h ^ *>i, then (A - Pl iy>x = 0. Hence if we write x in the form (38.13), we have for a 7^ 1 f a (A)x = 0. As a takes the values 2, . . . , p, it follows from (38.14) that X2 = = x p = 0. Consequently we have : [38.2] // x is any non-zero vector such that (A - Ip a ) c x = (c ^ rj, then x is a vector of the root-space of p a . Suppose now that a is not a root and that (A - I*Yx = 0. Since cr is not a root, there exist polynomials M (p) and N(p) such that Since this is an integral identity in one variable and is a direct consequence of the rules of addition and multiplication, it will hold when a matrix is substituted for p, so that we have M(A)(A - 7tr) c 150 IV. THE ADJOINT GROUP Since , we have is invariant under transformations of any G\ of the group. If we denote by \*X a f, where the X's are constants, the symbol of a particular Gi, and put (39.1) v a = X*r, then the v's are canonical parameters of this G\. If u a are canoni- cal parameters of a given G\ and tZ those for the Gi into which it is transformed by the above Gi, it follows from (29.4) that (39.2) U* = cl(v)u\ where the er's are functions of the t;'s, such that (o) T =o = 5?. These are equations of the transformations of the u's into the w's, involving r parameters t>. We shall show that they form a group. In fact, the equations ::'i. :!'/',> to (29.7) are (39.3) T V T U T; 1 = TV If we put similarly T V T^ = 3T Bl> then T Zi = T*T v TT; l T;t = T v T u T^ t \ where (39.4) v\ = ^(f^i), these being the equations in canonical parameters analogous to (4.6) in general parameters. Hence we have: 39. ADJOINT GROUP OF A GIVEN GROUP 151 [39.1] Equations (39.2) define a group and the equations connecting the parameters are the same as for the given group G r . This group T is called the adjoint of the given group. It is not necessarily of order r, as will be shown later; that is, all of the parameters v a are not necessarily essential. From the equations m m m~l /rr_ m m fp~^ 7^_ * v-L u\J- v *~~ * UD J- v * HI-*- v * t* 2 ) we have rrt^ '7 T _ T 7 nn ni ^ r P T* T 7 ^ T 7 IP T' 'T 7 ^ * 5 2 -t t*i J- v-* u 2 -* v -L v-L m* v -* t>^ M 2 -* u\l v Hence each transformation of the adjoint group establishes an automorphism of the given group, that is, an isomorphism of the group to itself in the sense of 32, and consequently we have : [39.2] The constants of structure of a group are invariant t when the canonical parameters undergo a transformation of the adjoint group. If in (29.14) we replace the symbol Xf by \X a f, we have in consequence of (7.2) that gl = X e c&e, and consequently in place of equations (29.20), we have dii a (39.5) ~ = u\*c b . dr When the expressions (39.2) are substituted in (39.5), we obtain (39.6) ^M _ ^ x . Cd -. dr Comparing this result with (14.12), we see that the functions ffl are those given by (14.13). Furthermore, these functions . In order that the adjoint group be of order r, it is necessary and sufficient that there do not exist any relations of the form (39.11) 0cSo = 0, where the g's are constants, as follows from (39.8). If we denote by C the matrix of r columns and r 2 rows (39.12) C - ||c||, the indices b and a indicating rows and a the columns, we have that if the matrix C is of rank s(^ r), equations (39.11) admit r s independent sets of solutions, and the order of the adjoint group is s. In this case there are r s linearly independent (constant coefficients) exceptional sub-groups G\ (29). From (39.8) and (7.3) it follows that u a tf = 0, and consequently the generic rank of the matrix ||T7a|| is less than r. Hence we have: [39.4] The adjoint group is always intransitive. In 16 it was seen that if XJ, . . . , Xmf generate a sub-group G m of G>, then c* u = for t, u = 1, , m\ z = m + 1, , r. Then from (39.10) we have: 39. ADJOINT GROUP OF A GIVEN GROUP 153 [39.5]. // Xrf, . . . , X m f generate a sub-group G m of G r , then Eif, . . . , EnJ generate a sub-group of the adjoint group, and conversely. Furthermore, if this G m is an invariant sub-group of G>, then Ca* = for a = 1, , r; t = 1, , m; * = m + 1, , r, and we have: [39.6] // Xif, . . . , X m f generate an invariant sub-group of G r , then Eif, . . . , Emf generate an invariant sub-group of the adjoint group, and conversely. Herein lies the significance and importance of the adjoint group, in that the determination of sub-groups and invariant sub- groups of a given group reduces to such determinations for the adjoint group, which is linear homogeneous in the canonical parameters. To a sub-group GI of G> with the generator v a X a f corresponds the sub-group >0i of the adjoint group with the generator v*Eaf. As in 37 the quantities v a are the components of a vector in a vector- space E r * such that each sub-space E\ represents a G\ of G> and also a #1 of the adjoint group, and each transformation in G\ and &\ is represented by a vector in E\. To each sub-group G m of G> there corresponds a sub-group of the adjoint group represented by the vectors of a sub-space E m . When the vectors of this E m are sub- jected to any transformation of the adjoint group, the E m goes into a new E' m unless the transformation is represented by a vector of E n , as follows from the first of the above theorems, or unless the sub-group G m is invariant under G>. Hence we have : [39.7] A necessary and sufficient condition that a sub-space E m of E r represent a sub-group G m of G r is that it be an invariant variety of all the transformations of the adjoint group represented by vectors of E m . A necessary and sufficient condition that G m be an invariant sub-group is that E m be an invariant variety of the adjoint group. If ui and u 2 are any two vectors of E r , then (ulX a , ulX b )f u{*X e f (a, b, e 1, , r), where Uu = c a iu*ul. * E r , Ei and E m are not to be confused with the symbols E a f of the adjoint group. 154 7F. THE ADJOINT GROUP Hence M J2 , denoted by (MI, 1*2), is a vector of the derived group of G r . Also if u\ and M 2 are vectors of a sub-group of G>, then un is a vector of this sub-group. Furthermore, if HI is any vector of an invariant sub-group of G r and M 2 is any vector of G r , then u\i is a vector of the sub-group. In consequence of the Jacobi relations (7.4) for any three vectors MI, M2, MS we have ((MI, ^2), M 3 ) + ((M 2 , M 3 ), MI) + ((MS, MI), M 2 ) = 0. As an application of this identity we consider the case when MI and M 2 are vectors of an invariant sub-group G m and MS is any vector of G>. Since the sub-group is invariant (M 2 , M 3 ) and (MS, MI) are vectors of G m , and consequently the second and third terms of the above identity are vectors of the derived group of (7 m , and hence their sum is a vector of the derived group. Consequently the first term, which is the transform of a vector in the derived group of (? m by a general transformation of G>, is a vector of the derived group, and consequently the latter is an invariant sub-group of G>. Since the second derived group of G m bears to its first derived sub-group, if it is not equal to it, the same relation that the latter bears to G m > it is an invariant sub-group of G>, and so on. Hence we have: [39.8] All of the derived groups of an invariant Sub-group of a G r are invariant sub-groups ofG r .* In particular we have: [39.9] All of the derived groups of a G r are invariant sub-groups of G r . Exercises 1. If in the general linear homogeneous operator a*x f pi non-homogeneous x a coordinates y = (or = 1, , n 1) are introduced, it becomes x n + aX ~ . Lie-Scheffers, 1893, 1, p. 564. 7. If ti 1 , . . . , u r are considered to be the homogeneous coordinates of a point in a flat space R r -i, each infinitesimal transformation is represented by a point in R r -i, and the operator (u a X a , iPXb)/ is represented by the point c a lu a v b . What does theorem [39.7] become under this interpretation? 8. Show with the aid of Ex. 11, p. 44 that, if the GI with the symbol u a Xaf is transformed by the transformations of the GI with the symbol iflXrf, the point of coordinates u a in R r -\ describes a curve whose tangent at the point u a is the line joining the latter to the point c Jwt^. Lie-Scheffers, 1893, 1, p. 471. 9. For the group G* of Ex. 7, p. 123 the invariant sub-group is represented in #3 by the plane u* = 0, and the adjoint group of G* leaves a conic invariant by Ex. 3 and theorem [31.1]. Show that X 4 /can be chosen so that E 4 f leaves the plane u* = point-wise invariant and deduce therefrom the result of Ex. 7, p. 123. Lie-Scheffers, 1893, 1, p. 573. 10. In the #2 of a Gs the equations w* = c a jwt^ define a correlation between the point w a and the line joining u a and v a , whose Pliicker coordinates are pab u atf> _ u b v a This correlation is singular unless G* is simple. 11. If a sub-group G* of a G* is not invariant, its representative plane Ri is transformed by the adjoint group of GI into w 1 planes, which envelope a cone, a general developable surface or all pass through a line. In each case there is an invariant configuration and consequently there is no simple GI. Lie-Scheffers, 1893, 1, p. 576. 40. The characteristic equation of a group. The rank of a group. The equation (40.1) A(u, P ) s |n?(ti) - P 5?| = 0, where r?? is defined by (39.8), is called the characteristic equation of the group, and A(w, p) the characteristic matrix. From 38 it follows that the latter is unaltered when the u's undergo a homo- geneous linear transformation. The classification of groups is based upon the properties of the characteristic matrix (cf. 46). Since the rank of ||i?b|| is less than r (39), we have (40.2) (-l)'A(tt, p) = p' - ti(u)p'~ l + fa(u)p'-* - ts - - + (-l)' To within sign at most $ q (u) is the sum of the principal minors of \\ril\\ of order q. Hence if the rank of the matrix is q, the functions 156 IV. THE ADJOINT GROUP \l/ 8 for s > q are zero and the equation (40.1) admits zero as a root of order r q at least. From (40.2) we have ft c _ * / o ft c I a 6 c a 6 r\ abe and so on. If we put / A f\ o\ 6 a (4U.5) ^>i = TJ , >2 = f]anbj V* = and so on, we have ^1 = ^1, ^2 = 2(^1 V2), ^3 = \V\ and consequently (40.4) vi = \[/i, pi = \l/l 2^2, ^3 = ^i 3^i^ 2 + 3^3. Hence 2 it follows that (40.5) m. If the vector-space E m is invariant also for r?(t>), then it is invariant for q(w), as follows from (40.8). Then from (40.8) and r,* p (u) = i* p (v) = for a > m we have rj p p (w) = T??(W)^(V) - i}l(v)t] q p (u) =0 (p,g = 1, ,m). But rjp(w) is the trace of the minor of the matrix tj(w) operating on the sub-space E m . Since the trace is independent of the choice of canonical parameters, we have : [40.1] // vj(u) and v)(v) leave a sub-space invariant, the trace of the minor of the matrix rj(w), where w* = c a lu a v b , operating on the sub- space is zero. If a G> admits a sub-group (? m , and uX a fis a generator of a G\ of G m , then u a is called a vector of (? m . If the basis of G> is chosen so 158 IV. THE ADJOINT GROUP that Xif, . . . , X m f is the basis of G m , then for any vector of G m we have u* = 0(2 = m + 1, , r). From this result and (16.5) it follows that ift(u) = for t = 1, , m, and the characteristic matrix A for u a is of the form v = 1, , ro; \ , 2 = m + 1, , r/ Since = cju% where t, v, s = 1, , m, it follows that ||i?i( w )ll is ^e matrix of the sub-group with respect to the vector u. Hence we have A = A m \rf w pdw\j where A,,, is the characteristic matrix of the sub-group G m . We may write ( l) m A m = p m ^m,l(u)p m ~ 1 + " * ' + (~l) W ~Vn,m-l(^)p. If G m is an invariant sub-group, then as follows from (31.3) ^l( u } = 0, and we have A = (-l)'-'V- m A m . In this case \l/ m , a = \[/ a , and zero is a root of the characteristic equa- tion of order r m + 1 at least. Hence we have: [40.2] // a G r admits an invariant sub-group of order m, the character- istic equation of G r for a vector of the sub-group has a zero root of order r m + 1 at least. Since in particular \l/\ and \l/z for A and A m are equal in this case, we have: [40.3] // G m is an invariant sub-group of a G T , the sum of the roots and the sum of the squares of the roots of the characteristic equation of G m with respect to a vector of G m are equal to the same quantities for the characteristic equation of G r with respect to this vector. We shall prove the following theorem of Killing:* * 1888, 3, p. 260. 40. THE CHARACTERISTIC EQUATION OF A GROUP 159 [40.4] The coefficients t K (u) in the characteristic equation of a group are invariants o] the adjoint group, that is, they satisfy the system of equations (40.9) Eaf = (a = 1, - , r). If we put e *e 76 = ^b P$&, then ,dA K dA dnl K e dA E a &(u, p) = TJ O = rj a = lac* \ du dy b du dy b = U d C d aC k l J = U d (CabCdl + C K e a C d b) e oyi dy b _ e * dA Since the last member of the above equation is zero. Hence E a &(u, p) = whatever be p and consequently the theorem is proved. We have seen that the functions ^i, . . . , \l/ r -i are homogeneous functions in the u's of orders 1, , r 1 respectively. Among them there may be relations so that only p of them are functionally independent; p is called the rank of the group. If the rank of \\ifi\\ is q, the equations (40.9) admit at most r q independent solutions; consequently (40.10) p < r - q. But, as we have remarked, if the rank of H^H is q, the equation A(w, p) = has a zero root of order r q at least. Hence as a corollary to (40.10) we have: [40.5] The rank of a group is equal to, or less than, the order of the zero root of the characteristic equation. In particular, when all the ^'s vanish and consequently zero is a root of order r, the rank of the group is zero. 160 IV. THE ADJOINT GROUP Since the roots are functions of the ^'s, we have : [40.6] The non-zero generic roots of the characteristic equation as functions of the u's are invariants of the adjoint group. Since the ^'s are functions of the roots, the number of independent fa is equal to the number of independent non-zero generic roots. Hence we have: [40.7] The rank of a group is equal to the number of functionally independent non-zero generic roots. If u a Xaf and v a X a f are the generators of a sub-group (7 2 of a given G r , then from (40.6) and (40.7) we have admits m exceptional sub- 162 IV. THE ADJOINT GROUP groups Gi (29), and these determine an invariant sub-group G m . Moreover, G m is Abelian and consequently integrable. Again if cJ 7* 0, the equation (41.5) c defined by (41.4), if the latter exists. Since the latter is an invariant sub- group of G r , it is an invariant sub-group of the sub-group defined by (41.6). An invariant sub-group is defined also by (41.7) c. ik v? = 0, where (41.8) CM = Ci k ,ghk = Ci h ,cS!iCk l M . From the definition it follows that c,-/* is skew-symmetric in i and j. That it is skew-symmetric in j and k follows from the following expression which is equivalent to the right-hand member of (41.8) in consequence of the Jacobi relations (7.4): Consequently c is integrable and the adjoint group is of order r, the adjoint group is integrable, since we have equations of the form (36.1) in the generators E a f. If the order of the adjoint group is s( admits r s independent exceptional sub-groups (cf. 29), we choose as the generators of the adjoint group the first s of the E a f which are independent (con- stant coefficients), and we denote them by E t f. If we take any two of this limited set, say E t and E u (u > t), then (E t , E u )f = CtuEif, 42. INTEGRABLE GROUPS 163 where in consequence of (36.1) I < u. If I is a value not taken on by t, then Eif = a v iE v f, where the a's are constants and v takes on the values of the set taken on by t which are less than I. Hence we have (E t , E*)f = d t :E v f, where u > t, v < u and the d's are at most linear combinations (constant coefficients) of c t * where 6 ^ v. Consequently conditions analogous to (36.1) are satisfied, and we have: [42.1] The adjoint group of an integrable group is integrable.* Since the adjoint group is linear homogeneous, we have in consequence of theorem [37.5] the following theorem of Lie:f [42.2] // a group G r is integrable, in the representative vector-space of the adjoint group there is at least one sequence of invariant sub- spaces EI, . . . , E r -i such that any E K is contained in E K +I. Consider any one of these invariant sub-spaces, say Eh. To the vectors of Eh and Eh +K for K = 1, , r h there correspond sub-groups Gh and Gh+ K of G r , and since Eh+ K contains Eh, Gh is a sub-group of G h + K . Moreover, by theorem [39.7], since Eh is invariant under all the transformations of the adjoint group, Gh is an invariant sub-group of G> and consequently of Gh+ K . Hence we have the theorem of Lie:J [42.3] // a group G r is integrable, it admits a sequence of sub-groups Gi, Gzj . . , G r -\ such that Gh for any h is an invariant sub-group ofG h+K ,forK = 1, , r - h. Hence the generators of an integrable group can be chosen so that (42.1) (X h , Xh+ K )f = Chh+lXJ (h = 1, , r- 1;Z= 1, ,*;\ \K = 1, - , r - h ) from which it follows that (42.2) Cal = for i > a or 6. * Lie-Scheffers, 1893, 1, p. 536. 11893, 1, p. 536. J1893, 1, p. 537. 164 IV. THE ADJOINT GROUP When these conditions are satisfied, we have t?,(u) = C&A* = Of or k > j. Hence the terms of the matrix ||ij*|| below the main diagonal are zero, and accordingly the roots of the characteristic equation are w*ci, . . . , w*'cr. Since these roots are invariant under the adjoint group, we have Ci/u* (c K e e u K ) = c^Cal = (e not summed). Consequently = (e not summed), and since c/wV is a vector of the derived group, and any vector of the derived group is a linear expression of such vectors, we have : [42.4] For an integrable group the roots of the characteristic equation with respect to any vector of the derived group are all zero. Since g^u' is equal to the sum of the squares of the roots of the characteristic equation of rj(u), we have the necessity of the follow- ing theorem of Cartan:* [42.5] A necessary and .* /.."''< /' condition that a group be integrable is that gijU l ui be zero for each vector of its derived group. That this condition is sufficient will be shown in 44. We shall prove by induction the following theorem of Engel:f [42.6] // all the roots of the characteristic equation for an arbitrary vector of a group are zero, the group is integrable. We assume that the given group admits an integrable sub-group G m , and choose the generators of (3> so that X i/, X%f, . . . , X^/are the generators of G m . We denote by E m the vector-space of G m and denote by E r -m the projection of E r with respect to E m . Since G m is integrable, the transformations of the adjoint group for vectors of E m leave E m invariant and at least one direction of E r -m by the results of 37. If we adjoin this direction to E m and denote by E m +i the resulting sub-space, we have that E m+ i is invariant for the * 1894, 1, p. 47. ILie-Engel, 2, p. 774; cf. Killing, 1888, 3, p. 289. 43. THE ROOT-SPACES OF THE MATRIX tj(w ) 165 transformations of the adjoint group for vectors of E m + i, and conse- quently these vectors represent a sub-group G m +\ of G>. In accord- ance with the results of 37, when a particular vector u of E m is used and v is a vector in E m +i but not in E m we have c*ttW = p(u)v k (mod # m ), where p(u) is a root of the characteristic equation for G> with respect to u } and by hypothesis is zero. Hence for all the vectors u of E m we have c<*w = (mod J0 m ) and consequently G m is an invariant sub-group of (j m +i. Since any Gi is integrable, we can by the repeated application of the above result establish the existence of a sequence of sub-groups as in (35.5), each invariant with respect to the sub-group of one higher order, and hence the theorem is proved. 43. The root-spaces of the matrix 77(1/0) for a regular vector U Q . The classification of continuous groups is based upon the canonical form to which the constants of structure of the group are reduci- ble by a suitable choice of the basis. This is accomplished by a study of the matrix ||r?2(i/o)|| = ||cSwj||, where UQ is a regular vector of the group, that is, one for which the generic roots 0, pi, . . . , p p of (40.1) are different. From 38 it follows that the parameters u a may be subjected to a linear homogeneous transformation such that in the new coordinate system the matrix ||i?6(^o)|| assumes the form (here the superscript indicates the row and the subscript the column) : (43.1) M a Pa lO . . . Pa 6*2 . . . 00 Pa 3 . . Op a < c I il o ^ v a ] P_ where po = 0, and the *s are one or zero, as the case may be. To each root there corresponds a vector sub-space of E r of the same order as the root which we call the root-space for this root. If in 166 IV. THE ADJOINT GROUP any system of parameters we denote by d, . . . , e Vo the funda- mental vectors of EQ of the root zero and by e, +i, . . . , e V9 + Vi those of EI for the root pj, and so on, then in the particular system of parameters for which (43.1) holds these vectors have the compo- nents da, where a takes the values for the given minors. From this we have that, if e a and e a +i are consecutive fundamental vectors of the root-space of the root p a , where a takes the range of values for the matrix M a , then (38) TjOuoX+i = p a e a +i + e a 00 a , where a/3 is the for the row of the vector e a . This vector equation holds in any parameter system. Since 175(^0)^0 = 0, it follows that UQ may be taken as the first fundamental vector of the root-space E Q . Suppose that el and el are fundamental vectors of the root-spaces E a and E $ respectively, which may be the same or different root- spaces; this means that a and b take values appropriate to these respective root-spaces. Then elX % f and eiX l f are the generators of two sub-groups GI of G r . Now we have (eiXt, eiXtf = u a k b X k f, where = and consequently are the components of a vector of E r , unless they are all zero. We seek the sub-space of this vector. We have (p a e a (Pa If e a is the first vector of E a , then e a d = and may be zero other- wise; if e a is not the first vector of E a , then u a -\b is a vector of the same sort as Uab. Hence if we write the above equation in the form we have ~ (p a a-2 6. 43. THE ROOT-SPACES OF THE MATRIX ij(u ) 167 Consequently we have where p ^ v a + v& since the ultimate effect of repeating the operation is to operate on the first vectors of the root-space and get c's which are zero. A similar result will follow, if in place of fundamental vectors e a and e b we take any vectors of E a and Ep respectively, since any such vector is a linear combination (constant coefficients) of the fundamental vectors of the root-space. From this result and theorems [38.2] and [38.3] we have:* [43.1] // u a and Up are vectors of the root-spaces E a and Ep respec- tively, which may be different root-spaces or the same one, the quantities (43.2) u k a p = c?jU a Up, u a p 35 (u a , Up) are zero, if p a + pp is not a root, and, if they are not zero, then p a + pp is a root and they are the components of a vector of the root-space of this root; moreover, if p a + pp is a root, this vector, which may be a zero vector, is a vector of the root-space of this root. Another way of stating this result is as follows: [43.2] // the matrix r?(?O for a vector u a of the root-space E a operates on a vector of Ep, the resulting vector, if it is not zero, is a vector of the root-space for p a + pp. When in particular u a and Up are vectors of EQ, so also is (u a , Up). Hence we have: [43.3] The generators of G> for the vectors of the root-space EQ for the matrix rj(u Q ) generate a sub-group ofG r . We call this the sub-group U. Again if u is a vector of EQ and u a of any other root-space E a , then (u, u a ) is a vector of E a , that is, the root-spaces for y(u) are the same as for 77(^0); consequently rj(u) operating on a vector of E a yields a vector of this root-space. In the special parameter system yielding (43.1) the fundamental vectors of E a have the components 5, where a takes the values appropriate to E a . Then r)}(u)da = X^Sft, where the X's are constants and 6 takes the values for E a . From these equations it follows that t\l(u) = X^6& and * Carton, 1894, 1, p. 41; Weyl, 1926, 2, p. 357. 168 7F. THE ADJOINT GROUP consequently T?i(w) = 0, unless i takes the values on the same range as a. Hence the matrix t)(u) consists only of principal minors, and each minor is of the same order as for 77 (w ), but none of the terms of one of these minors is necessarily zero. Thus the matrix rj(u) is of the form (43.1), where now (43.3) M a = a+1 o+l r) 04 " 1 i?SI a a+va (a = VQ + vi + Accordingly the characteristic determinant of rj(u) is equal to the product of the characteristic determinants of these principal minors, and the order of any determinant is equal to the order of the corre- sponding root-space. Hence the roots of the equation obtained by equating to zero the characteristic determinant of the matrix M a are generic roots of the equation (40.2), and in fact all the roots are equal. For suppose that two of them were different so that the equation were of the form (p p a (u)Y(p p a (w))". But when u is replaced by wo, the determinant is (p p a (^o)) Ka , and conse- quently P(WO) = Pa(tto), which is contrary to the hypothesis that UQ is a regular vector. Hence for any vector u of the sub-group C7, the characteristic determinants of the respective minors of the matrix rj(u) are (43.4) ", (P , (P - It should be remarked that if u is a special vector two or more of the roots are equal. However, the vectors corresponding to each minor are the same in all cases. But only when u is regular will we speak of them as the root-spaces of r/(w). Accordingly we have: [43.4] The root-spaces for the matrix ri(u), where u is a regular vector of the sub-group U, are the same as for the matrix i?(w<>). Since the trace of the matrix M a is equal to v a p a , we have (43.5) ?aPa( w ) ^(w) = Cblu b (a not summed), where a takes the values appropriate to the root-space E a , and b takes the values 1, . . . , ?o, since the vector u is a linear combina- 44. CANONICAL FORM OF THE MATRIX 7j(w) 169 tion of the vectors of components 6& for b = 1, , VQ. Hence we have: [43.5] Each non-zero root of the characteristic equation for i)(u) is a linear homogeneous function of the components of u. * Incidentally we have that if (u, u a ) is zero, where u a is a vector of the root-space E a and u is a vector of the sub-group U y then u is not a regular vector. For, if u were a regular vector, then u a is a vector of its zero root-space and the latter would be of order greater than j>o. Hence we have: [43.6] // T}(u)j where u is a regular vector of the sub-group U, operates on a vector of any root-space, the resulting vector is in this root-space. Hence theorem [43.1] holds for the matrix TJ(W), where u is any regular vector of the sub-group U. Since the roots of the characteristic equation for the matrix Afo are zero, we have in consequence of theorem [42.6]: [43.7] The sub-group U as determined by any regular vector of a group G r is integrable. Since U is integrable, its derived group is either the identity, or an invariant -ub-tii-.-up of U. The vectors of this sub-group, if it exists, are linear combinations of vectors of the form ti = (u, u f ), where u and u' are vectors of EQ. Since each root-space is invariant under rj (u) and rj(w'), the trace of the minor of 77(12) for the root-space of any root is zero by theorem [40.1], that is, v a p a (u) = (a not summed), and consequently all the roots are zero. Hence we have: [43.8] The characteristic equation of a group with respect to a vector of the derived group of U is p r = O.f 44. Canonical form of the matrix 17(11). Cartan's criterion for integrable groups. Since the sub-group U is integrable, the corre- sponding sub-group of the adjoint group is integrable, and in consequence of theorem [37.5] when this sub-group is applied to any one of the root-spaces, say the one of root p a , there is an invariant * Carton, 1894, 1, p. 38; Weyl, 1926, 2, p. 358. t Cartan, 1894, 1, p. 42. 170 THE ADJOINT GROUP direction, jEj, an invariant E z through it and so on. If we take the first of these as the first fundamental direction of the root-space, a vector determining with the latter the invariant E% as the second fundamental vector, and so on, then the principal minor of the matrix TJ(ZI) for the root-space E a is of the form (cf. 37) (44.1) Pa Pa Pa .... Pa where the terms above the main diagonal do not enter into the discussion. Since the vector-spaces E\, E 2 , . . . in E aJ which led to the above result are invariant for the sub-group of the adjoint group corresponding to the sub-group C7, they are invariant for every one- parameter group of this sub-group of the adjoint group, whether the vector of this one-parameter group be regular or special. We have seen (43) that if u a is any vector of the root-space E a and (u, u a ) = 0, then u is a special vector of the sub-group U. Suppose this condition holds for all vectors u a of this root-space. Then in particular it holds for the first fundamental vector e a . But for a regular vector u of [/, we have (u, e a ) = p a (u)e a . Since by hypothesis (u, e a ) = and the vector e a is independent of u, it follows that p a (u) = 0. Hence we have: [44.1] // u is a vector of the sub-group U such that (u, u a ) = for all the vectors u a of the root-space E a , then p a (u) = 0, that is, u is a special vector of U. If e'j e" , . . . , e ( " a) are the fundamental vectors of the root- space of the root p a , we have (cf . 37) 0,(t) - 7p a )e' - 0, (i,(tO - 7 Pa ) V = 0, , (n(u) - Ip a y<*e ( ^ = 0. These equations are equivalent to 44. CANONICAL FORM OF THE MATRIX r?(w) 171 e" where tt, w"', ... are vectors of the root-space E a . Hence if we take any vector of the root-space, that is, a linear combination (constant coefficients) of e', . . . , e ( " a) , we find from the right-hand members of the above equations the vector of the root-space upon which ij(u) operates to yield the given vector. Hence we have the following converse of theorem [43.6]: [44.2] // u a is a vector of the root-space E a , there is a unique vector u a of this root-space such that u a = rj(u)u a , where u is any regular vector of the sub-group U. The vectors defined by (43.2) are evidently vectors of the derived group of G>. From theorem [43.2] we have that, if such a vector belongs to the root-space J0 , it arises either from two vectors of E*, or from the vectors of two root-spaces whose roots differ only in sign. Suppose we denote two such roots by p a (u) and p a '(u), and vectors of these spaces by u a and u a >, respectively, then U = (u a , w a >) is a vector of E Q . The characteristic equation for TJ(#) has the same root-spaces as for 17 (u), and the roots for the spaces of u a and u a > will differ in sign, since they are p a (u) and p a '(w), respectively. If pp(u) is any root of the characteristic equation of 77 (u) other than p a or p a /, and we operate with rj(u a ) on any vector u& of the root-space E$ of root pp, we have that the resulting vector is zero, or that it is a vector of the p a + p ft root-space and that p a + p$ is a root in accordance with theorem [43.1]. Similarly, if we operate with ri(u a f) 9 we obtain a zero vector, or pp p a is a root. Since it follows from the above observations that if neither p + p a nor Pft p is a root of the characteristic equation of r)(u), then 77(11) operating on every vector of the root-space Ep is zero. Hence by theorem [44.1] p ft (u) = 0, and u is a special vector of the sub-group U. Suppose then we consider the possible sequence of roots 172 IV. THE ADJOINT GROUP (44.2) p ft - MP, ' * , P0 - 2p, p0 - P, P/j, P/J + P, P/s + 2p a , , p ft + Xp a , it being understood that p + (X + l)p a and p/s (M + l)p are not roots, and denote by E$ the totality of the root-spaces of these roots. Then E ft is invariant under rj(u a ) and rj(u a f). Hence by theorem [40.1], the trace of the matrix TJ(W) in Ep is zero. For the root-space of each root the trace is equal to the product of the root by the order of the root. Hence the trace is of the form 0p0(w) + bp a (u), where a and b are integers. Consequently we have : [44.3] // u a and u a > are vectors of two root-spaces whose roots p a and p a > differ only in sign, and u = (u a , u a >) is a general vector of [7, then pp(u) is a rational multiple of p a (u). We are now in position to establish the sufficiency of Cartan's criterion for an integrable group (theorem [42.5]). We have observed that if a vector u of the derived group G> of G> is in EQ, it is a linear combination of vectors (w, w'), where u and u' are in U , and of vectors (u a , w a /), where u a and u a ' are vectors of root-spaces whose roots differ only in sign. The characteristic equation with reference to a vector of the first kind has only zero roots by theorem [43.8]. The non-zero roots of the characteristic equation with respect to one of the vectors of the second type are all rational multiples of one of them. Hence if g^u V, that is, the sum of the squares of the roots, is zero for a vector of the second kind, all of the roots are zero. From (43.5) it follows that any non-zero root for a vector u which is a linear combination of vectors of E Q is the sum of the roots of the same root-space for the different vectors of the combination. Hence if gnu { u j ' is zero for all vectors of the derived group which are in EQ, it follows that all the roots of the characteristic equation for any such vector are zero. Since G r is integrable when all the roots of its characteristic equation are zero by theorem [42.6], we have only to consider the case when some of the roots are not zero in which case we have a set of root-spaces for a general vector UQ. If the first derived group G> t coincides with (? r , then each vector in E is a vector of the derived group and if gfywW vanishes for each of these vectors, then by the preceding paragraph all the roots are zero for the corresponding characteristic equation, and thus in particular for 45. SEMI-SIMPLE GROUPS 173 the vector MO, which is contrary to the hypothesis that not all the roots are zero. Hence G' TI is a proper sub-group of G>, and is an invariant sub-group. Since G' fi is an invariant sub-group of G r , if the generators of this sub-group are chosen as in 40, it follows that the sum of the squares of the roots of the characteristic equation of G' fl is equal to the sum of the squares of the roots of the characteristic equation of G r for a general vector of G' fl . Hence we may apply to G fl and the second derived group G' r ( the same consideration which showed that Gr t was a proper sub-group of G>. Accordingly by continuing the process we get a sequence of derived groups ending in the identity, and consequently G> is integrable. The results of the preceding paragraph were due to the fact that G' TI is a proper invariant sub-group of G> and did not involve the fact that it is the derived group of G>. Consequently we have as a corollary the criterion of Cartan:* [44.4] A necessary and sufficient condition that an invariant sub-group G m of a G r be integrable is that gijU { u* vanish for all the vectors of the derived group of G m . As an immediate consequence of this theorem we have that the invariant sub-groups defined by (41.4) and (41.6), if they exist, are integrable. For, in either case (fc/wV is zero for all vectors of the sub-group, and consequently for its derived group. Consider now the invariant sub-group defined by (41.7), that is, (44.3) c^ 0, dj k If ui and u z are two vectors of this sub-group, we have 0. Since u is a vector of the derived group of the sub-group, the latter is integrable by the above theorem. Incidentally we have that either the sub-group (44.3) is Abelian, or its derived group is the sub-group (41.6). 45. Semi -simple groups. By definition a semi-simple group is one which does not contain an integrable invariant sub-group other than the identity. Since a simple group (31) is one which has no * 1894, 1, p. 47. 174 7F. THE ADJOINT GROUP invariant sub-groups, other than itself and the identity, it follows that a simple group of order greater than one is semi-simple, so that theorems applying to the latter apply also to simple groups. Since the sub-group (41.4), if it exists, is invariant and integrable, we have that a semi-simple group does not admit any exceptional sub-groups G\. Since the sub-group (41.6), if it exists, is invariant and integrable (44), the rank of the matrix \\ga\\ is r for a semi- simple group. Cartan has shown that this condition is also suffi- cient. In fact, if a G> admits an integrable invariant sub-group G my either the latter is Abelian or one of its derived groups is Abelian. Since all of the derived groups of G m are invariant sub- groups of G r by theorem [35.5], in either case there is an invariant sub-group of G r which is Abelian. Suppose its generators are X P f (p = 1, , 0- Then, since c p h a = for a = 1, , r and h > t by (31.3), we have Ik I a I ^ = 1) " ' " > PJ 1 Oip = C lk C p i = C iq C p Q i = C iq C p 9 8 I ' _ ! . ,) \P> Q> s i> i l / But Cp\ = 0, since the sub-group is Abelian, and consequently the rank of the matrix \\ga\\ is less than r. Hence we have the theorem of Cartan:* [45. 1J A necessary and sufficient condition that a group be semi- simple is that the rank of the matrix \\gu\\ be r. When the rank of the matrix ||0 ; || is r, the quantities g i} may be used to endow the vector space E r with a metric, so that we may speak of lengths of vectors and angles between them. Thus if two vectors u and v are such that gijuV = 0, we say that they are orthogonal. Also if gr t -/wV = 0, we say that u is a null vector; a null vector is self-orthogonal (cf. 47). As a consequence of the preceding we prove that: [45.2] A semi-simple group is the direct product of a set of simple groups, the representations of which in E r are mutually orthogonal. The theorem is trivial when the group is simple, there being only one group in the set, namely the group itself. If the group is semi- simple, by definition it admits at least one invariant non-integrable sub-group. If it admits more than one, there is at least one G p * 1894, 1, p. 52. 45. SEMI-SIMPLE GROUPS 175 For p < r which is not contained in an invariant sub-group other than G T \ also p > 1, otherwise the GI being integrable, the group jf r would not be semi-simple. Let u a for a = 1, , p be p independent vectors representing G p , and denote by E p its repre- sentative vector-space. The equations gnUaV' = define r p independent vectors orthogonal to E p which determine an E r -p orthogonal to E p . Since QH is invariant under the adjoint group and E p is invariant, it follows that E r - P is invariant, and consequently represents an invariant sub-group G T - P . E p and E r - p have one, or more vectors, u in common only in case gai^u' = 0. [n this case G p and G r ~ P would have in common an invariant sub- group of G r by theorem [31.3] and it would be integrable in conse- quence of theorem [44.4] which is contrary to the hypothesis that G r is semi-simple. Hence G r is the direct product of G p and G r - P . Moreover, G r - P is simple; otherwise it would contain an invariant sub-group. The product of the latter and G p would s;ive an invariant sub-group of G> containing G p , contrary to hypothesis. Since G p and G r - P are invariant sub-groups of G> and do not have a sub-group in common other than the identity, and if Xhf for h = 1, , p is the basis of G p and X 8 f for s = p + 1, , r the basis of G r - P , then we have (31) Chk = 0, c, h t = 0, ( = f a = 1, - , r;ft, k = 1, , , t = p + 1, , r and consequently b a b h I h A = Ck a C a b = CkhCab = CkhC 8 l = U, (A, ft, f, m = 1, , p). From the first of these equations it follows that the determinant g a b\ for a, b = 1, , r is equal to the product of the determi- nants \g h k\ and |0. J, and consequently both of these determinants are different from zero. From this result and the second equation it follows that G p is semi-simple. If it is simple, the theorem is proved ; if not, we proceed with G p as with G r and ultimately establish the theorem. 176 IV. THE ADJOINT GROUP Consider any vector u of the sub-group U and a vector u a of the root-space E a . By theorem [44.2] it follows that there is a vector u a of this root-space such that r?(w)tZ a w a Hence we have giiu'uZ = t ,uV'*VuL = c k iiU*u k u l a = 0, since Ckn is skew-symmetric in all of its indices. Accordingly we have: [45.3] For a semi-simple group the vectors of any root-space E a are orthogonal to all the vectors of EQ. If UQ is a vector of a root-space other than E Q , we have (45.1) = c K iiu l u* a U0 = guu l ($i* a Ufi = guu l (u a , U0Y. If p a + pp is not a root, then (u a , Up) = (43) and u a and u ft are orthogonal. If it is a root say p y T& 0, then the last term is g^u l u\ 9 which is zero by theorem [45.3]. Consider finally the case when pp p at in which case we call it p a >, as formerly. If this case did not arise, or if it does arise and gnu^ul' = for every vector of E a ', we should have giflfjf = 0, where v is any vector of E T1 since in the above discussion u a and u ft may be vectors of the same root- space. Hence we should have 0 and for any u^ of the root-space E a there is a u a > of the root-space E a > such that gijU l a u } a > ^ 0; the vectors of any two root-spaces are orthog- onal, except when their roots differ only in sign. Since the fundamental vectors of each of the non-zero root- spaces are orthogonal to all of the fundamental vectors of E Q) if we put (45.2) QU = flf^e'e}, we have (45.3) g ap = (a = 1, , v ; p = VQ + 1, , r). If we define quantities g*' by (45.4) fflfi - a*', 45. SEMI-SIMPLE GROUPS 177 we have also (45.5) jp* = 0. Referring to (43.5), we see that if we put (45.6) v aa = -^, v ap = (a = 1, - , j/ ; p = VQ + 1, ' ' , r), then, since p a (u) is a scalar under change of basis and u is a contra- variant vector, Vai is a covariant vector. If we put (45.7) v* = gt'Vaj, then v are the contravariant components of this vector. Since in consequence of (45.5), (45.8) t = g**v ai = g* a v aa = 0, we see that v a is a vector of E Q , which we call the root-vector for the root p a . Then (43.5) assumes the form (45.9) Pa (u) = v aa u* = g a bU"v b a . In the special coordinate system of 43 g a b = g a b, so that the sum of the squares of the roots of the characteristic equation for a vector u of the sub-group U is g a bU a u b . Also from (45.9) we have that the i, . . . ,P sum of the squares of the roots is ]x v a v aa v ab u a u b . Since these two expressions must be equal for all vectors u of U, we have The rank of the matrix \\g a b\\ is i> and this is at most equal to the rank of \\v a b\\, that is, the number of independent root-vectors, say Z.* However, since these root- vectors are in J57 , 1 cannot be greater than VQ. Hence we have : [45.5] The number of linearly independent root-vectors is equal to the order of the sub-space E Q , that is, the number of zero roots of the characteristic equation for a regular vector. * Cf, Bdcher, 1907, 1, p. 79. 178 IV. THE ADJOINT GROUP From (45.9) it follows that a necessary and sufficient condition that there be a vector u of E Q such that all the roots p a (u) are zero is that v aa u a = for every a. This is impossible since the rank of \\v) . None of these are null vectors. For, if one were, the sum of the squares of the roots of its characteristic equation would be zero by (40.5) and hence each root would be zero. Then u would be orthogonal to all the root- vectors by (45.9), which has just been seen to be impossible. Let e a be the first fundamental vector of the root-space E ay in which case we have (45.10) ri(i*)ei = We know that there is at least one fundamental vector of the root- space E a > not orthogonal to e aj say e a >. If we multiply the above equation by g K ie l a > and sum for *, we have Chuu h e* a e l a > = p a g K ie l a >eZ = p a cr a (a not summed), where ), we see that u is not a zero vector. Comparing this equation with (45.9) and observing that for the basis of 43 jjhk = ghky we see that (45.11) u = (e al e a >) = ff a v a (a not summed), that is, u and v a have the same direction. Furthermore, as shown in the preceding paragraph 0-/wV ^ 0. In consequence of (45.11) and (45.9) for the vector u we have ^aPa(^) = We consider now the sum of the Jacobi relations ~ ~" where a and a' are the indices of the fundamental vectors e a and e a > referred to above in the set of fundamental vectors of E r , and p takes on the values of this set for the root-spaces of the above roots. Since u defined by (45.11) is in E , we have c a q a > = for q > VQ. Similarly, if u p is a vector of the root-space of a root mp a then (e a >, u p ) is a vector in the root-space of (m l)p a , if this is a root, and (e a} u p ) is a vector of the root-space (m + 1) p a , if this is a root. Accord- ingly the above equation reduces to b V i t* V i w' t>' rv /7 \ Coo'C&p + (ViCtto + Cv'aCu'a' = (6 = 1, ' ' ' , ^ ), where u, v, u' , v' run through the indices belonging to the root- spaces of the following roots; w:0, p a , , (s - l)p a ; (s + t)p a , , (s + w - l)p a ; ":p> ' , SP ; (s + t+ l) Pa , -,( + w)p a ; w':2p a , , sp a .; (s + < + l)p a , -,( + ^)p a ; v ; :pa, ',(*- !)P ; (s + OP, -,( + w - l)p. In consequence of these ranges and the skew-symmetry of the c's the above equation reduces to b p V b /Li \ Caa'Cbp = CabCa'v (0 = 1, * * * , VQ)- Since e = d v a , it follows from (45.10) that c a l = unless k = a, and consequently the above equation reduces to (45.12) cJta? - c>:e (6 = 1,.-., ). The vector u defined by (45.11) is equal to c>. Consequently the left-hand member of (45.12) is the trace of the matrix of rj(tZ) in the root-spaces of the given set of roots, which by (43.5) is equal tO [Ki + 2*2 + + SK, + (S + f)K,+t +'+( + U>)Ki+w]pa(u). * Carton, 1894, 1, p. 55; Weyl 1926, 2, p. 364. 180 IV. THE ADJOINT GROUP From (45.10) for u replaced by u we have that the right-hand member of (45.12) is p a (u). Equating these two expressions, we have that KI = 1 aad that the other /c's are zero. If there had been more than two groups of roots, the result would not have been altered and hence the theorem is proved. 46. Classification of the structure of simple and of semi-simple groups. Many of the results which have been presented in this chapter were derived in the process of the classification of semi- simple groups and in the determination of the normal forms of the members of the various classes. In 1885-1889 Lie* showed that there were four large classes of simple groups, those with the same structure as the general projective group in n variables; as the projective group of a linear complex in 2n 1 variables; and as the projective group of a hypersurface of the second degree in 2n and 2n - 1 variables (cf. Exs. 14, 15, 16, p. 184). In 1889 Killing,! by making use of the characteristic equation of a group (40), showed that in addition to these four classes of simple groups, there are only five other possible structures of simple groups, their orders being 14, 52, 78, 133 and 248. Cartan,J after deriving many of the results referred to in the preceding sections, established the existence of these five special types and showed that they are distinct. Weyl studied the same problem and gave a more geometrical setting to the problem. We shall give an outline of certain consequences of the results of the preceding sections which serve as a geometrical basis for classification. We denote by e(= $1) for a = 1, , Z(= v$) the independent vectors which determine the root-space E Q of the zero root of the characteristic equation for a general vector. By theorem [45.7] the root-space of a non-zero root is one-dimensional, and if p a is a root, so also is p a (= p a ')> that is, the roots go in pairs. We indicate by e v a = 6^ and e % a >(= 5/) vectors of these root-spaces and understand that the indices a and a for the various roots are chosen, so that ,, take the values I + 1 to r. If we put (cf . 45) (46.1) u a = (e a , e a >), * 1885, 1, p. 130; 1886, 1, p. 413; 1889, 3, p. 325. 1 1889, 1, 48. 1 1894, 1, pp. 68-95. 1925, 4; 1926, 2. 46. CLASSIFICATION OF STRUCTURE OF SIMPLE GROUPS 181 then by (45.11) (46.2) u a = = 1, g afi = (a = 1, - - , i; \ \, |8 = Z + 1, - , r; ft * a'), from which we get (46.4) <7* = 2t& = X">,0 aa = 0,0* = 0(|8 ^ a'),fT' = -1. a a Since the sub-group U is Abelian by theorem [45.6], we have c a l = for a, 6, e = 1, , I, and by (41.8) and (46.3) c a & = 0. From (46.1) and (46.2) we have (46.5) u a a = c a a a > = -v*, c& = (/3 = I + 1, , r). Hence we have (46.6) Caa'a = U aa = ^aa, and in consequence of (46.4) (46.7) Co2 = v aa = u aa (a not summed). By theorem [43.1] cj& = unless p a + p/s = P 7 . From this result and (46.4) we have c a p y = unless p + p/s + p 7 = 0. It remains to be shown that these quantities are not zero under these conditions. In fact, Weyl,* by making use of a sequence of roots (44.2) has shown that l)p a (u a ). Furthermore, Weylf showed that by a suitable choice of basis we have c a >p y > = c afty , and consequently (46.8) (c afty )* - -iX( M + l) Pa (tO. * 1926, 2, p. 372. 1 1926, 2, p. 374. 182 IV. THE ADJOINT GROUP It can be established that p a (u a ) is negative, so that c a/?7 is real, and Weyl* has shown that all the root- vectors are linear combinations with rational coefficients of any I independent ones. When the basis is chosen so that these are real, all of the coefficients c* are real, and QabU a u b is positive definite. Referring again to a sequence (44.2) and remarking that each of these roots is simple, we have from the argument fnlluuini: (44.2) that (x + M + i),,(tO + -- - PM = o, and consequently (46.9) P ,(uJ = -^^p a (J. L From (45.9) and (46.2) for a a = 1 we get (46.10) p0(tO = -fyoC P(M) = -0aC By a combination of these equations and (46.9) Schoutenf was enabled to find the possible angles between root-vectors and their lengths. If the configuration of root-vectors of a group is called its root-figure, the problem of classification of the structures of semi- simple groups reduces to that of finding all possible and distinct root-figures. By theorem [45.2] a semi-simple group is the direct product of simple groups, the representations of which in E r are mutually orthogonal. Hence the set of vector-spaces of these component simple groups are orthogonal, and consequently their root-figures. Therefore when root-figures are found which cannot be decomposed into mutually orthogonal figures, they are the root-figures of simple groups. The problem of finding simple groups by the determination of all possible root-figures on the basis of Schouten's results was carried out by Graham, f who obtained from this point of view the results which Cartan had established by algebraic processes. In this investigation use was made of the fact that the reflection of a root- vector with respect to any other root-vector is also in the root- figure. These reflections for the root-figure of any group generate * 1926, 2, p. 368. t Unpublished lectures delivered at the University of Leyden. t Not yet published. 46. CLASSIFICATION OF STRUCTURE OF SIMPLE GROUPS 183 a discrete group, which is a sub-group of the linear transformations leaving the root-figure invariant. This sub-group is the discrete group S of Weyl.* In his study of semi-simple groups Weyl made use of the theory of representation of continuous groups. This theory for finite groups is due to Frobeniusf and Burnsideft and was extended to con- tinuous groups by SchurJ and developed further by Weyl.Jt It consists in an association of a non-singular homogeneous trans- formation, say A (a), with each set of values of the parameters of* of a G rj indicated by y' = A(a)y, which satisfies the law of com- position (4.6) of G r , denoted in this case by A(a 2 )A(ai) = A(a 2 ai). For example, the adjoint group is a representation of a G>, when the parameters of the latter are canonical. When the representation is transitive, it is said to be irreducible, in the sense that there is no sub-space invariant under all the transformations of the repre- sentation. In certain cases a reducible representation may be considered as the sum of two or more irreducible representations. In such cases one of the principal problems concerns itself with the establishment of a complete set of irreducible representations of G r such that any representation of G r may be considered as the sum of certain members of the complete set. For a full treatment of this subject the reader is referred to the authors indicated. Also it should be remarked that recently the theory has been fruitful in the study of physical and chemical problems. Exercises 1. If a sub-group G m of a G> is integrable, it is contained in at least one sub- group of order m + 1 of G>. Lie-Scheffers, 1893, 1, p. 564. 2. If a G r has an Abelian sub-group G>_i, it contains an invariant Abelian sub-group Gr-it and hence is integrable. Lie-Scheffers, 1893, 1, p. 584. 3. If a G r is of rank zero, it contains at least one exceptional sub-group Gi. Killing, 1883, 3, p. 288; Umlauf, 1891, 4, p. 40. * 1926, 2, p. 367. 1 1897, 2 and 1899, 1; ttl9H, 2, pp. 231-242, 269-279. 1 1905, 5. 1925, 4 and 1926, 2. Cf. Weyl, 1931, 1; also Wigner, 1931, 2. 184 IV. THE ADJOINT GROUP 4. If a G r is of rank zero, its first derived group is at most of order r 2. Killing, 1888, 3, p. 288, Umlauf, 1891, 4, p. 40. 5. If a G r is of rank zero, every sub-group G t is Abelian. Umlauf, 1891, 4, p. 35. 6. A G r contains at least oo r -< Abelian sub-groups Gi. Lie-Engel, 1893, 2, p. 756. 7. In consequence of Ex. 11, p. 155, Ex. 6, p. 137 and the definition of an integrable group it follows that a G* which does not contain a simple sub-group (7| is integrable. Lie-Scheffers, 1893, 1, p. 574. 8. A group G r is integrable, if and only if it does not contain a simple sub- group Gj. Lie-Engel, 1893, 2, p. 757. 9. The direct product of two semi-simple groups is semi-simple. 10. If a group of order r m is isomorphic with a group G r of rank zero, the former is of rank zero. Umlauf, 1891, 4, p. 35. 11. A necessary and sufficient condition that a G r be integrable is that cfc,M c *i = ccfo k Z (i,j, k, X, /z, v = 1, , r\ Cartan, 1894, 1, p. 48. 12. The general linear group has the symbols pi and x*'p/, and has three invariant sub-groups with the symbols Pi ; pi, x*pi ; p, x*pj, z>, - x*pj (i, j not summed; i 5* j). Lie-Engel, 1888, 1, vol. 1, p. 562. 13. The general linear homogeneous group in n variables has the symbols X*PJ, and has as invariant sub-groups the G\ with the symbol x'p t and the group with the symbols X*PJ, x { pi X'PJ (i 7*j', i, j not summed). The latter group is simple. Lie-Engel, 1888, 1, vol. 1, pp. 560-1. 14. The general projective group in n variables has the symbols (cf. Ex. 9, p. 43). p t , x

x*, y i (i = 1, , n) with the symbols 46. CLASSIFICATION OF STRUCTURE OF SIMPLE GROUPS 185 Pi - 2/*>, q* + x*r y r, A + zr (qi = ; r - j, a?*?/ + x^, a;p/ - y'qi, y (t ^ j), gp - y*A, zq> + x*A, zA, where A = x*pi + y*'g + zr, is simple and leaves invariant the linear complex defined by dz -f x^dy* - y*dx* = 0. Lie-Engel, 1888, 1, vol. 2, p. 522. CHAPTER V GEOMETRICAL PROPERTIES 47. Riemannian spaces. In the preceding chapters we have considered each set of values of the variables x i as the coordinates of a point in an n-dimensional manifold F n , and the transformations of a group G r as insnsforminir continuously a given point into other points of F n . This geometry is a geometry of position and thus far, except in the case of semi-simple groups, there has been no basis for the determination of magnitude nor for a comparison of directions at two different points. In this section we define mag- nitude and parallelism, and in subsequent sections make use of these definitions in their relation to the theory of continuous groups. Riemann* generalized the idea of element of length of euclidean 3-space, defining the element of length of a V n by means of a quadratic differential form, thus ds 2 = gijdx^x^ where in general the g'a are functions of the z's, and the form is positive definite. Because the general theory of relativity and other physical theories make use of a quadratic differential form which is not positive definite, we take as the basis of the metric of V n a real fundamental quadratic form (47.1)

J* 1 /if* 1 a? 1^^ P* - ^ fe ^ d t behave like constants under covariant differentia- tion. If covariant differentiation is applied to T}\ ; ; ; , k , we obtain the second covariant derivative of components T}\ \ ; ; };,**. These quantities are not symmetric in the indices fc and I as in the case of ordinary second derivatives. But on eliminating the second derivatives from the two expressions, the resulting equations may be put in the form (An i A^ (47.10) These are known as the identities of Ricci and, when covariant differentiation is used, these identities must be used in place of the usual condition of integrability for ordinary differentiation.! In euclidean 3-space referred to cartesian coordinates the vectors of a vector-field of components * are parallel, if the f's be constants, in which case their first covariant derivatives are zero, and consequently they are zero in every coordinate system, that is, (47.17) *'" m+ *M =0 ' * 1926, 3, p. 28. 1 1926, 3, p. 30. 190 V. GEOMETRICAL PROPERTIES We consider these equations in a general V n . When we express the condition of integrability of these equations, we obtain (47.18) ?/2JU = 0. Consequently the system (47.17) is completely integrable for a euclidean F and accordingly there exists a field of vectors parallel to any given vector; its components are the solutions of (47.17) determined by the components of the given vector as initial values for the values of the x's of the point where the given vector is. If V n is not euclidean, equations (47.18) are not satisfied identically, and consequently parallelism as understood in euclidean space does not hold in a general F n . However, if we consider any curve C in a given V n defined by equations where t is a parameter, the equations <' i-.f-f^iwf- admit a solution determined by values of the 's for t = 0. When such a solution is known, we have a set of vectors of components * at each point of the curve. Following Levi-Civita* we say that these vectors are parallel with respect to the curve C. In this sense parallelism is relative, whereas in euclidean space it is absolute. From (47.15) it follows that for a solution of (47.19) gat*? is a constant, and consequently the 's differ by a constant factor at most from those of a unit vector, if the initial values are chosen so that it is not a null vector. Two vectors at a point are said to have the same direction, if corresponding components are propor- tional. Accordingly, if a set of functions * satisfy (47.19), the vectors of components I* where \l/ is any function of t, should be interpreted as parallel with respect to C. From (47.19) we have <*!* , -frti \dx* -/Hog ft - -"-" * 1917, 1; another geometrical interpretation of parallelism as thus defined is given by Levi-Civita; cf. also, 1926, 3, pp. 62-65, 74. 47. RIEMANNIAN SPACES 191 d log \f/ Eliminating -- from these equations, we have as the general equations of vectors parallel with respect to a curve (dropping the bars) dx i Since are the components of the tangent to the curve C, a QL necessary and sufficient condition that the tangents be parallel with respect to C is that C be a curve satisfying the equations _oix dxi'/dW , , i} dx h dx k \ dx^dW (i (47 - 21) -*Ur + M * a") - *(-*- + If s is the arc of C, then is a unit vector and from (47.19) the as equations which C must satisfy are (47.22) +{}.. . ds 2 l ds ds These curves, called the geodesies of the space, may be said to be the straightest curves in V n and are the generalization of straight lines in euclidean space. In fact, if the coordinates in the latter d 2 x { are cartesian, equations (47.22) become = 0, which are the equa- ds 2 tions of the straight lines. From the form of (47.22) it follows that an integral curve of (47.22) passes through each point in V n in an arbitrary direction. They can be shown to be characterized by the property that for two points sufficiently near one another there is only one curve of the congruence through these points and the first variation of the arc between the points is zero.* When the fundamental form (47.1) is positive definite, this is the shortest distance between the points; when the form is indefinite, it may be the shortest or the greatest distance according to further distinguish- ing properties of the form. 1926, 3, pp. 48-50, 53. 192 V. GEOMETRICAL PROPERTIES 48. Linearly connected manifolds. Linear connection deter- mined by a simply transitive group. Let ( be the components of the n vectors of a simply transitive group G n in a F n . Then func- tions are defined uniquely by (21.9) for q = n, and as in (21.12) we have the functions (48.1) AJ* = li^j? = _#g? (a , ,-, j t k = 1, , n). If the x'a undergo a non-singular transformation into coordinates x fi and 'a are the components of the vectors in the z"s, we have <9rr (48-2) I' - ?.- If we denote by A')* the functions in the "s analogous to (48.1), we find that /.OON (48 - 3) We remark that these equations are similar to (47.9). From (21.15) we have (48.4) A)* - Aj y = C.J&& (a, 6, e, i, j, k = 1, , n), and consequently A}* are not symmetric in the indices j and fc, unless the group is Abelian (13), whereas {* m } are symmetric in I and ra as follows from (47.11). Although (48.3) have been obtained from (48.1), we consider the general case when Aj m and A'$b are two sets of functions in the relations (48.3) in their respective coordinate systems. We say that each set of these functions defines the same linear connection; they are called the coefficients of the connection. Comparing (47.13) and (21.17) we see that they are similar in form, so that the tensor of components A< ; * is called the curvature tensor for the given linear connection. Analogous to (47.20) are the equations A set of functions *' which satisfy these equations for a curve C, defined by expressing the x'& as functions of a parameter t, are said 48. LINEAR CONNECTIONS. SIMPLY TRANSITIVE GROUPS 193 to be the components of vectors at points of C parallel with respect to C. And the curves defined by dxJ ( d * xi A < dxk dxl \ dxi ( d * xi . A * dx " dxl \ n A - == a* a (48 * 6) are called the paths of the manifold; they are a generalization of geodesies of a Biemannian space in that they are the straightest lines.* Thus by means of a linear connection we are enabled to compare directions at different points, but are in no position to handle metric properties. From (21.19) it follows that for the linear connection defined by (48.1) in terms of a simply transitive group we have (48.7) A*,-* = 0, that is, the curvature tensor is zero. When any linear connection satisfies this condition, we say that the space is flat or that the connection is of zero curvature. In consequence of (48.7) the equations (cf. equations (21.13)) (48.8) g + j*Aj, = are completely integrable, and consequently a solution is determined by arbitrary initial values of the f's. From (48.5) it follows that any two vectors of such a vector-field are parallel with respect to every curve joining their points of application and consequently we say that they are absolutely parallel. As in the case of euclidean space, there are n such independent vector fields, say fi, and from 30 it follows that they are the vectors of the group reciprocal to the given group. If we define quantities Aj* by the equations (48.9) A}* = Aj,, it follows from (48.3) that A/* are the coefficients of a second linear connection. In consequence of (48.1) and (48.9) we have (48.10) ~? + ilj* = 0. ox* * Cf. 1927, 1, pp. 4, 5, 13, 14. 194 V. GEOMETRICAL PROPERTIES These equations are necessarily completely integrable and conse- quently the curvature tensor of components A<#, defined by express- ions analogous to (21.17) is a zero tensor. Comparing (48.10) with (48.8), we see that the vectors & are parallel absolutely with respect to the connection A)*- Hence we have: [48.1] For the two linear connections determined by a simply transitive group the parallelism is absolute.* From (48.8) and (48.9) it follows that fAQ 1 1 \ I y^" >^f * (48.11) A,, = -h- = ^ where f" are defined by equations of the form (21.9), that is, (48.12) Jfti = *J, rfrf = 81. Thus Ay* bear to the reciprocal group a relation analogous to that of Ay* to the given group. If we denote by F)-* the symmetric part of the A}*, that is, (48.13) ri = i(Ay* + Aj y ), it follows from (48.3) that the F's satisfy equations of the form (48.3) and thus are the coefficients of a third linear connection. Since the F's are symmetric in the indices j and fc, they are the coefficients of a symmetric linear connection.! In consequence of (48.9) Fy* is the symmetric part of Ay* also. We define quantities g ij ' and g^ by (48.14) g* = rirf, gt, From these equations and (48.12) we have (48.15) jty = dl, and (48.16) g<*fi = f>, g^ = ^. From the second set of these equations and (48.14) it follows that 9i*tb = 5 a &. Consequently if g if are taken as the components of the fundamental * Cf. 1925, 1; also MattidU, 1930, 2. 1 1927, 1, p. 55. 49. LINEAR CONNECTIONS. SIMPLY TRANSITIVE GROUPS 195 tensor of V n the vectors f are unit- vectors and any two are orthog- onal to one another.* From (48.14) and (48.8) we have (48.17) + g*'Ai k + g^L = 0, ox in consequence of which and (48.15) we have (48.18) T ~ ff**A* - 0i*A% = 0. ox Hence if we define covariant differentiation with respect to the linear connection by equations obtained from (47.14) on replacing {]k} by A*,A; and the comma with a solidus, we have (48.19) g**\ k = 0, g it p = 0. From (48.1), (48.18) and (21.9) we have (48.20, , the significance of which will be seen in 51. 49. Simply transitive groups determined by linear connections of zero curvature. We have seen that when the curvature of a linear connection is zero, there exists an ennuple of independent absolutely parallel vector-fields. We seek the conditions that these are the vectors of a simply transitive group. We denote by 12}* the skew symmetric part of the coefficients A}* of the linear connection, that is, (49.1) 12}, = i(A}* - Aj,). From these equations *and (48.13) we have (49.2) A;* = r}* + 12;*. From these expressions and (21.17) we have (49.3) A^-* = & i]k + 4-*, * 1926, 3, p. 38. 196 F. GEOMETRICAL PROPERTIES where fAQ A\ & _ ar <* dr <' a. r 1 r* r' r* (49.4) tfijk = -^ : - + lali/ - I\,r w and (49.5) ok* = ok,, - flj,, - oi,nji - n!,-Gj, - 2oJ*nJ, ; as defined in 48, a solidus followed by an index indicates covariant differentiation with respect to the A's. B h ijk as defined by (49.4) are the components of the curvature tensor of the symmetric con- nection of coefficients r#; they satisfy the identities* (49.6) B h l]k + Eh, = 0, &lik + B>iki + E>ku 0. When A*,* = 0, we have from (49.3), (49.5) and (49.6) (49.7) fl}/|* + flj*, + oS*,y + 2(oXi + OJ 4 n}, + OiiOji) = O.f If we denote by { the components of the n fields of absolutely parallel vectors and define ? by (21.9), and put (49.8) <4 = the quantities cj are scalars and (49.9) c a e b + c b e a = 0. From (49.8) it follows that (49.10) Q',|i = 0, when and only when the c's are constants. When (49.10) is satisfied, it follows from (49.7) that (49.11) fl[yti + fl/*rf + oLOyi = 0, from which and (49.8) we have the Jacobi relations (7.4). Further- more, from (49.1), (48.1) and (49.8) we have * 1927, 1, p. 55. t Cf. Einstein, 1929, 2, p. 6. 49. LINEAR CONNECTIONS. SIMPLY TRANSITIVE GROUPS 197 In consequence of the second fundamental theorem we have that are the vectors of a simply transitive group. Hence we have: [49.1] A necessary and sufficient condition that a linear connection of zero curvature determine a simply transitive group is that (49.10) be satisfied. We have seen in 48 that the coefficients Ay* and Ay* of the linear connections determined by a simply transitive group and its reciprocal are in the relation Ay* = A*y. From this and (49.2) it follows that Ay* == Fy* 12y*. If, conversely, the coefficients of two linear connections of zero curva- ture are in this relation, we have equations (49.7) and the equations obtained from the latter on replacing each fiy* by its negative. Consequently we have (49.11) and (49.12) Ofci* + Of*,* + lt,y = 0. Since B h ijk are the same for the two connections, we have fi{y* = ojy*, where fl{y* is obtained from fij,* on replacing each 2}* by its negative, from which, in consequence of (49.5), we have fl{j|* fl*| y = 0. From these equations and (49.12) we obtain (49.10) and conse- quently we have : [49.2] A necessary and sufficient condition that an asymmetric linear connection of zero curvature determine a simply transitive group is that the connection of coefficients A}* = A*/ be of zero curvature; in this case the second connection determines the simply transitive group reciprocal to the given one. When A*,* = and (49.10) hold, it follows from (49.3), (49.5) and (49.11) that (49.13) B}y* = OJftO&. Conversely, if this condition is satisfied, we have (49.11) satisfied in consequence of (49.6). Hence if (49.10) is satisfied, it follows from (49.5) and (49.3) that A*y* = 0. Accordingly we have: [49.3] A necessary and sufficient condition that an asymmetric linear connection determine a simply transitive group is that (49.10) and (49.13) be satisfied. 198 F. GEOMETRICAL PROPERTIES If we indicate by a semi-colon covariant differentiation with respect to the F's, that is, if we replace {#} by T& and the comma by a semi-colon in (47.14), we have (49.14) Consequently for a simply transitive group we have (49.15) 12^;, = 0. Conversely, if this condition and (49.13) are satisfied, we obtain (49.10) from (49.14) and (49.11). In consequence of (49.15) we have from (49.13) (49.16) Efl jk ]i = 0. 60. Geometry of the group-space. Since the parameter groups of a group are simply transitive and reciprocal (30), the results of the preceding sections may be applied to define two asymmetric linear connections and a symmetric one for the group-space. In fact, we have made use of the coefficients of the first two connections in 8, defining them by (50.1) As thus defined, the vectors A% are absolutely parallel with respect to the first connection and AS with respect to the second. Moreover the respective curvature tensors of components Lp yS and LJ3 7 $, defined in the I/s and L's by (21.17), are zero tensors, that is, (50.2) Lfa = 0, L a y > = 0. From the results of 48 it follows that L 7 = I/ 7/3 . Hence the paths for the two connections, as defined by (48.6), are the same for both and are the integral curves of the equations <> + 50. GEOMETRY OF THE GROUP-SPACE 199 where (50.4) r| Y = *(!& + it,) are the coefficients of the symmetric connection determined by the parameter groups. For each path we have where p is a determinate function of t. If we define a parameter s by i - " the preceding equations become f +<- Thus along each path a parameter s, called the affine parameter, can be chosen so that the equations of the paths are (50.5). Com- paring these equations with (47.22), we note that s is the analogue of the arc in a Riemannian space. In this section we consider the geometrical properties of the group-space S determined by the linear connections thus defined. Many of the results are due to Cartan and Schouten,* who devel- oped this theory following the announcement by the authorf of the results of 48. They introduced the terms (+)-parallelism, ( )-parallelism and (O)-parallelism for the connections with the coefficients Z/JJ 7> L% y and T% y respectively; we use this terminology. We recall that the first two of these parallelisms are absolute, but not the third. We refer to these connections as respectively the (-{-^-connection, the ( ^connection and the (0) -connection. Because of the conditions (8.4) we have that at the point of coordinates a? of S, representing the identity in G>, the vectors A" and A" coincide. Consequently the vector fields A" and A" throughout S may be thought of as obtained from the r vectors A(a )( = Aa(a )) by (+)-parallel and ( )-parallel displacement respectively. We have related the vectors at a general point by the equation (8.14). Comparing these equations and (39.2) we see that the vectors of the first set at a point are transformed into those of the second set by the adjoint group. * Cf. 1926, 1; Cartan, 1927, 2; Schouten, 1929, 1. t 1925, 1. 200 V. GEOMETRICAL PROPERTIES If a point of coordinates a" is transformed into a point of coor- dinates a' a by a transformation of the second parameter group, it follows from (6.7) and (9.2) that (50.6) ^ = AS(a')Al(a). The transform of the vector A & a by a transformation of the second Q la. parameter group is Af j which reduces by means of (50.6) to Aa (a') ; that is, the vectors of each vector-field A% are transformed into one another by the second parameter group. Similarly those of the second set are transformed into one another by the first parameter group. Hence we have: [50.1] The vector-fields A" and A" are transformed into them- selves by the transformations of the second and first parameter groups respectively. In 12 it was shown that the trajectories of the first parameter group are integral curves of equations (50.5), and that the equations of the path through the point of coordinates a a are (12.1). Since the paths for the two connections are the same, it follows that the equations of the above path as a trajectory of the second parameter group are expressible in the form (50.7) a' a = a" + te*A a a a + , where necessarily e*A" a (d) = *Al(a), AJ = A2~ oa The paths which are (+)-parallel to the given one form a congruence which ordinarily is not the same as the congruence of paths which are ( )-parallel to the given one. From theorem [50.1] it follows that the curves of the first congruence are transforms of one another by the second parameter group and of the second congruence by the first parameter group. A one-parameter group of G> is defined, as we have seen in 11, by taking for the a's the values given by (12.2) for assigned values of the e's; these are the equations of a path through the origin , 50. GEOMETRY OF THE GROUP-SPACE 201 the point <*o which defines the identity in G>. If 6* is any point of S not on this path, the equation (50.8) 2V = T a T b defines a trajectory of the first parameter group through 6. It is an integral curve of .da = A a a (b f }Al(a)e b Al(a) = e a AS(V). Since similar results hold for the second parameter group we have : [50.2] The trajectories of a one-parameter sub-group of the first (second) parameter group are ( + )-parallel ((-)-parallel) to the trajec- tory through the origin (oj) which determines the sub-group. Consider two ordered point-pairs a\ b { and a'*, b fi of the group- space S such that the transformations of the group satisfy the condition (50.9) T b T~ l = TvT?. When a* is the origin a* , it follows from (50.9) that IV = T b T a ,, from which and (50.8) we have that the point b' a is the transform of the point a ta by a transformation of the first parameter group. If the point b a describes a trajectory of the first parameter group, in which case we have a one-parameter sub-group of this group, the point b fa describes a trajectory of this group, and the two trajectories are (+)-parallel, as follows from theorem [50.2]. The same result holds for every point-pair a', b' in the relation (50.9) to a , &, and any two, such point-pairs are in the relation (50.9). Since the paths are the generalization of straight lines in euclidean space, we may speak of the segments ab and a'V of the paths as vectors. In the terminology of Cartan* two vectors in the relation (50.9) are equipollent of the first kind. When the two ordered point-pairs a\ b* and a", b fi are such that (50.10) T~ l T b = TjT b >, * 1927, 2; the reader is referred to this paper for a complete treatment of the subject. 202 V. GEOMETRICAL PROPERTIES it follows by reasoning similar to the above, that db and a'b f are segments of paths of the second parameter group. In this case the vectors a& and a'b' are said to be equipollent of the second kind. If each term of equation (50.9) is multiplied on the left by TV 1 and on the right by T a / we obtain T~ l T a > = T^ l T b >. Consequently we have:* [50.3] // ab and a'b' are equipollent of the first kind, aa' and bb' are equipollent of the second kind, and vice-versa. As a consequence of theorem [50.2] we have: [50.4] // two vectors are equipollent of the first or second kind, they are segments of (+)-parallel or (-)-parallel paths. Suppose that G p is a sub-group of G> and that any one parameter sub-group of the former is a linear combination (constant coefficients) of the symbols e a X a f for s = 1, , p. In terms of general parameters a a a given path of the first parameter group for G p through a point a a is defined by equations (12.1) when the e's are replaced by a suitable combination of e. The locus of all these paths through the point a a is a sub-space of p dimensions, say S p . In terms of canonical parameters the equations of the S p through the origin, say SQ P , are (cf. (12.6)) (50.11) u a = c where \l/(t) is suitably chosen, we have that the new functions " satisfy the equations (50 - I3 > f + #% - - We say that a solution a of these equations determined by initial values ? defines at each point of C the vector obtained from f o by (O)-parallel displacement of ?. If S p is a sub-space of p dimensions, the coordinates a a can be chosen so that S p is defined by (50.14) a a = (. Since similar results follow for ( )-parallelism, we have:* [50.7] A necessary and sufficient condition that the manifold of the paths through the origin in a bundle of directions of .order p represent a sub-group of order p is that the manifold be flat, and that the directions at any point (+)-parallel or (-)-parallel to tangential directions at any other point be tangential. The results of 48, 49 can be applied immediately to the group- space 8. If we denote by tip y the skew-symmetric part of L 7 , we have from (8.8) (50.16) tt% 7 From equations analogous to (49.13) we have Bfiy SS B% ya = 120 e fi^, a , from which and from (50.16), (5.7) and (40.5) we have (50.17) 8^ where g fty is thus defined. If the group G> is semi-simple, the determinant \g^\ is of rank r by Cartan's criterion. (44) and likewise the rank of \g a(i \. Conse- quently g aft serves as the fundamental tensor of a Eiemannian * Cf. Cartan and Schouten, 1926, 1, p. 813; Schouten, 1929, 1, p. 266; Carton, 1927, 2, p. 18. 206 V. GEOMETRICAL PROPERTIES metric for the group space. From (49.16) and (50.17) it follows that g<*0> t ^ 0, in which case it can be shown that the coefficients FjJ 7 are the Christoffel symbols as defined by (47.11), and consequently the curvature tensor is the Riemannian curvature tensor. Further- more Bp y is the Ricci tensor Rp y and in this case equations (50.17) are R fty = Y^g^ so that the space is an Einstein space.* Hence we have the theorem of Cartan and Schoutenif [50.8] For a semi-simple group the (0) -connection of the group-space is Riemannian, the fundamental tensor being (50.18) g a , and it is an Einstein space. If we put A a \ a = g a pA%, we have where c eae is defined by (41.8), Since these constants are skew- symmetric in all the indices, we have (50.19) A a ^ y + A aM = 0, and from (50.18) (50.20) gafiAZAt = g aa . These results are interpreted by theorem [52.6]. For any group equations (50.18), in which g ab are any constants such that the determinant \g a b\ is not zero, may be used to assign a Riemannian metric to the group-space. If we calculate the Christoffel symbols for g aft , we obtain From the above results it follows that for a semi-simple group the F's are equal to the Christoffel symbols, and consequently the Riemannian geodesies coincide with the paths. Furthermore, we have + gOi = o. * 1926, 3, pp. 113-114. 1 1926, 1, p. 810; cf. also, Schouten, 1929, 1, p. 271. 50. GEOMETRY OF THE GROUP-SPACE 207 If we multiply these equations by g $8 and sum for ft and 6 we obtain Oi, = 0, and from (50.16) it follows that c 2 = 0. In order that the Rie- mannian geodesies for any group coincide with the paths the same results must hold.* Exercises 1. Show that the components of the curvature tensor satisfy the identities R*ik = ~R h ikji Rijk + R]ki + #*/ = 0. 1926, 3, p. 21. 2. Show that /EJJ ?A . = and that /, = R^ jh is a symmetric tensor of the second order; it is called the Ricci tensor. 1926, 3, p. 21. 3. If we put Rhijk = QhiRljk, then Rhijk Rihjk = Rhtkj Rjkhi, and 1926, 3, p. 21. 4. When Rh\jk = p(ghkg%j ffhjg\k), P must be a constant, and F is called a space of constant curvature. 1926, 3, pp. 25, 83. 5. When the fundamental quadratic form of a Riemannian space is positive definite, none of the principal minors of any order of the matrices \\gij\\ and Hflf 1 ' 1 '!! are zero. 1932, 2, p. 202. 6. Equipollence of the first or second kind (50) possesses the following properties : 1 A vector equipollent to a zero vector, that is, one whose extremity coin- cides with its origin, is a zero vector; 2 Any vector is equipollent to itself; 3 If a vector is equipollent to a second vector, the latter is equipollent to the first; 4 If two vectors are equipollent, so also are their opposites; 5 Every point in space is the origin of one, and only one vector, equipollent to a given vector; 6 Two vectors equipollent to a third are equipollent to one another; 7 If ab is equipollent to a'V and be to 6V, the vector ac is equipollent to a'c'. Cartan, 1927, 2, p. 4. * Cartan and Schouten, 1926, 1. 208 V. GEOMETRICAL PROPERTIES 7. If in a space S there is defined an equipollence of vectors possessing the seven properties of Ex. 6, then S may be regarded as the group-space of a group for which the equipollence is of the first kind for the group. Carton, 1927, 2, p. 7. 8. A necessary and sufficient condition that two groups of the same order be isomorphic is that a point correspondence can be established between the group-spaces of the two groups transforming an equipollence of either kind of one space into an equipollence of either kind of the second space. Cartan, 1927, 2, p. 11. 51. Groups of motions. Equations of Killing. Each trans- formation of the group of motions in euclidean 3-space preserves lengths and angles, and thus leaves invariant the metric properties of the space. We generalize this conception and say that a Rie- mannian space admits a group of motions into itself, when each transformation leaves invariant the metric properties of the space. In order to obtain the conditions for a group of motions, we consider infinitesimal transformations. If a V n with the fundamental form (51.1) is subjected to an infinitesimal transformation (51.2) x ri = x i + {''SI, we have (51.3) MX* = ddx i = , dx j dx k Consequently in order that S

of motions is n 1, the minimum invariant varieties for ordinary points are hypersur- faces, that is, varieties of order n 1. If we choose the coordinate system so that these hypersurfaces are the coordinate spaces x 1 = const., and the other coordinate spaces are orthogonal to them, then gu ^ and g\j = for j = 2, , n. In this case o = and equations (51.4) for i = j = 1 reduce to * 7 = for dx* k = 2, , n. Since the rank of the matrix ||&|| is n 1, we have that g\i is a function of z 1 alone. By choosing x l suitably we have (51.10) holding and consequently we have: [51.8] When the minimum invariant varieties of a group G r of motions are hypersurfaces, they are geodesically parallel. 52. Translations. A particular sub-class of motions are those for which the trajectory of each point is a geodesic. They are called translations and are an evident generalization of translations in euclidean space. If the coordinates are chosen so that the components of the translation have the values (51.6), then the trajectories (51.8) are to be geodesies. In this case we have from equations (47.21) and (47.11) that In consequence of (51.7) this condition reduces to (52.2) gfl& = (j, k = 2, , n). From (47.10) it follows that the determinant of these equations is equal to gn/g, where g is the determinant |gr * The cosine of the angle between the vector f and C is equal to ,-7- as For this to be a constant we must have + frrr '' ds ) dx k ds ds dx>'dx k Since this condition must be satisfied for every geodesic, we must have (51.5) and, since (* is a unit or null vector, the motion is a trans- lation. As a consequence of theorems [52.2] and [52.3] we have: [52.4] The trajectories of two G\s of translations meet under constant angle.'f * In the case of a null vector this is taken as the definition of the cosine of the angle. Cf. 1926, 3, p. 38. t Bianchi, 1918, 1, p. 501. 53. DETERMINATION OF GROUPS OF MOTIONS 213 From theorems [51.5], [52.1] and [52.4] we obtain: [52.5] // & for a = 1, , p are the vectors of translations, so also is c a a, where the c's are arbitrary constants. From (50.19) and (50.20) it follows that the infinitesimal transformations of the first parameter group in the group-space S of a semi-simple group are translations. Since similar results hold for the second parameter group, we have the theorem of Cartan and Schouten:* [52.6] The infinitesimal transformations of the first and second parameter groups of a semi-simple group are translations of the group-space. 63. The determination of groups of motions. Theorem [51.2] supplies a criterion for determining whether or not a given Rie- mannian space admits a one-parameter group of motions. Instead of proceeding in this manner, we consider under what conditions equations (51.4), or their equivalent (51.5), admit one or more solutions. Differentiating equation (51.5) covariantly, we obtain (53.1) &,* + ,-,,* = 0. If these equations are satisfied, so also are &>/* + Zliik + %i,kj + Zk,ij (%j,ki + *>/) = 0- From (47.16) it follows that (53.2) &,# {,-,*/ = ZhRijk, so that the preceding equations are equivalent to 2fc,/ + &(!&/ + By* + !&/) = 0. In consequence of the identities of Ex. 1, p. 207 these equations reduce to (53.3) and when these equations are satisfied, so also are (53.1) and (53.2). * 1926, l, p. 811. 214 V. GEOMETRICAL PROPERTIES The conditions of integrability of these equations obtained from the Ricci identities (cf. (47.16)) are (53.4) h(Rkij,l ~~ Rhi,k) + Zi,hRjkl 0. If we write (53.3) in the form (53.5) and observe that by definition (53.6) = &{?,} + &,,, we see that the solution of (53.3) reduces to the solution of the system of equations (53.5), (53.6) in the n(n + 1) quantities i and ,-,,-, the latter being subject to the conditions (51.5); con- sequently the system is mixed, equations (51.5) being the set F Q in the terminology of 1, and (53.4) are the set FI. If (53.4) are a con- sequence of (51.5), then since there are n(n + l)/2 of the latter, the general solution of the above system involves n(n + l)/2 parame- ters. If (53.4) is not a consequence of (51.5), we differentiate (53.4) and proceed as in 1. The question of whether there is any solution, and if so how many parameters are involved reduces to the question of the consistency of a sequence of equations as in 1 (cf. Ex. 2, p. 12). Moreover, if there are p independent solutions, any solu- tion is a linear combination (constant coefficients) of these. We consider further the case where (53.4) is a consequence of (51.5). As a first consequence we have (53.7) Rkijji RUM = 0, and from the other terms of (53.4) we have ZhypffiRkij tikKlij + tfRikl $Rjkl) = 0, from which because of (51.5) it follows that (53.8) biKkn oiKiij bkRuj + &kR*ij + tfRiki SjRiki diRjki + SiRfii = 0. 53. DETERMINATION OF GROUPS OF MOTIONS 215 Contracting for I and p, we have in consequence of Exs. 1, 2, p. 207 (53.9) / n 1 Multiplying by g h i and summing for h, we have (cf. Ex. 3, p. 207) (53.10) fii/ = - -(gjiRik - guR,k). n 1 Multiply (53.10) by g ki and sum for k and i; the result reduces to so that from (53.10) we have R (53.11) R mj = __ (g/tgit ~ gg,*)i where JK = g* ; 72,-/. Hence F n is a space of constant curvature (cf. Ex. 4, p. 207). From (53.9) we have which values satisfy (53.8) identically, and also (53.7) in conse- quence of (47.15). Hence we have: [53.1] When and only when V n is a space of constant curvature, the equations of Killing admit solutions involving n(n + l)/2 parameters; in all other cases there are fewer parameters. We shall show that, if Xif and X z f are the symbols of two one- parameter groups of motions, so also is (Xt, Xi)f. If we denote by ^ the vector of the latter, we have /'KQ 10N H >*^ 2 t*^* 1 *' t*^ (53.12) * = & - & = 51 {2,* - {2?!,*- If we put & = g + gij&imp = ( . . i Vi J = 1, ' ' ' ,n where A/ TOP are defined by (21.17). When q = r and r ^ n, we have from (21.19) that (54.5) are satisfied identically, and conse- quently (54.3), which are the only conditions in this case, are com- pletely integrable. Hence a solution is determined by arbitrary initial values of the gr's, which when n > r may be arbitrary func- tions of the variables x r+1 , . . . , z n , and we have:* [54.1] Any r-parameter continuous group in n variables such that the generic rank of the matrix |||| is r( ^ n) is the group of motions of a Riemannian space whose fundamental tensor g^ involves n(n + l)/2 arbitrary functions. Whenn = r, that is, when the group is simply transitive, the solution involves n(n + l)/2 arbitrary constants, as was shown by Bianchi.t In particular, if the group is simply transitive and Abelian, the coordinates may be chosen so that = & (13), and then from (51.4) we have that the g's are constants and therefore V n is euclidean.J We met with the above result in another form in 48; in fact (48.20) are the equations of Killing, the g's being defined by (48.14) in terms of the vectors of the reciprocal group. The determination * Cf. Fubini, 1903, 1, p. 54. 1 1897, 1, p. 291. JCf. Bianchi, 1918, 1, p. 521. 54. EQUATIONS OF KILLING of the latter involve n 2 arbitrary constants, but from the definition of the g's it follows that the latter involve only n(n + l)/2 arbitrary constants. Before proceeding with the general case when q < r, we consider intransitive Abelian groups. As shown in 13 the coordinates can be chosen so that (54.6) _ - q being the generic rank of the matrix (54.1). From (21.5) we have that the functions <> a i are zero, so that from (21.11) it follows that the functions is intransi- tive, if for particular values of x 1 , . . . , x n , say xl, the equations (54.4) are consistent and we take a set of values of gu for i = l,-'',n;Z = l,-'*,g from these equations as the initial values of gn such that ||gTi/|| is of rank q and any arbitrary initial values for g and let P be an ordinary point not in F. Consider now the V q +i consisting of infinity of V q s including V q and the V q through P; this evidently is an invariant variety of G r and in particular of the subgroup G ff leaving V q point-wise invariant. Any motion in G> induces a motion in V q +i which sends geodesies into geodesies and preserves angles and distances (51). The * 1903, 1, p. 40; the proof of the second part of the theorem as here given is due to the author, cf. 1932, 2, pp. 199-201. 222 V. GEOMETRICAL PROPERTIES point P is on a geodesic in V q +\ normal to Fj. This geodesic remains fixed under the group induced in V q +\ by G a1 because angles are preserved and geodesies go into geodesies. Hence P remains fixed, because distances are preserved. Since P is any point, G ff must be the identity and the first part of the theorem is proved. In order to prove the second part of the theorem we make use of the results of 54 and the formulas of 21, noting in particular that the coordinates x { are those for which (21.8) hold. We seek a non- singular transformation of coordinates (55.1) x l = v l (x' 1 , , z' n ), x" = x' i = i, , = x 9 the inverse of (55.1). From the equations (48.2) and (21.8) we have h = 0, and Assuming that the new coordinate system has the desired property and differentiating the first set of (55.3) with respect to x' ff , on making use of (21.13), the second set of (55.3) and the fact that the determinant |^i| is not zero, we obtain / ,N dx * (55 - 4) ^ For the transformation (55.1) we have = dx m dx' "' dx m Consequently if we multiply (55.4) by ^ and sum for m we obtain * In these and subsequent equations h, I, m, p, X, /* take the values 1 to q; 's are not independent. By Ex. 5 p. 207 this determinant cannot be zero, if the fundamental form is positive definite. Conse- quently the jacobian dx l is not zero in this case, and accordingly a non-singular transformation is defined by x't = ^, x'* = x 9 (I = 1, , g; s = q + 1, - , n). In this new coordinate system (dropping primes) we have from the equations (21.11) for

2 3/2 (53). Hence the group must be transitive, and the sub-group of stability (18, 20) of any * Cf. Bianchi, 1918, 1, p. 510. t Cf. 1909, 1, p. 155. t Cf. Bianchi, 1918, 1, p. 542. 58. MOTIONS IN A LINEARLY CONNECTED MANIFOLD 229 point P is of order 5 3 = 2. If there were such a 2, the points at a constant geodesic distance from P would constitute a minimum invariant variety, and thus we should have a family of geodesically parallel invariant varieties. This is the case just considered, and from the form of the transformations (56.4) and (56.7) it follows that all the transformations of such a G 2 are of order zero (18), and consequently there cannot be an invariant point.* We are now in position to prove the following theorem due to Fubini.t [57.11 A V n for n > 2 cannot admit a complete group of motions of order %n(n + 1) 1. We prove this theorem by induction, assuming it to hold for a V n -i- If a V n admits a G r with r = %n(n + !) !, it must be transitive; otherwise by theorem [55.1] a variety of order n 1, or less, would admit a group of this order, which is impossible since a F n _i can admit at most a group of order n(n l)/2. If G> is transitive, there is a sub-group of order leaving a point P fixed, which is a group of motions of < l F n _i's, the loci of points at constant geodesic distance from P . But this is contrary to the assumption that the theorem holds for a V n -\. Since we have shown that the theorem holds for a F 3 , the proof is complete. 58. Motions in a linearly connected manifold. We have remarked in 50 that, if a point of coordinates a a in the group- space S of a group G> is transformed into a point of coordinates a' a by a transformation of the second parameter group of G>, we have equations (50.6), that is, da' a , /pro * \ A a /^f\ A //*\ Differentiating these equations with respect to a 7 , we can reduce the resulting equations by means of (50.1) and (58.1) to * Cf. Bianchi, 1918, 1, p. 540. 1 1903, 1, p. 54. 230 F. GEOMETRICAL PROPERTIES These equations are similar to (48.3) but in this case (58.3) L't(a') = Ll(a'), that is, the L"s are the same functions of the a"s as the L's are of the a's. Consequently the geometry of S in the neighborhood of each of the two points as determined by the (+)-connection is the same. Accordingly we say that each transformation of the second parameter group determines an automorphism of the space S. If the coefficients A}* of any linear connection are such that there exists a solution of equations r) 2 r' /eo A\ (58 ' 4) we say that the solution determines an automorphism in V n . A solution of these equations transforms paths into paths as follows from (48.6). When such a solution involves one or more arbitrary continuous parameters, we say that the space admits a motion in the sense that paths are carried continuously into paths as in the case of a motion in a Riemannian space (cf. [51.3]). Thus the group-space S admits the second parameter group as a group of motions. Furthermore the trajectories of this group are the paths of S, and thus the motions are a generalization of translations in a Riemannian space (cf. 52). In particular, when G r is simple or semi-simple, by theorem [52.6] we have that these motions are translations in the Riemannian sense. In like manner it can be shown that the first parameter group establishes an automorphism of S for the ( )-connection, whose coefficients are L% y = L" ft (cf. 48, 50). Hence we have: [58.1] The first and second parameter groups of a G r are groups of translations of the group-space S in the sense that they determine automorphisms of the geometry of S as defined by the ( ^connection and (-{-^-connection respectively of S. We remark that in accordance with Ex. 2, p. 216 a group of motions in a Riemannian space is a group of automorphisms of the space. From (50.4) and (50.16) we have (58.5) L 7 = Tg 7 + icoUJA^l?. When these expressions are substituted in (58.2), the resulting equations are reducible in consequence of (58.1) to 58. MOTIONS IN A LINEARLY CONNECTED MANIFOLD 231 If we proceed in like manner with the equations for ( )-connection analogous to (58.2), we obtain (58.6). Hence we have: [58.2] The first and second parameter groups of a G r are groups of automorphisms of the geometry of the (Q)-connection of the group-space and they are groups of translations of the space. We turn to the general case of motions, that is, equations (58.4). Denoting by T]k and tf ik the symmetric and skew-symmetric parts of A}*, we may replace (58.4) by (58.7) ^l + rf^^.rlM^ dx j dx k dx j dx k dx l and r)r'I r)r /w * r)r' (58.8) OU*')^; ^7 = 05*(*)7T dx j dx k dx l In order that the infinitesimal transformation (58.9) x fi = + ?tt be a motion, we have on substituting in (58.7) and (58.8) and neglecting terms in the second and higher powers of 5t - ' dx j dx k dx l and ^Qin d ** 4. o d? 4. o* d? o* af * n (58.11) -^ + 12 W + fy* ~ Q jfc = 0. dx l dx j dx k dx l If the coordinate system is chosen so that {* = 6\, we have that = aa; 1 ax 1 When these conditions are satisfied, equations (58.4) are satisfied by the finite equations (51.8) of the group Gi generated by the infinitesimal transformation (58.9). Hence we have: [58.3] When a space with a linear connection admits an infinitesimal motion, it admits as a group GI of motions the group generated by the infinitesimal one; and 232 V. GEOMETRICAL PROPERTIES [58.4] A necessary and sufficient condition that a space with a linear connection admit a group G\ of motions is that there exist a coordinate system in which A** are independent of one of the coordinates, say x l ; then the curves of parameter x l are the trajectories of the motion. Indicating by a semi-colon covariant differentiation with respect to the r's, equations (58.10) may be written in the form (58.12) ^ = ?B} kh , where B] k i are defined by (49.4); these equations are a generalization of (53.3) (cf. Ex. 3, p. 216). Because of (49.6) the Ricci identities (cf. (47.16)) are satisfied identically. Differentiating (58.12) covariantly with respect to x l and substituting in the Ricci identity we obtain (58.13) ?B} kl . h - &# + &Bi + &B} W + ^B] kh = 0, in consequence of (49.6) and of the generalized identities of Bianchi (58.14) B] k i th + B] lhlk + B'w = 0.* Equations (58.11) may be written in the form (58.15) !&.,{* + QU& + <&& - 4l;* = 0. The problem of finding one-parameter groups of motions reduces accordingly to the solution of the system of equations *!! - tf w dxi " *'* " * h " (58.16) * in the n(n + 1) quantities f* and f! y , subject to the conditions (58.15), which is the set F Q of the mixed system consisting of the latter and (58.16). Let l and 2 be two solutions of these equations, then (58.17) ? = ^ - ^ = M, _ &<, are the components of the vector of the commutator (JCi, JC 8 )/. * Cf. 1927 1, p. 56. 58. MOTIONS IN A LINEARLY CONNECTED MANIFOLD 233 We shall show that this also is a solution. From (58.17) and (58.12) we have, in consequence of (49.6), (58.18) & - tf;rfM - ,fcV + &B* M . Differentiating these equations covariantly with respect to x k and in the reduction making use of (58.13) and (58.14), we find that (58.12) is satisfied. From (58.15) we have by covariant differentiation and reduction by means of (58.12) If we multiply by 2 and sum for Z, then subtract the equation obtained by interchanging the sub-scripts 1 and 2, and in the result make use of the Ricci identity the resulting equations are equivalent to the conditions that and ;/ given by (58.17) and (58.18) satisfy (58.15). Accordingly we have for motions in a space with a linear connection theorems [53.2] and [53.3]. If we substitute the expressions (58.9) in (58.4), we obtain the equations resulting from (58.10), when the T's are replaced by the A's. In consequence of (49.1) the resulting equations may be written in the form (58.19) ({{, + 2l4f% - ?Aj tt = 0, where, as in 49, a solidus followed by a subscript indicates covariant differentiation with respect to the A's. Equations (58.11) may be written (58.20) + n}4* - ojik^ = o. In order that the mixed system (58.15) and (58.16) in the n(n + 1) quantities * and !,- admit a solution involving n(n + 1) parameters, it is necessary that Oji = 0, B] kt = 0, that is, the connection is symmetric and of zero curvature. In this case there exists a coordinate system for which F** = 0,* 1 1927, 1, p. 81. 234 V. GEOMETRICAL PROPERTIES and consequently in this coordinate system equations (58.12) reduce to ; : = 0, so that dx'dx k a { + aV, and the group (? n of motions; the vectors (* of the latter are linear combinations (constant coefficients) of the vectors of the given group multiplied by a scalar. Knebelman, 1930, 3, p. 281. 7. If a Riemannian space V n admits an intransitive group of motions G r , there exist n r functionally independent spaces conformal to V n admitting the same group of motions. Knebelman, 1930, 3, p. 282. 8. A necessary and sufficient condition that a space with a symmetric connection be the space of a group is that its group of automorphisms admit a simply transitive sub-group of translations. Cartan, 1927, 2, p. 95, 9. Show that, if * is a set of solutions of (58.12), the transformations of the group G\ with the symbol *p,- carry paths into paths. Eisenhart and Knebelman, 1927, 3, p. 42. 10. The fundamental form whose coefficients are defined by (48.14) is positive definite. If we put where each e a has an assigned value which is plus or minus one, the funda- mental form may be made indefinite. The above expressions satisfy (47.10) and from them follow 0 and that for the vectors f J and f J of Ex. 15, p. 139 ft = ftA*, h - KA f where ftrj = r2f2 = *& Show also that ij-a JL / 2 \ .a v o * o /i I $ 7 i * /3 c , a I I A /37 = ~ ? ^ = ~2 \ ^ ^ "" ^ V^ a/57 / and Consequently for the connections of coefficients A$ y and A0 the hypersurfaces f(x) = const, and their tangential uC hyperplanes are transformed into the hypersurfaces f(x) const. and their tangential I hyperplanes, where (59.7) /(x) = In this case p> may be called the components of the covariant normal to these hypersurfaces. It is evident that, if any two hypersurfaces are tangent at a point P, their transforms are tangent at F, the transform of P. We consider a generalization of the transformations (59.1) and (59.5) of the form (59.8) * i - f | ;*; >*;; ?" > *-> Pi = ti(x l , , x n ;pi, , Pn) in the 2n variables x* and p ly such that equations (59.4) hold, that is (59.9) t'iri**' + ir \dx l dpi for arbitrary values of the differentials dx i and dp^ A transforma- tion (59.8) satisfying this condition is called a homogeneous contact transformation; the significance of this term will appear later.* From (59.9) it follows that a necessary and sufficient condition for such a transformation is that & and ^ t - satisfy the equations dip (59.10) t dx pi From the second set of these equations it follows that the rank of the jacobian must be less than n. If the rank is zero, we have the case (59.1) and (59.5). If the rank is n r, on the elimination of the p's from the first set of equations (59.8) we obtain r inde- pendent equations (59.11) F a (&, ,*; * l , -,*)= (a = 1, - , r). * The following presentation of this subject is substantially the same as made by the author several years ago, 1929, 4. 240 VI. CONTACT TRANSFORMATIONS From (59.11) it follows that the differentials at a transformed point corresponding to differentials at a point P(x*) are related by the system of equations \pi *\ET (59.12) -dx< + dx* = 0. dx 1 dx j Since the conditions (59.4) must be satisfied as a consequence of (59.12), we must have (59.13) pi = p^, Pi = -p<*^, dx 1 dx 1 where the p's are parameters such that, when these expressions for Pi are substituted in the first set of equations (59.8), the latter are satisfied identically because of (59.11). From the second set of equations (59.13) it follows that the rank of the jacobian matrix dF a ~ must be r. Consequently r of these equations can be solved dx 1 for the p's, which as thus obtained are linear functions of r of the p's, and when these are substituted in the remaining n r equa- tions, we obtain n r equations involving x\ x { and p t , the latter entering linearly and homogeneously. Since on solving these new equations and (59.11) for the x's, we obtain the first set of (59.8), it follows that & are homogeneous of degree zero in the p's, and consequently (59.14) pi = 0. dpi Also from the first set of equations (59.13) we have that the func- tions \f/i(= pi) are homogeneous of degree one in the p's. In order that the first set of equations (59.10) admit a unique solution in the ^'s, it is necessary and sufficient that the jacobian of the t dpi ^ d

= (t (59.16) dx>' = 0, where the X's are so defined. From the first of these sets of equa- tions we have dp h ' " dx l Jdtf ' Hence because of (59.15) the conditions of integrability of the first set of equations (59.16) are Multiplying by p/, summing for j and noting that from the second set of (59.16) we have _ n l -. i \^ = ~ ,^; u ; ^^ Pi ^ A u > to f d^ we find that X a = \ hi , that is, X* 7 is symmetric in the indices. From (59.16) we have d

dx i Since \ i] is symmetric in i and j, we have the identities (59.17) d Pi Conversely, if ^* are any functions of x i and p* satisfying (59.14), (59.15) and (59.17) and ^ are functions defined by the first set of (59.10), then these functions are such that 242 VI. CONTACT TRANSFORMATIONS Because of (59.15) it follows that these functions fa satisfy the second set of (59.10). Hence we have: [59.1] A necessary and sufficient condition that a set of functions 's with respect to the x's be of rank n and that (59.17) be satisfied.* 60. Geometrical properties of homogeneous contact trans- formations. We consider now the geometrical significance of equations (59.11) and begin with the case when there is a single equation (60.1) F(&, , x n ; x l , - , z n ) = arising from the elimination of the p's from the first set of equations (59.8). In this case a point P(x l ) is transformed into the points of the hypersurface 2, whose equation is given by (60.1) when the x's are given the values at P. Now equations (59.13) become dF dF (60.2) ft = p-, Pt = -p-, Each choice of p, in (59.8) determines an elemental F n ~i at P, and for this choice there is a determined point P on the hypersurface 2. Moreover, from the first of (60.2) it is seen that the transform of the vector p t is the covariant normal to the hypersurface 2 at P. Also if the x's in (60.1) are given the values at P, this equation in the x's defines a hypersurface 2, all of whose points transform into P. In particular, it follows from the second of (60.2) that the vector p chosen at P is the covariant normal to 2 at P. Consider a hypersurface S, whose equation is (60.3) f(x\ - , *) = 0. To each of its points there corresponds a hypersurface 2 by means of (60.1). When, in particular, we put p> = - in the first of (59.8) these equations define a hypersurface S, the x's being defined in * This theorem was established in a different manner by Lie-Engel, 1888, 1, vol. 2, p. 137. 60. GEOMETRICAL PROPERTIES 243 terms of the x's as parameters, which are subject to the condition (60.3). Thus each point P(x i ) of S is transformed into a definite point P(x i ) of S. A displacement on S is given by & - ( d 1 in (59.11). From (59.13) it follows that the jacobian matrix of the F's with respect to the x's is necessarily of rank r, and likewise the matrix of the F'a with respect to the x's. Geometrically this means that a pointP (z*) is transformed into the points of a space of n r dimensions, say 2rj-r, and that a point P(x l ) is the transform of the points of a space of n r dimensions, say S n _ r .* Each choice of pi at a point P determines an elemental V n -i at P, and for this choice there is a unique point P of 2 n _ r defined by the first set of equations (59.8). ftp Any displacement in S w - r at P satisfies the conditions -dx 1 = dx % and consequently p t dz' = 0, as follows from (59.13). Hence pi are the components of a covariant normal to S n _ r , in the sense that a covariant vector X in a V n which satisfies the condition Xete* = for all displacements at a point in a sub-space of V n is called a * These observations are consequences of the fundamental theorem on the existence of solutions of a set of equations involving implicit functions, cf. Fine, 1927, 4, p. 253. 244 VI. CONTACT TRANSFORMATIONS covariant normal to the sub-space at the point. When pi are chosen, ' are determined as we have seen and then the correspond- ing Sn_r, and from the second set of (59.13) it is seen that pi are the components of a covariant normal to this S n -r at P. As in the case when r = 1, each point of a hypersurface (60.3) is transformed into a S n _ r and each of these varieties is tangent to the hypersurface whose equation is obtained, when we replace p by - in the first of (59.8) and eliminate the x'a from these equations (JX and (60.3). We call the space S M -r defined by (59.11) for a point the funda- mental variety of the transformation for the point. Also we say that two sub-spaces of orders p and q respectively (p ^ q) of a space V n are tangent at a point P, if every covariant normal to the former at P is a covariant normal to the latter at P. Accordingly we have: [60.1] When the points of a hypersurface and all the elements at these points undergo a homogeneous contact transformation, the fundamental varieties of the transformation for the points are tangent to the hyper- surface which is the transform of the points for elements tangential to the given hypersurface; and the transformed elements are tangential to the hypersurface. As a corollary of this theorem we have that, if two hypersurfaces are tangent at a point, that is, have the same covariant normal at the point, the envelopes of the two sets of varieties S n - r for the points of the two given hypersurfaces will be tangent at the point which is the transform of the given point. The foregoing results justify the term contact transformations. Consider a sub-space S n -j> of n p dimensions, defined by the equations (60.4) f,(x\ - , *) = (a = 1, , p). With each point P(x ) of 5 n ~ P there is an associated "S n -r. If in the first set of equations (59.8) we put (60.5) Pi = u' d ., where the U'B are parameters, we obtain the coordinates of a point P(x i ) of 2- r . Since the functions * and \l/i satisfy the identities (61.12) {*><, **} = {*,, *,} = 0, {*,, dp, that is, C is homogeneous of first degree in the p's; this follows at once from (62.3), since (62.1) defines a homogeneous contact trans- formation. Differentiating (62.5) with respect to x i and p,-, we obtain respectively dC (62 ' 6) 5* - which are equivalent to (62.2) because of (62.4). Conversely, if C is any function of the x's and p's homogeneous of first degree in the p's t and we take * and 17,- as defined by (62.4), the conditions (62.2) are satisfied, because (62.6) are a consequence of (62.5). Accordingly we have:* * Lie-Engel, 1888, 1, vol. 2, p. 263. 252 VI. CONTACT TRANSFORMATIONS [62.1] Every infinitesimal homogeneous contact transformation is defined by equations of the form (62.7) *' = * + fit, p t = p. - ^St, dpi ox* where C is homogeneous of first degree in the p's; moreover, any such function C determines an infinitesimal homogeneous contact trans- formation. The function C is called the characteristic function of the transforma- tion. If we form the differentials of (62.7), we obtain dx* = dx { + I ~d xi + ( 3 2 C . 8 2 C \ api == api ( .ax ~T~ ! apk Jot. Equations (62.7) and (62.8) define the extended infinitesimal trans- formation of the elements x*, p, dx 1 and dp*. For this extended transformation p v dx { is invariant in consequence of (62.6). Hence in accordance with the general theory of continuous groups the quantity pidx { is invariant under the finite group Gi generated by the extended infinitesimal transformation, and the finite equations* of Gi are given by the integrals of the equations (62 . 9) ** = *, ** = _ , dt dpi dt dx i say (62.10) & = ^(x; p; 0, ft = iM&; P; 0, and their differentials. Consequently (62.10) define a one-param- eter group of contact transformations. From the foregoing considerations we have : [62.2] The most general one-parameter group of homogeneous contact transformations is given by the solution of equations (62.9) in which C is an arbitrary analytic function of the x's and p's which is homogeneous of degree one in the latter. * Cf. Lie-Engel> 1888, 1, vol. 2, p. 212. 62. INFINITESIMAL HOMOGENEOUS TRANSFORMATIONS 253 Any integral of (62.9) satisfies the condition C = const. This means that in the 2n-dimensional space of coordinates x i and p* each integral curve, or trajectory, lies in one of the hypersurfaces C = h. When in the right-hand members of (62.10), the quantities x i and pi are given particular values, these equations define such an integral curve, and the hypersurface is determined by the value of C for the particular point. Since C is homogeneous of degree one in the p's,

' dft- ~~ dft- 6V* If C' is the transform of C, we have V ^c _ v ac _ /6V " \dx* dpj "" 6^J ^/ dx 7 * / V a^j _ V w*\ dC^ \dx> dp, dp,- dx'J dpi dC' In consequence of (61.13) this reduces to p In like manner the dp* right-hand member of the second set of equations is equal to M - dx' Hence we have: [62.3] A group Gi of homogeneous contact transformations is trans- formed into another group Gi by any homogeneous contact transforma- tion and the equations of the new group are integrals of the equations ^ = *2L $ - _^ dt " dp-' dt ~ dx'*' where C r is the transform of the characteristic function of the given group. From (62.1) and (62.4) it follows that the symbol Af of an infinitesimal homogeneous contact transformation expressed in terms of the characteristic function C is given by <62.,4) 4, _(J +...,.. Exercises 1. The homogeneous contact transformation determined (61) by the equation x* -f 1 = has the equations $i as -- -, JH Pi** 2. If F(x\ p) and G(x* t p) are transformed by a homogeneous contact trans- formation into F(x; p) and G(x; p), then p< = p.~ and (F, G) = (?, G). dp* dpi 3. If u is a solution of the equation (F, f) =0 and v is any function such that (F, v) is a function of F, then (u, v) is a solution. 63. NON-HOMOGENEOUS CONTACT TRANSFORMATIONS 255 4. A necessary and sufficient condition that the two one-parameter groups of homogeneous contact transformations determined by functions Ci and 2 be commutative is that (Ci, C 2 ) = 0. 5. If r(^n) independent (constant coefficients) infinitesimal homogeneous contact transformations with the symbols (<7, /) for a = 1, , r are such that (C a , Cft) =0, there exists a homogeneous contact transformation in vari- df ables ' and p* in terms of which the symbols are (C , /) = ~' ojC Lie-Engel, 1888, 1, vol. 2, p. 267. 6. A necessary and sufficient condition that r independent (constant coeffi- cients) characteristic functions Ci, . . . , C r determine a group G r of homo- geneous contact transformations is that (C , C 6 ) = Cat Ce, where the c's are constants satisfying (7.3) and (7.4). Lie-Engd, 1888, vol. 2, p. 300. 7. Show that the functions Ci - Pi + p 2 , C 2 = x*pi + x*p 2 , CB = Wpi + Wpt determine a G* of homogeneous contact transformations. 8. Show that the functions determine a 6r 3 of homogeneous contact transformation with the same constants of structure as the 3 of Ex. 7, and that it is transformed into the latter by the transformation ix* \/p 2 _ __ V Pi Lie-Engel, 1888, 1, vol. 2, p. 312. 63. Non -homogeneous contact transformations. In his develop- ment of the theory of contact transformations Lie considers first transformations of the form (63.1) Z = (z, x l , , z n ; pi, , Pn), X i = ?<(*, re; p) f P, = j,t(z, x] p), such that the equation (63.2) dZ - P i dX i = is satisfied whenever (63.3) dz - p^ = 0. 256 VI. CONTACT TRANSFORMATIONS Consequently the analytical problem reduces to the determination of functions ^>, *(x l , , x n ; p 2 , , p n ), (63.5) fa = }i(x\ , x n ; p 2 , - - , p n ), where the p's and ^'s are such that the relation (63.6) p4x { = -dx l + p a dx a (a = 2, - , n) holds for independent values of the differentials dx i and dp a . The resulting conditions are (63.7) *g--l, *g-P ^ = ( = 2,...,n). Comparing these equations with (59.10), we see that if we take any homogeneous transformation and in its equations put pi = 1, we obtain a non-homogeneous transformation. Suppose conversely that we have a set of functions satisfying (63.7), and in the ^>'s we put (63.8) p a = - 63. NON-HOMOGENEOUS CONTACT TRANSFORMATIONS 257 and denote the resulting functions by , Pn satisfying these equations and such that the jacobian of the = (-1) ax 1 d

"& and \l/"& we have in the notation (63.11) {*<) IQ\ (63.13) / Equations (63.11) and (63.13) follow directly also, if we solve the d\I/j d dpi and substitute in (61.13). These results apply to the transformation in the form (63.5). In order to obtain them for (63.1) and (63.4), we let i in (63.5) take the values 0, 1, . . . , n, and put ^ = Z, x = 2, p Q = 1, p = l/^o and P a = ^a/Vo for a = 1, , n. In equations (63.11) and (63.13) now k and I take the values 0, 1, . . . , n. Making use of these equations, we obtain from (61.13) for a contact transformation (63.1) the following: [Z, X] = [X a , X ft ] = [P a , P ft ] = 0, [P a , X*} = (a * j8), (63.14) (P a , X] = p, [P ttf Z] = P P a (a = 1, , n). In these equations the parentheses are defined by (63.11) with a taking the values 1, . . . , n.* If the rank of the jacobian matrix for equations (63.5) is n r, on the elimination of the p's from the first n of these equations we obtain equations F.(x l , , x; Z 1 , , x n ) = (* of the transformation requires the solution for the x's of the equations F 9 = and (61.4) in which pi = 1. In so doing it is understood that the variables are num- dF bered in such a way that the determinant ; for l (x 2 , , x n ; p 2 , , p n ) and the functions

Pn satisfying (64.2) and

a , say F(if> 2 y . . . , v? n ). In this case we have from (64.1) n = ' so that the transformation is essentially of the homogeneous type. We remark also that for the restricted case equations (63.13) become d\b (64.7) (* al ^) = * - p, > (^, ^ a ) = ?, (fc, ^) = 0. From these equations it follows that theorem [61.5] holds for a restricted contact transformation. The jacobian of

is n 1 * Cf. Jaeobi, 1837, 1, p. 67. 264 VI. CONTACT TRANSFORMATIONS dH dC (65.2) - C , dpi dpi dp t dpj dp,dpj From the second set of these equations it follows that the hessian of H with respect to the p'a is equal to* (65.3) C dC dC dpi dC dp n dpi dpi dpidp n dC d*C d*C dp dpidp n dpi If we multiply the last n rows of this determinant by p\ y . . . , p n respectively and add the resulting terms of all but the last row to the corresponding terms of the last row, the terms of the latter become C, 0, . . . , in consequence of (62.5) and (62.6). Consequently a necessary condition that the hessian of H with respect to the p's be of rank n is that the determinant formed from (65.3) by omitting the first column and last row be of rank n. If the columns of this determinant be multiplied by pi, . . . , p n respectively and the terms of the first n 1 columns be added to the last, the terms of the latter become C, 0, . . . , 0. Hence if this determinant is to be of rank n it is necessary that the determinant for dj ft = 1, , n 1 be of rank n 1. If we apply the above process to (65.3) using any one of the last n rows as we used the last, we find that it is necessary that every minor of order n 1 of the hessian of C with respect to the p's be of rank n 1. But this is a consequence of one minor being of rank n 1. For, if fora, = 1, , n 1 is of rank n 1, it follows from the equations (i = 1, , n; a = 1, , n - 1) * Cf. Fine, 1905, 1, p. 605. 5. TRANSFORMATIONS OF MAXIMUM RANK 265 that each of the determinants of order n 1 of the matrix is of rank n 1. When then we apply the same reasoning to the 2 C equations p v - = for a equal to n and any n 2 of the num- dp l dp ff bers 1, , n 1, we are led to the conclusion that, if the hessian of C with respect to the p's is of rank n 1, every minor of order n 1 is of rank n 1. Accordingly we have: [65.1] When and only when the rank of the hessian of C with respect to the p's is n l t the rank of the hessian of H with respect to the p's is n. Without loss of generality we take (62) (65.4) C = 1. Then equations (62.9) may be written because of (65.1) in the form (65.5) *' - **, *< - -** v ' dt dp t dt dx* If we assume that C satisfies the condition of the preceding theorem, the hessian of H with respect to the p's is of rank n and consequently the first set of equations (65.5) can be solved for the p's as functions dx i of the x's and x's, where x l = Since H is homogeneous of at degree two in the p's, it follows that the solutions p are homogeneous of first degree in the x's. When these expressions for the p's are substituted in H, we get a function 3C of the x's and x's } which is homogeneous of degree two in the x's. Thus we have (65.6) 3C(z; x) = H(x\ p). Differentiating 3C with respect to x\ we have in consequence of (65.5) Hence by Euler's theorem d3C dv (65.8) 25C = = z'z*- From this equation and (65.7) we have 266 VI. CONTACT TRANSFORMATIONS dJC I/ dpj\ 1 IdW M=A Pi + 'M) = ~2 Pi + 2W' and consequently d3C (65.9) P< = - From (65.6), (65.5) and (65.8) we have _ , _ ..j __ dx i ~~ dx i dpj dx i ~~ dx { X dx i ~~ do;' dx { Consequently s+s-* If we denote by 6 the function of the x's and x's when the expressions for p t from the solution of (65.5), that is, (65.9), are substituted in C, we have (65.11) 3C = the latter being a consequence of (65.4). From (65.9) we have ae (65.12) p, = - and from (65.10) dC dC (65.13) + = 0. v ' dx* dx l When now the expressions (65.12) are substituted in the second set of equations (62.9), we have in consequence of (65.13) Thus in the quantities x { and x i equations (65.12) and (65.14) replace (62.9). Conversely, if we have any function 6 of the x's and 's, homo- geneous of degree one in the 's, such that (65.15) G(x; x) = 1, and the rank of the hessian of 6 with respect to the x's is n 1, equations (65.9) can be solved for the x'a as homogeneous functions of degree one in the p's, and by repeating the above processes we 66. GEOMETRICAL PROPERTIES. WAVES 267 obtain the first set of equations (62.9), and the second set follow from (65.12) and (65.14). If we take a particular point P(x { } of our space, the first set of equations (62.7) defines a near-by point, which varies for each choice of the p's subject to the condition (65.4). The equation of the locus of this point is obtained on the elimination of the p'& from the first set of (62.7) and (65.4). If the hessian of C with respect to the p's is of rank n 1, this elimination can be accom- plished by the process which led to (65.11). Consequently, if y i are coordinates with P as origin so that y* = x { #*, we have from (62.7) and (62.9) y i = x { dt and hence the equation of the locus is (65.16) e(x; y) = , since (3 is homogeneous of the first degree in the x's. Also from this property of 6 and from (65.12) it follows that Consequently pi are the components of the covariant normal to the hypersurface (65.16), which we call the elementary hypersurface of the transformation for the point P. If X 1 are current coordinates, the tangential hyperplane at a point is defined by Pi (X< - 2/<) = 0, or (65.18) piX* = w, where in consequence of (62.7) and (62.5) (65.19) w = p> it = CSt. dpi Consequently (65.18) with w given by (65.19) is the tangential equation of the elementary hypersurface.* 66. Geometrical properties of continuous groups of maximum rank. Waves. Consider the finite equations (66.1) ^= * Cf. Vessiot, 1906, 1, p. 262; also, Levi-Civita and Amaldi, 1927, 6, pp. 452, 462. 268 VI. CONTACT TRANSFORMATIONS of a group Gi of homogeneous contact transformations for which the hessian of the characteristic function C with respect to the p's is of rank n 1. When the functions where a = 1, ,n 1, da with respect to the x's is not zero. Hence the equations (66.7) i's and ^'s constitute a solution of equations (62.9). In fact, if we substitute (66.9) in (66.7) and (66.8) and differentiate the resulting identities with respect to t, we obtain /AA-m ***** 1 n dV d*' n (66.11) 1 = 0, 7 = 0. dx l dt da a dx* dt Also if we substitute the function

a? + *5ft From the first set of these equations, (66.11) and (66.4) we obtain djp_ /dx* _ sc \ _ ay /d& ___ ^c\ 5i< \^ " dp* ) ~ f ^^ i aa a V dt "" apv " ' which are equivalent to the first set of equations (62.9), since the determinant of these equations is a non-vanishing jacobian. From this result, (66.10) and the second set of equations (66.12) we have dp, __ ay d& _ ay ac ^c eft ax i 'ax ; eft ax i ax y ap> ax f as was to be proved. If we denote by x*' and p the values of x* and pi when = 0, we have from (66.7) and (66.8) ~v(x aa and from (66.10) (66.13) a* = - a) " p ^i* (x; a)= ' There are n 1 independent equations (66.14); they define the n 1 a's as functions of the x's and p's, homogeneous of degree zero in the p's. Then a n and the 6's are given by (66.13) as functions of these quantities and from (66.9) we have the first set of (66.1). For t = we have from (66.6) and (66.10) 66. GEOMETRICAL PROPERTIES. WAVES 271 Consequently the second set of (66.1) is given by substituting the expressions for the x's, a's and b's in terms of the x'a and p's in (66.10') fr = cf foP>^' which is homogeneous of degree one in the p'a. From (66.7) and (66.13) we have (66.15) dx j dx* the X'Q being subject to the relation F = 0. If the x's are eliminated from (66.21), the first of (66.13) and F = 0, we get a relation iKa 1 , - , a"" 1 ) = a n . Consequently (66.20) is the envelope of (66.7) for the a's in this relation. Conversely, if we have any such relation, the envelope of = b dpi dp n where p is the factor of proportionality. From (66.10) and the sec- ond set of (66.12) we have , *v J . c dpi = /i : .dx' = p .' ^ ax 1 ^ dz Consequently the curves (66.23) are integrals of (62.9). In order to obtain the curves through a point P(x), we replace (66.23) by d(p(x\ a) d . as 2 as as Consider now the group G\ of contact transformations for which the characteristic function is (67.3) C = Vjffiwi. In this case we have equations (62. 13). When the trajectories are non-singular, it follows from (62.13) and (67.1) that t = s. Hence we have (*7A\ <, i (67 ' 4) * = *"' * = ~ 2 In consequence of (47.10) and the first set of (67.4) we have dx 1 (67.5) Pi = g*,- as Substituting this expression in the second set of (67.4), we obtain equations (67.2). If the form (67.1) is not definite, there are real values of the p's such that (67.6) g*p.pj = 0. As in the preceding case we obtain from (62.13) equations (67.2) with s replaced by t, and from the first set of (62.13) and (67.6) it dx { dx> follows that 0,j -- = 0, that is, the curves are minimal at at geodesies. Hence we have: [67.1] For the group G\ with the characteristic function (67.3) the non- singular trajectories are the non-minimal geodesies of the Riemannian space with the fundamental form (67.1) and t = s; and the singular trajectories are the minimal geodesies of the space. 67. APPLICATION TO GEODESIC S 275 For the case of non-minimal geodesies equation (66.6) is From the results of 66 it follows that, if Ox { d!> Consequently the form (67.1) becomes the hypersurfaees //" = const, being the geodesic hyperspheres with P(x) as center. Suppose that the x's and p's, as defined by (67.5), are subjected to any homogeneous contact transformation (59.8) of maximum rank and that C is the transform of C. Then C is homogeneous of degree one in the p's. From the results of 62 it follows that equa- tions (67.1) are transformed into (67.10) = -*'. '^ - -*. us df)i dx dx l If we define functions Tf l by then ^ i; is homogeneous of degree zero in the /Vs and it follows from the second set of equations (62.6) that (67.12) r ., rix> dx k where [jfc, /] are defined by (47.11) in terms of 0,,, since Thus the trajectories defined by (67.10) are the geodesies of the space with the generalized Riemannian metric rfs 2 = g^dx l dx 3 into which the given Riemannian space has been transformed. * 68. Application to dynamics. Consider a conservative dynami- cal system for which neither the constraints nor the potential energy V involve the time t. Suppose that there are n independent variables a- 1 ; then the kinetic energy is given by _ 1 where the g's are functions of the x's, and involve the masses of the system, and the Lagrangian function is * Spaces with a generalized Riemannian metric have been studied by Finsler, 1918, 2, and by Berwald, 1925, 6, vol. 34, p. 213. 278 VI. CONTACT TRANSFORMATIONS Substituting in the equations of Lagrange d/dL\ dL / _ i __ o V dx< ' we find that the resulting equations may be written (68.1) gx* + ^ = 0, ax* in consequence of (47.11). These equations admit the first integral (68.2) \ga^ + V = E, where E is the energy constant. If we put (68.3) Pi = gat*, x i = g<*pj, we have from equation (68.2) gt'ptft = 2(E - F). Hence if we take for the characteristic function of a group Gi of homogeneous transformations the quantity* (68 - 4) c " the equations of the transformation are /fl0rN dx { 1 .. dpt (68 ' 5) = Tf' Pi > ITs = where the parameter s is given by (cf . 67) (68.6) ds* = 2(E - V}g ii dx i dx^ The Hamiltonian function of the dynamical system is given by * This problem for the motion of a particle was discussed by Lie, 1889, 2, pp. 146-156; and the more general problem by Vessiot, 1906, 1, pp. 266-268. 68. APPLICATION TO DYNAMICS 279 // = p^ - L = -g*'p t p, + F, and consequently the Hamiltonian equations are dx { .. dpi (\ dg' k dV = n % 1n > = I ~~ ' 7, y Pn j* V o ^ , at at \2 dx l These equations are equivalent to (68.5) with the relation (68.7) J = 2(E - F) at holding along any trajectory. Thus we have the known result that the trajectories of energy E of the given system are in one-to-one correspondence with the geodesies of the space with the fundamental form (68.6). From the results of 67 it follows that, if (p is any solution of the equation (j (D (j(D //*o ON _ t y * i ~ __ o / T/T ^__ T/^ the orthogonal trajectories of the hypersurfaces

-S-7ETTT5- Comparing this with (68.9) we have (68.12) u = mv From (68.10) it follows that the spaces with metric (68.6) and ds Q are conformal. Consequently in the latter metric the trajectories through a point P(x) are orthogonal to the hypersurfaces (67.9). If we take K = E, the equation of these hypersurfaces is (68.13) f(x; x) - Er = 0. From (68.7) it follows that along any trajectory the relation between the time t of the particle and r of the light are given by (68.14) ^ = which is in conformity with (68.12). Exercises 1. Show that ~r y = y H / -> P = p define a Gi of restricted contact transformations, each of which sends a point into a circle, and that the characteristic function is \/l + P 2 . 2. Find the contact transformation in 3-space determined by 2 +z +x + yy = 0, and show that it transforms a point into its polar plane with respect to the paraboloid x 2 -f 2/ 2 4- 2z = 0. 3. If equation (59.4) with pi = 1 is written in the form dx* - Pad** = p(dx l - Patdx") (a - 2, , n), equations (63.16) are in this case 69. FUNCTION GROUPS 281 dW "W" ~ /dW dW and the symbol of the GI generated by this infinitesimal transformation is Lie-Engel, 1888, 1, vol. 2, p. 255. 4. If an infinitesimal non-homogeneous contact transformation with the characteristic function W undergoes a general contact transformation (63.1), the characteristic function of the resulting transformation is equal to pW. Lie-Engel, 1888, 1, vol. 2, p. 277. 5. If A \f and A zf are the symbols of infinitesimal non-homogeneous contact transformations with the characteristic functions W\ and TF 2 , then (Ai, A z )f is the symbol of the transformation with the characteristic function [Wi, JFa], 6. The symbol of the G\ of restricted non-homogeneous contact transforma- tions with the characteristic function W(x; p) is (W,f) + (-*) 7. If AJ and Ai and $2 are any two functions of these r functions, it follows from the identity (69.1) (*i,*i) r a ft that ($1, $2) is a function of the set. Hence starting with a set of functions, we get by the above process a set each of which is a function of r independent ones and such that (F a , F ft ) for any two functions of the set is a function of these r independent ones. In consequence of (69.1) any function of functions of the set may be added to it and it will continue to possess the above property. Such a set of functions is said to constitute a function group of rank r, r being the rank of the jacobian of the functions of the group, and the r independent functions are a basis of the group. In conse- quence of (69.1) we have that any r independent functions of the group are a basis, and as we change these functions we change the basis but not the group. If a sub-set of functions of a function group form a function group, we say that the latter is a sub-group of the given group. Suppose that two groups of ranks r and s have p functions in com- mon; in this case it must be possible to choose the bases F a and G so that F = G ff ((T = 1, , p}. Then we must have (F,,G T ) = *w6Fi, ' ' ' ,*V) = *(+i, ...,(?,. Hence we have: * Cf. Lie-Engel, 1888, 1, vol. 2, pp. 17&-210. Also Vivanti, 1904, 1, pp. 255-264. 69. FUNCTION GROUPS 283 [69.1] // two function groups have functions in common, the latter form a sub-group of each group. Consider the system of differential equations (69.2) (F a ,f) = 0, where the F's determine a basis of a function group. From the Jacobi identity (61.15), we have (69.3) (F a , (F PI /)) - (F ff (F a , /)) = ((F a , F f ), /). Since (F aJ F ft ) is a function of the F's, it follows from (69.1) that the right-hand member of (69.3) vanishes because of (69.2) and conse- quently equations (69.2) form a complete system. Hence they admit 2n r independent solutions, f ( of the basis functions F i, . . . , F r we have 284 VI. CONTACT TRANSFORMATIONS and consequently, if $ is to be a singular function, it must be a solution of the system of r equations d$ (69.5)

0, the functions v m+l , . . . , m+q are functions of the reciprocal group. Furthermore, if any one of these functions, say > m4 " 1 , is omitted from the functions of the given group, the remaining ones determine a sub-group of the latter. Then

n) and are the functions of a group of rank 2n of which the original group is a sub-group. If q = in (69.8), we adjoin the reciprocal group and get (69.9). Hence we have: [69.6] A non-commutative function group of rank r is a sub-group of a group of rank 2n whose basis can be chosen so that the functions of the basis satisfy (69.9) identically. In consequence of theorem [64.1] and equations (64.7) we have that these functions determine a restricted non-homogeneous contact transformation. Suppose that we have two function groups of the same rank and with the same number of singular functions, and denote their respective bases by F a (x, p) and F' a (x f , p'}. Because of theorem [69.6] we have that for the former a basis can be chosen so that we have a group of rank 2n for which the functions of the basis satisfy the equations (69.9) identically. In like manner we obtain a similar group of rank 2n, whose basis functions H i, , H q (2m + q = r), in which the ^'s are of degree one, the

^ x ) = 6 X , 290 VI. CONTACT TRANSFORMATIONS where X, ju = 1, , m + q\ a, ft = 1, , w, or X, /i = 1, , m; <*, /9 = 1, , m + q, where 2m + Q = r, the a , m+q satisfies (70.4). If we take the sub-group obtained by omitting

-unnm and having no singular functions. If we add to this group its reciprocal group with basis in canonical form we have a group of rank 2n, whose basis satisfies the identities (70.5) (?, *0 = (fc, fc) = 0, (fc, ^) = d{ (i,j = 1, - - - , n). If we proceed with a group having the basis (70.4) with X, IJL = 1, , m; a, /3 = 1, , m + q, we arrive at the same result. Hence we have:* [70.5] A non-commutative homogeneous function group of rank r is a sub-group of a homogeneous group of rank 2n, whose basis can be chosen so that equations (70.5) are satisfied identically, the 1888, 1, vol. 2, p. 225. 70. HOMOGENEOUS FUNCTION GROUPS 291 singular functions of degree zero as follows from theorem [61.5]. Suppose we have two homogeneous groups of the same rank and with the same number of independent singular functions and the same number of independent singular functions of degree zero, and that the bases of corresponding groups of rank 2n are taken in the form (70.5) and similar equations in a & defined by (69.6) satisfy identically the equations the second set being a consequence of (61.15). *Lie-Engel, 1888, 1, vol. 2, p. 226. 292 VI. CONTACT TRANSFORMATIONS 7. If a p are given functions of r quantities F a such that the equations of Ex. 6 are satisfied, there exists a number n such that the functions F a are expressible in 2n variables x l t . . . , rc n ; pi, . . . , p n , they constitute a function group and satisfy (69.6) ; their determination requires the solution of a system of linear partial differential equations of the first order. Lie-Engel, 1888, 1, vol. 2, pp. 235-242. 8. Show that the functions v>o0 = c a y F yi where the c's are constants satisfying (7.3) and (7.4), meet the conditions of Exs. 6, 7 and thus define a function group. 9. Show by means of the results of 29, 39, that, if Ci, . . . , C r are the characteristic functions of a G r of homogeneous contact transformations and x* =

193 > 195 / Lagrangian function, 263, 278. Length, element of, 186. Levi-Civita, 190, 267, 294, 295. Lie. 36, 39, 54, 55-58, 66, 80, 84, 91, 106-108, 119, 122, 123, 131, 137, 145, 154, 155, 163, 164, 180, 183- 185, 242, 248-252, 255, 258-261, 278, 281, 282, 285, 287-292, 293. Lie group, 40. Linear, complex, 185; group, 43, 138, 143-150, 153, 163, 184, 234, 292; see also Connection. Matrix, equation, 142-144, 148-150; characteristic, 155; trace of a, 156, 157. Mattoli, 194, 295. Maurer, 23, 293. Maximum invariant sub-group, 122, 127-131. Metric, Riemannian, 187, 206; gener- alized, 277. Michal, 43, 295. Minimum invariant variety, 67-71, 211, 221, 223, 235. Mixed, group, 58; system of differ- ential equations, 4. Moore, 15, 294. Motion, in 2-space, 42, 80, 98, 226- 228; in 3-space, 43, 80, 228; infinitesimal, 209. Motions, group of, 57, 208-221, 230- 237; in a linearly connected manifold, 229-235, 237 ; intransi- tive, 210, 216^226, 235. 236; multiply transitive, 221; ol maxi- mum order, 215, 224, 236; simply transitive, 218, 235, 236. Multiply isomorphic groups, see Iso- morphism. 300 INDEX Normal, covariant, 239, 242, 243. Null vector, 174, 187, 212. n; t , 196-198, 205-207, 231-235, 237. Operator, linear. 6; Poisson, 6. Order of a transformation, 64, 66, 216. Orthogonal vectors, 174, 195;ennuple, 236. p, 42. p<, 6, 238, 239. Parallelism, absolute, 190. 193, 194, 199, 217, 234, 235; relative, 190, 191, 193, 199; (+)-, 199-205; (-)-, 199-205; (0)-, 199, 204, 205. Parameter, affine, 199, canonical, 46; essential, 9-12, 56. Parameter-group, 31, 32, 44, 55, 72, 93, 98, 114, 124, 125, 198-203, 213, 229-231. Parenthesis of Poisson, 250, 254, 261, 281-286, 2897-292. Partial differential equations, see Differential equations. Paths, of the group-space, 45, 47, 198-203, 206, 207; of linearly connected manifold, 193, 230, 236; parallel, 202, 203. Pfaffian equations, 93-95. Point, ordinary, 64; singular, 64. Points, equivalent, 67; invariant of, 108. Poisson, operator, 6; parenthesis of, 250, 254, 261. 282-286, 289-292. Primitive group, 80, 123. Product, of two groups, 120-122, 174, 184; of transformations, 14, 263. Projection of vector-spaces, 144, 145, 164. Protective group, general, 184; in 3-space, 123; on the line, 66, 138; of the plane, 43, 65, 66. 42. Quadratic form, 186, 207, 208, 226, 236. R, 215. #* , 188, 207, 215. R Mik , 207, 215. ft,, 206, 207, 215. Rank, of a group, 159, 160, 183, 184; of a function group, 282. Reciprocal groups, 114-117, 193, 194, 197, 218, 235; see also Function group. Relative invariant of a group, 63, 66, 67, 69, 93, 95, 107. Representation of a group, 183. Ricci, identity, 189, 232; tensor, 206, 207, 215. Riemann, 186, 293. Riemannian metric, 187, 206; space, 187-191, 206, 218, 221, 230, 235, 236; tensor, 188, 206, 207. Robertson, 138, 139, 234, 237, 296. Root-figure, 182. Root-space, 147-150, 165-172, 179. Root-vector, 177. Rotation, in 2-space, 34, 36; in 3-space, 43, 85. Scalar, 26, 30. Schouten, 182, 199, 203, 205-207, 213, 295. Schreier, 40, 294. Schur, 25, 52, 53, 183, 293, 294. Semi-group, 15, 17, 18, 21, 28, 29, 31. Semi-simple group, 173-184, 205, 206, 213, 230. Series of composition, 127-134; nor- mal, 128. Similar groups, 76. Simple group, 118, 119, 122, 123, 132, 154, 162, 173, 174, 180-182, 184, 185,230. Simply isomorphic groups, see Iso- morphism. Simply transitive group, 72, 78, 113- 115, 117, 118, 192-197, 218, 235. Singular, functions, 283, 284, 286, 287, 291, 292; trajectories, 253, 268, 272-274. Skew-symmetric indices, 32. Space, flat, 188, 193, 203-205; of constant curvature, 207, 215, 216, 227, 228, 237; see also Euclidean, group-space. Structure of a group, 28; see also Constant. Sub-group, 59-61, 153, 155, 158, 160; exceptional, 113, 118, 152, 161, 162, 183; invariant, 118-124, 129-134, 137-139, 144, 153, 154, 158, 161-163, 173, 174, 183, 184, 203; maximum invariant, 122, 127-131; of a function, 62; of a function group, 282-286, 290; of stability, 65, 83; self-conjugate, 118; C7, 167-172, 176-178. Surface of constant curvature, 227, 228, 235. Symbols of a group, 28. Symmetric linear connection, 194, 198, 233, 234, 236. Systatic group, 84, 123. Systatic variety, 84, 85. INDEX 301 System, holonomic, 263, 277; of imprimitivity, 80-85, 117, 118, 137. T., 14. Tangent hypersurfaces, 239, 243, 244; varieties, 244. Tangential hyperplane, 239, 267. Tensor, 187, 189; contravariant, 187; covariant, 187, curvature, 188, 192; fundamental, 187; Ricci, 206, 207, 215; Riemannian, 188, 206, 207. Thomas, 47, 296. Total differential equations, 1. Trace of a matrix, 156, 157. Trajectories, of a group, 35, 40, 44, 45, 253, 274; of a motion, 209, 210, 212, 235; dynamical, 279; of wave motion, 268, 272; singu- lar, 253, 268, 272-274. Transform of a transformation, 18, 109-113, 150, 155. Transformation, admits a transforma- tion, 110-113, 122; continuous, 14; infinitesimal, 36; invariant under a transformation, 110, 112; non-singular, 15; of coordinates and parameters, 26-27, 34, 35; of a vector, 140. Transformations, commutative, 110, 112, 113; product of, 14. Transitive group, 71: multiply, 72, 117, 221; simply, 72, 78, 113-115, 117, 118, 192-197, 218, 235, 236; Wold, 108. Translation, 34, 80, 83, 211-213, 217, 230, 231, 235-237. U&, V*. U&, / 51-55. Umlauf, 183, 184, 293. F n , 13. Variety, fundamental, 244; tangent, 244; see also Invariant, systatic. Vector, base, 141; contravariant, 25, 26; covariant, 27; null, 174, 187, 212; regular, 157; special, 157; transformation of, 140; unit, 140, 187, 195, 212; zero, 150, 187. Vectors, equipollent, 201-203, 207, 208; orthogonal, 174, 195; see also Parallelism. Vector-space, 140-145, 147, 153, 163, 164, 170. Vessiot, 267, 278, 294. Vivanti, 282, 294. Wave, 268, 271; -front, 271, 272. Weyl, 167, 169, 179-183, 295, 296. Whittaker, 250, 295. Wigner, 183, 296. Zero, connection, 199, 206; curvature, 195-198, 217, 233-235; displace- ment, 204; parallelism, 199; ten- sor, 194; vector, 150, 187.