GEOMETRY
OF
DIFFERENTIAL
Vo I . B l i z n i k a s
and
Z. Yu.
EQUATIONS Lupeikis
UDC 513.7:517.9
Introduction G e o m e t r i c s t r u c t u r e s on differentiable manifolds and, in p a r t i c u l a r , on jet s p a c e s (or other fibered spaces) a r e defined by v a r i o u s methods and a r e studied not only in differential g e o m e t r y , beginning with the work of R i e m a n n and Christoffel, but also in differential topology. Along with t h e s e g e o m e t r i c s t r u c t u r e s , which lead to the notion of R i e m a n n s p a c e s or other m e t r i c s p a c e s , the r e c e n t y e a r s h a v e s e e n an intensive study m a d e of v a r i o u s s p a c e s of support e l e m e n t s (in L a p t e v ' s terminology), of a l g e b r a i c s t r u c t u r e s on diff e r e n t i a b l e manifolds and other tangent bundles of finite o r d e r of c e r t a i n f i b e r e d s p a c e s (these s t r u c t u r e s a r e defined by t e n s o r fields or s y s t e m s of t e n s o r fields), etc. Only those g e o m e t r i c s t r u c t u r e s which yield connections in fibered s p a c e s a r e defined not by t e n s o r fields but by the fields of d i f f e r e n t i a l - g e o m e t r i c obj e c t s of higher o r d e r . Among the g e o m e t r i c s t r u c t u r e s defined by fields of d i f f e r e n t i a l - g e o m e t r i c objects of h i g h e r - o r d e r , an i m p o r t a n t place is occupied by those g e o m e t r i c s t r u c t u r e s which a r e a s s o c i a t e d with specific s y s t e m s of differential equations. The study of manifolds or of fibered s p a c e s with s t r u c t u r e s g e n e r a t e d by a given differential equation s y s t e m is of i n t e r e s t botl~ f r o m the local as well as the global point of view. The development of the g e o m e t r y of differential equation s y s t e m s is c l o s e l y tied in with the developm e n t of the g e o m e t r y of Riemann, F i n s l e r , C a f t a n s p a c e s and with the t h e o r y of v a r i o u s connections in f i b e r e d s p a c e s (see the s u r v e y a r t i c l e s by Bliznikas [14], Laptev [60], Laptev [64], L u m i s t e [65], Shirokov I109], and the p a p e r s by Ku [25-29], Yen [276]). H i g h e r - o r d e r connections (see Bliznikas [5-7, 11, 13, 17]), t e n s o r connections, and,in p a r t i c u l a r , affinor connections (see Hombu [173-176]), and b i v e c t o r connections (see H o k a r i [167-171]) a l s o find application in the g e o m e t r y of differential equation s y s t e m s . The study of v a r i o u s manifolds and submanifolds, i m m e r s e d in a p r o j e c t i v e , affine, Euclidean, or other space, as well as the investigation of v a r i o u s g e o m e t r i c objects of specific s p a c e s lead to the study of differential equation s y s t e m s (for e x a m p l e , s e e Vuiichich [23], D e m a r i a [133], H a y a s h i [165], Matsumoto [210, 211], Miller [216], Obata [218], P r v a n o v i t c h [222], R u s c i o r [225], Sauer [226], Segre [227], T a m ~ s s y [245], T s a n g a s [253], Vala [255], V a r g a [256], Vincensini [264], Weise [271], Yen [276], and others). It should be noted that the initial r e s e a r c h e s on the g e o m e t r y of differential equation s y s t e m s w e r e connected with the t h e o r y of the i n v a r i a n t s of differential quadratic f o r m s (Christoffel, Lipschitz, Ricci, K r a m l e t , N o e t h e r , Weft, and others) and with the w o r k s of Lie touching on the i n v a r i a n t p r o p e r t i e s of differential equation s y s t e m s (see Lie [206, 207]). Lie e s t a b l i s h e d the connection between c e r t a i n a s p e c t s of the t h e o r y of differential equations and of line g e o m e t r y whose foundations w e r e c r e a t e d by PlUcker, as well as with the g e n e r a l t h e o r y of c u r v e s . He succeeded not only in significantly advancing integration methods for differential equations but also in investigating c e r t a i n c l a s s e s of equations r e l a t i v e to a specified group and in giving an effective g e o m e t r i c int e r p r e t a t i o n of the r e s u l t s obtained. It is n e c e s s a r y to r e m a r k that L i e ' s g e o m e t r i c a p p r o a c h to the t h e o r y of s y s t e m s of partial differential equations not only p e r m i t t e d h i m to develop and extend the r e s e a r c h e s , existing in his day, of Jacobi M a y e r , Monge, and o t h e r s on the t h e o r y of differential equations, but also to pose new p r o b l e m s , to open up new d i r e c t i o n s , and by this, to r e n d e r a c o n s i d e r a b l e influence on the development of g e o m e t r y . LieVs ideas w e r e developed in the r e s e a r c h e s of Vessio, Z u r a v s k i i , and o t h e r s ; m o r e o v e r , it was p r o p o s e d that the Vession s y s t e m s of differential equations, admitting of s i m p l e finite nondegenerate groups (Lie groups), be n a m e d Lie s y s t e m s . T r a n s l a t e d f r o m Itogi Nauki i Tekhniki (Algebra. Topologiya. G e o m e t r i y a ) , Vol. l l , ' p p . 209-259. 9 76 Plenum Pubhshmg Corporation, 22 7 West 17th Street, New York, N. Y. 10011. No part o f thts pubfieation may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, mierofilming, recording or otherwise, without written permission o f the publisher. A eopy o f this article is available from the publisher for $15.00.
591
The m o s t b r i l l i a n t a n d profound development of L i e ' s gifted ideas is due to Caftan. Caftan* e x a m i n e d the m o s t g e n e r a l c a s e s of differential equations, i . e . , the c a s e s when the c o r r e s p o n d i n g s i m p l e group is infinite. T h e s e groups, c o r r e s p o n d i n g to specific differential equation s y s t e m s , a d m i t e i t h e r of an i n t e g r a l i n v a r i a n t of m a x i m a l d e g r e e (theory of Jacobi m u l t i p l i e r s ) or of a r e l a t i v e l i n e a r i n t e g r a l i n v a r i a n t (theory of equations r e d u c i b l e to canonic form) or of an invariant P f a f f equation (equations r e d u c i b l e to f i r s t - o r d e r p a r t i a l differential equations. Cartan, in his investigations touching on the t h e o r y of continuous groups (Lie groups and infinite groups), on the P f a f f p r o b l e m , on the t h e o r y of g e n e r a l i z e d s p a c e s , on the t h e o r y of diff e r e n t i a l and i n t e g r a l i n v a r i a n t s , on the t h e o r y of s y s t e m s of e x t e r i o r differential equations, often dwelled on the m o s t i n t e r e s t i n g a s p e c t s of the g e o m e t r y of differential equations; m o r e o v e r , he a p p r o a c h e d the g e o m e t r y of differential equations e i t h e r f r o m the point of view of i n t e g r a l i n v a r i a n t s or f r o m the point of view of g e n e r a l i z e d s p a c e s . One of the m e r i t s of C a f t a n is p r e c i s e l y that his w o r k on the g e o m e t r y of differential equations w a s b a s e d on the g e o m e t r y of differential equations as an independent direction in t h e g e o m e t r y of g e n e r a l i z e d spaces. The g e o m e t r i c p a p e r s on the t h e o r y of connexes, whose initiator was Clebsch, found application in the t h e o r y of differential equations. The application of the methods of the t h e o r y of i n v a r i a n t s and of a l g e b r a i c g e o m e t r y to the t h e o r y of connexes proved to be useful both for g e o m e t r i c as well as for the analytic a s p e c t s of the t h e o r y of differential equations. The t h e o r y of only a t e r n a r y convex was worked out in C l e b s c h ' s p a p e r s . The subsequent development of this t h e o r y is due to Sintsov (see [94]) who c o n s t r u c t e d the t h e o r y of a q u a t e r n a r y connex (an e l e m e n t is a point-plane) and e s t a b l i s h e d the connection of its principal coincidence with the t h e o r y of f i r s t - o r d e r p a r t i a l differential equations, i.e., with the fundamental ideas of L i e ' s integration t h e o r y (see Lie [206, 207]), T h e s e fundamental r e s e a r c h e s on the t h e o r y of connexes led to the n e c e s s i t y of developing the g e o m e t r y of p a r t i c u l a r c l a s s e s of differential equation s y s t e m s , i.e., to the g e o m e t r y of P f a f f and Monge manifolds (see B e r z o l a r i , Bonsdorf, Veneroni, I s s a l i , K a s n e r , Kowle, L a z z e r i , Lilienthal [208, 209], Ogura, R o d g e r s , Sintsov [94], Voss [265,266], and others). T h e s e r e s e a r c h e s a r e connected with the m o d e r n t h e o r y of nonholonomic and s e m i n o n h o l o n o m i c r u l e s manifolds. In the investigation of differential equation s y s t e m s c o n s i d e r a b l e attention was paid to the study of t h e i r group p r o p e r t i e s . By a group p r o p e r t y of a differential equation s y s t e m S we m e a n a p r o p e r t y of this s y s t e m which r e m a i n s unchanged when the dependent and independent v a r i a b l e s o c c u r r i n g in s y s t e m S a r e subjected to t r a n s f o r m a t i o n s f r o m s o m e t r a n s f o r m a t i o n group G. If such a p r o p e r t y holds for a given differential equation s y s t e m S, then we say that s y s t e m S a d m i t s of group G. Under such t r a n s f o r m a t i o n s any solution of s y s t e m S is led once again to a solution of this Same s y s t e m . T h e s e group p r o p e r t i e s can be used not only for the c o n s t r u c t i o n of v a r i o u s c l a s s e s of p a r t i c u l a r solutions of the given s y s t e m S but also to effect a group c l a s s i f i c a t i o n of the differential equation s y s t e m s t h e m s e l v e s . Such p r o b l e m s , it a p p e a r s , w e r e f i r s t put forth at the end of the l a s t century by Lie (1885-1895). Lie and his students worked out the a n a l y t i c a l tools and studied a wide c i r c l e of applications of the t h e o r y of continuous t r a n s f o r m a t i o n groups, which a r o s e when solving the p r o b l e m s mentioned for differential equation s y s t e m s . An i m p e t u s for the subsequent development of this t h e o r y was given by the applied a r e a s of m a t h e m a t i c s dealing with differential equations or with the g e o m e t r i c p r o p e r t i e s of model s p a c e s . In applied p r o b l e m s , as a r u l e , we encounter differential equations of c o n c r e t e f o r m , admitting of nontrivial t r a n s f o r mation groups. T h e s e p r o p e r t i e s a r e p o s s e s s e d , for e x a m p l e , by the equations of h y d r o d y n a m i c s , the equations of the t h e o r y of e l a s t i c i t y and p l a s t i c i t y , the equations of combustion theory, the equations of detonation, the equations of m a g n e t o h y d r o d y n a m i c s and of other applied physical or m e c h a n i c a l t h e o r i e s . Only individual r e s u l t s exist in this direction (see Ovsyannikov [75-87], I b r a g i m o v [32-42]) and intensive investigations a r e being c a r r i e d out at the p r e s e n t time. B e s i d e s the a b o v e - m e n t i o n e d p a p e r s , to w o r k connected with the study of groups of analytic t r a n s f o r m a t i o n s of differential equation s y s t e m s we can r e f e r the p a p e r s by Evtushik [30], D u m i t r a s [137], Spencer [231], Hangan [161-162], and others. Thus, the g e o m e t r y of differential equation s y s t e m s can be i n t e r p r e t e d in m a n y ways, but t h e r e e x i s t profound r e l a t i o n s between all the t h e o r i e s even though they a r e not equivalent.
*E. Cartan, O e u v r e s C o m p l e t e s , P a r t i e II, Vols. 1,2; P a r t i e III, Vol. 2, G a u t h i e r - V i l l a r s , P a r i s (1953).
592
Let us note other features of the origin of the g e o m e t r y of differential equation s y s t e m s . Many a s pects of the local c l a s s i c a l theory of s u r f a c e s a r e connected with the study of. the s t r u c t u r e of geodesic lines (of the e x t r e m a l surface) whose differential equations have the form (Fik is the Christoffel object, n = 2) 9 d~s dxk d~x___~+ r~(x) ds 2
o (i,/ . . . . .
ds
The geodesic tines of any affinely-connected space also eral), i.e., to any affinely-connected space there always It is not difficult to note that if the differential equation manifold we can define affine connections for which the These connections have the form ~i
rik =
Fi
,
1,~ ..... n)
(0.1)
are d e t e r m i n e d by the same s y s t e m (n ~ 2, in genc o r r e s p o n d s the differential equation s y s t e m (0.1). s y s t e m (0.1) is given, then on the differentiable integral c u r v e s of s y s t e m (0.1) a r e geodesic lines.
i
'
jk ~- a j p k + a ~ pi,
(o.2)
where Pi is an a r b i t r a r y coveetor field. If we r e c k o n that s v a r i e s by the l a w ~ = as + b (a, b are a r b i t r a r y constants), then for the given differential equation s y s t e m (0.1) there exists a unique (torsion-free) affine connection whose geodesic lines coincide with the integral c u r v e s of the differential equation s y s t e m being examined. It is obvious that to any affinely-connected space t h e r e c o r r e s p o n d s a s e c o n d - o r d e r line-element space ~(2) n in which s u r f a c e (0.1) is given as the distinctive "absolute" of this space. Since the absolutes of a space a r e invariant r e l a t i v e to t r a n s f o r m a t i o n s (0.2), then, in general, the inverse c o r r e s p o n d e n c e is not unique. If linear t r a n s f o r m a t i o n s alone a r e admissible t r a n s f o r m a t i o n s for p a r a m e t e r s, then the abovementioned c o r r e s p o n d e n c e is unique. Affinely-eonnecte d spaces were studied f r o m this point of view by T h o m a s [248] and T h o m a s [249, 250], Veblen, and others. If we note that the quantities
9
dxk dxn ds
H i = V~h(x) ds
(0.3)
f o r m a d i f f e r e n t i a l - g e o m e t r i c object defined on a f i r s t - o r d e r l i n e - e l e m e n t space L ), then the g e o m e t r y of an affinely-connected space is equivalent to the g e o m e t r y of space L(n1) with the fundamental differentialg e o m e t r i c object H I. Cartan noted (see [48]) that the geodesic lines of a p r o j e c t i v e l y - c o n n e c t e d space have the form
d~ua_~ .~_ pZ ( f~, t, @-t ) - - d u dt Z~(ua,t,~t
(0.4)
where p a and ~ a r e a r b i t r a r y s e c o n d - d e g r e e polynomials in duff/dr. If the differential equation s y s t e m (0.4) is given, then the c u r v e s defined by the integrals of s y s t e m (0.4) can always be looked upon as the geodesic lines of a (torsion-free) p r o j e c t i v e l y - c o n n e c t e d space; m o r e o v e r , the projective connection of this space is defined nonuniquely (to within n (n + 1)/2 functions,of n arguments), i.e., a part of the components of the object of projective connection (of c e n t r o p r o j e c t i v e connection in general) can be chosen a r b i t r a r i l y . If the R i c c i t e n s o r of the c u r v a t u r e t e n s o r of a (torsion-free) p r o j e c t i v e l y - c o n n e c t e d space equals zero, then in C a r t a n ' s t e r m i n o l o g y (see [48]) such a p r o j e c t i v e l y - c o n n e c t e d space is called a space of n o r mal projective connection. It turns out that a space of n o r m a l projective connection always c o r r e s p o n d s uniquely to differential equation s y s t e m (0.4). Thus, a unique (local) c o r r e s p o n d e n c e exists between differential equation s y s t e m s (0.4) and spaces of n o r m a l projective connection (the p a r a m e t r i z a t i o n of the integral c u r v e s is a r b i t r a r y ) . In other words, spaces of n o r m a l projective connection play for the differential equation s y s t e m s (0.4) the v e r y same r o l e that Riemann spaces (with Levi-Civita parallelism) play in the theory of nondegenerate quadratic differential forms.
593
The differential invariants of the e l e m e n t a r y differential equation
dx~
w e r e examined by T r e s s e [252]. Cartan [121] proved that the integral c u r v e s of differential equation (0.5) can be looked upon as the geodesic lines of a l i n e - e l e m e n t space with a n o r m a l projective connection. We r e m a r k that the dual spaces p o s s e s s the same property. T h e s e investigations by T r e s s e and Caftan p r e cede the investigations of Koppisch [193] (Koppisch's papers predate the p a p e r s of T r e s s e , but Koppisch's dissertation [193] was published after T r e s s e ' s dissertation [252]). Koppisch studied, f r o m an analytic viewpoint, the c o r r e s p o n d e n c e between the solutions of two c l a s s e s of s e c o n d - o r d e r differential equations. The g e o m e t r i c interpretation of these r e s e a r c h e s is the following: the r e l a t i o n s existing between the families of geodesic lines of two mutually-dual l i n e - e l e m e n t spaces with n o r m a l projective connection w e r e studied. Analogous questions w e r e analyzed f r o m the analytic point of view by Yoshida [277], Zorawski [278], K a i s e r [183], P o d o l ' s k i i [91, 92], Neuman [217], and others. C a r t a n ' s g e o m e t r i c ideas w e r e c a r r i e d over to the multidimensional c a s e by Douglas [135, 136] who took it that the geodesic lines of a space a r e the solutions of the s y s t e m d~x~ L H ~ ( x k , t, dxk ~ = O, dt 2 dt ]
(0.6)
where H i a r e a r b i t r a r y homogeneous functions in dxk/dt. 9
t
,
The investigations of ]3ark [113], Berwald [114], Veblen [257-263], Weszely and Szflagyl [272]; Gaukhman [24], Kawaguchi and Hombu [185], Laptev [62], Slebodzinski [228-230], Stepanov [95-98, 100-102], Chern [122, 123], and others r e l a t e to the g e o m e t r y of such differential equation s y s t e m s . A cycle of papers by ]31iznikas [9-11, 17], Bompiani [116, 117], Kosambi [194-196], Moor [214, 215], Rzhekhina [93],Stepanov[99, 104], Takano [244], Udalov [105-107], Homann [172], Hombu [173-175], and others examines the g e o m e t r y of n o r m a l s y s t e m s of h i g h e r - o r d e r differential equations. Vasil'ev [22], t31iznikas [13, 16], Kil'p [54, 55], Lupeikis [66-73], P e t r u s h k e v i c h y u t e [90], and others devoted their papers to the g e o m e t r y of quasilinear s y s t e m s of differential equations. Many c l a s s i c a l p r o b l e m s in the t h e o r y of s u r f a c e s r e d u c e to the investigation of the p r o p e r t i e s of solutions of s e c o n d - o r d e r partial differential equations. If the s y s t e m ( a ~ /3) O~ x I Ou~ Ou~
Fv
O#
cz~ O-'--~-+
F ~ Oxk Oxn = 0 , kt~Ou-T Ou-T
where F~fl is the intrinsic object of affine connection of the surface, F~h is the object of affine connection of a Riemann space, has a solution of maximal dimension, then this solution defines a s u r f a c e in the R i e mann space, whose tangent coordinate lines a r e conjugate. This example is a special case of a s y s t e m of the f o r m (pk = 8xk/su~, etc.): r
X'
i k + no~l...o~p+ ` (Xk ,U "l~, p,Z...~J a) == 0 .
(0.7)
au~. . . . au%+ , (i,] . . . . . i
1,2..... n; a, 1~. . . . . 1,2, ..., m; a----l, 2 .... , p), k
where H~(_+l, a r e a r b i t r a r y functions (not n e c e s s a r i l y s e c o n d - o r d e r polynomials in p~ when p = 1). Such J d i f f e r e n t ~ equation s y s t e m s w e r e f i r s t studied f r o m a g e o m e t r i c viewpoint by Douglas. A generalization of these ideas of Douglas is due to Kawaguchi and Hombu [185]. The g e o m e t r y of such differential equation s y s t e m s has been examined by B i e b e r b a c h [115], Bortolotti [118], Ku [198-200], Ishihara and Fukami [180], Katzin and Levine [184], Mikami [212], Okubo [219, 220], Su [232-240], Suguri [241, 242], T a r i n a and Artin [246], Hashimoto [164], Hokari [167-171], Hua [178], Yano and H i r a m a t s u [274-275], and others. In connection with the development of the global t h e o r y of fibered spaces, of the t h e o r y of jets in the sense of E h r e s m a n n , and of other aspects of modern g e o m e t r y , t h e r e e m e r g e d investigations on the t h e o r y
594
of differential equations of differentiable manifolds, and also investigations on the global aspects of the g e o m e t r y of s y s t e m s of differential equations of various o r d e r s (Kumpera [201], Kuranishi [202-205], and others). We should r e m a r k that the p a p e r s on the g e o m e t r y of differential equations s y s t e m s , which a r e r e viewed in this survey, w e r e a c c o m p l i s h e d by various methods of d i f f e r e n t i a l - g e o m e t r i c investigations and a feature of the p r e s e n t s u r v e y is that the local r e s u l t s on the g e o m e t r y of differential equation s y s t e m s a r e presented by a single method of m o d e r n d i f f e r e n t i a l - g e o m e t r i c investigations - L a p t e v ' s method [61]. In this s u r v e y we s y s t e m a t i c a l l y examine the t h e m e s of only those papers which appeared in print f r o m 1953 on, while e a r l i e r p a p e r s a r e mentioned only under the n e c e s s i t y of a m o r e complete presentation of the development of the g e o m e t r y of differential equation s y s t e m s in its h i s t o r i c a l aspect. The s u r v a y ' s authors have made an attempt to unify to some extent the t e r m i n o l o g y and to p r e s e n t c e r t a i n r e s u l t s in a single invariant f o r m . The p a p e r s are concentrated around questions connected with the g e o m e t r y of differential equations. By the latter in this survey we mean a c o n c r e t e c i r c l e of p r o b l e m s which can be identified by the geometric concepts encountered in the t h e o r y of generalized spaces with fundamental-group connections and, in p a r ticular c a s e s , with the usual concepts of the t h e o r y of submanifolds of Klein spaces. We have left aside important sections of the g e o m e t r y of differential equations, which a r e connected with the fundamental p r o b l e m s of the general t h e o r y of r e l a t i v i t y (see P e t r o v [89]), the t h e o r y of differentialg e o m e t r i c methods in the calculus of v a r i a t i o n s (see Vagner [270], Winternitz [273], Kabanov [46], C a r a th6odory [120]), the t h e o r y of motions in generalized d i f f e r e n t i a l - g e o m e t r i c spaces (see E g o r o v [31]), the global aspects of the basic p r o b l e m s in the t h e o r y of equations on complex manifolds (Schwartz, Grothendieck, Dolbeau, Malgrange, H i r z e b r u c h , and others), and the a s p e c t s of differential topology, having a direct connection with the main p r o b l e m s of the g e o m e t r y of differential equations. The authors endeavored to pick out the fundamental directions of investigations in the g e o m e t r y of differential equations, which a r e being taken both in our own native as well as in the foreign schools of geometry. w
Fibered
Space
JP(Vm,
Vn)
Let Vn and V m be differentiable manifolds of c l a s s Cw (dim Vn = n, dim Vm = m) whose local c o o r dinates a r e the f i r s t integrals of the fully-integrable s y s t e m s r 1 = 0, 0 cx = 0(i, j , . . . = 1, 2, . . . , n; cL fi, . . . = 1, 2 , . . . , m). ~ 9 .fla a r e s y m m e t r i c relative to the The s t r u c t u r e equations for the 1 - f o r m s c0i a r e (wjI" .. Ja' 0ill. lower indices of the 1-form) n~o~ = cok /~ok, D(o} = o~1/~oJ k k~+ cok/\o)ik, f
(1.1)
a
O~}..j,
=
~! (~ - s)! ~...i~A~'(J~+~. (a =
The 1 - f o r m s 0 a, 0~, " ' "
1,2 .....
]a) k ,_,.,k ' w ,A,.,~ ~wir..lalr
p).
O~:t. " .fla are connected by analogous s t r u c t u r e equations.
Let us consider the set c P (Vm, Vn) of point functions ( f , y), where f is a mapping of c l a s s C p, i.e.,
f : (ya) _ (xi). Let u
(vm, v,,) = c; (vm,
YEV m
T w o e l e m e n t s (f, g), (g, g) ~Cp( V~, V~) a r e s a i d t o b e p - e q u i v a l e n t a t p o i n t (g~)~ V~, i f f ( y ) = g (y) and the partial derivatives of functions f and g coincide at point ( y ~ ) ~ V ~ up to o r d e r p (inclusive). This establishes a p-equivalence r e l a t i o n in the set of mappings f : Vm - - Vn taking (y~) into (xi). The c l a s s of p-equivalent elements is uniquely defined by any element of thos c l a s s .
595
The c l a s s of p-equivalent e l e m e n t s in c P ( V m , Vn) is called a p-th o r d e r jet (or a p - j e t in the s e n s e of E h r e s m a n n (see [138-144])). We denote the p - j e t g e n e r a t e d by mapping f at point (y~)~ V~ by jPyf, while we denote the set of all p - j e t s f r o m Vm into Vn by JP (Vm, Vn), i.e.,
J" (v,,,, v.) = U if, f. YeVm
This set is a f i b e r e d space. If 1 < m - n - 1 , then a r e g u l a r p - j e t is called an m - d i m e n s i o n a l s u r face e l e m e n t (see Kawaguchi [186, 189], C r a i g [124-131], Oliva [221]). I f m = 1, the p - j e t is called a c u r v i l i n e a r e l e m e n t of p-th o r d e r . We shall use the following notation:
Kn,m =J'(V,n, (
Vn) , Pr e < n ;
)L(nm
JP(V1, V,),
f.~P) = .P (r(, v.). Since the differential equations of mapping f : V m -~ Vn can be w r i t t e n as (o = f ~~O =,
(1.2)
by prolonging s y s t e m (1.2) s u c c e s s i v e l y , by virtue of s t r u c t u r e equations (1.1) and of the analogous equations for the 1 - f o r m s 0, we obtain dr& + ,~'~,,-k ''~ - - -t~ ~r o~ = f~, o6, ~k m i - U / : ~ n o ) i .
.
.
.
.
,
.
.
.
.
.
0v __t:iO v
__9~i .
.
.
.
.
.
.
.
.
.
.
-
-
(1.3) .
.
.
.
a
0%...% - - d~%...% +
'
='
9 Z s!a' C%z"" ks s=l
1
Z
- - 9 •]s [ ]l!..
(]l+.,.+/s=a)
a
• [(%"~1,)'"
f:;,+
o6(%
~ .(a--s)!
..+,,_~ . . . . , ) - -
.% f,%+r"%) t~ =Z %...%vr
S~I
(f~=,"'%l = 0
for any
a > 1).
If col = 0, 0 ~ = 0, then the differential equation s y s t e m (1.3) t a k e s the f o r m (a = 1, 2, . . . . p): 0 ~t -_-O ,
0 ~6 i = 0 .... ' 0 %.-% i =0.
(1.4)
It is fully i n t e g r a b l e and its f i r s t i n t e g r a l s f o r m a d i f f e r e n t i a l - g e o m e t r i c object r e l a t i v e to the Lie group GLP(n, R) x GL p (m, R) [GL p (n, R) is a differential group of o r d e r p for the differentiable manifold Vn] , i.e., the differential equation s y s t e m (1.4) defines the s p a c e of r e p r e s e n t a t i o n of the differential groups GLP (n, R) and GLP (m, R) (see L a p t e v [62, 63]). The components of this d i f f e r e n t i a l - g e o m e t r i c object a r e the local c o o r d i n a t e s of a p-jet, i.e.,
J~
%}.
'
The differential equation s y s t e m oi
0,0 ~
0, O~
0,
0~
=o
(1.5)
is fully i n t e g r a b l e and its f i r s t i n t e g r a l s a r e the local c o o r d i n a t e s of the f i b e r e d space JP (Vm, Vn). The s t r u c t u r e equations of s p a c e JP (Vm, Vn) have the f o r m (a = 1, 2 . . . . . p) D~o~ - - c ok ~/ \ O)ik, DO~-O~AO~, a l k " ... D 0 %..%= Z %,.. ~cA 0~f..~al + ~ c--=l
596
'
k + OVA el, ..%v,
(1.6)
where gi$r..fic
dO~r..% Of~ I "~ pr '
~%...%k = a
_
l
...ft.,
h
(il+-...+is~a)
0 i t~ 1
(1.7)
~___
9( a - - s ) t
aa ?
v(% -.% %+r"%) 13"
If n = 1, we i n t r o d u c e the following notation: [,
f (a)
1 , . . 1 :"== t a
and i ~a ] o)k~. ks
~(a) i _ df(a)~ +
(al-i-,..+a s=a)
s= 1
The s y s t e m (a = 1, 2 . . . . .
~
1 [(a~)k~...:(%) ~ alI...as!
(1.8)
p) o)i= O,
@(a) i = 0
is fully i n t e g r a b l e 9and I its~ f i r s t i n t e ~g r a l s a r e the l o c a l c o o r d i n a t e s of a p a r a m e t r i z e d line e l e m e n t of h i g h e r o r d e r ( o r d e r p) (x t, f ( )l . . . . . f ( p ) i ) , i.e., aor(ep ) point of the s p a c e L (p) (the s p a c e of h i g h e r - o r d e r line e l e ments). The s t r u c t u r e equations of s p a c e L~ have the f o r m (a = 0, 1 . . . . . p) D ~ (a)~
-~ ,~(c)k^ ~(a)i
~
(1.9)
/ \u(c)k,
where
w',
~(o)i
~(a) i (b) I =
u(o)i ~ r
o~(a) t O[(o) i '
'.,(o)k ~ ~(k
Otot
t
~
0/(0}k
(1.10)
Ok,
It turns out that tct) i • ( (r a ) ki --- 0 (c > a), .q, -(b) i
a!
b! (a -- b)!
a?~"- ~ ' (a >~ b).
(1.11)
The 1-forms o(a)~= u(b)
~(a) il
(b)k/ol~ 0, o(a)i=0
(1.12)
a r e the i n v a r i a n t 1 - f o r m s of the Lie g r o u p GL (n, p, R); m o r e o v e r , Do(a) i U(b)]
~-
a(c) k ^ a(a) i uib) j / \ U ( c ) k .
Group GL (n, p, R) is the t r a n s f o r m a t i o n g r o u p of the f r a m e s of a t a n g e n t s p a c e for ~ P ) . ing s u b g r o u p s : GL (n, R) C GL (n, 1, R) C " .
It h a s the follow-
~ GL (n, p - - 1, R).
597
The infinitesimal t r a n s f o r m a t i o n s of the f r a m e v e c t o r s {~((0)i,-~(1)i . . . . . -e(p)i} of the tangent s p a c e T(P) for finp) have the f o r m de(a)i
= 0 ( ~(a), ) k / (c)k .
The e l e m e n t s of the t e n s o r product (T*(P) is the dual v e c t o r tangent space) 4 T (p) 4 T *(p)
a r e said to be q t i m e s e x c o v a r i a n t and r t i m e s e x c o n t r a v a r i a n t , the concept of which was introduced by C r a i g (see [124-131]). The d i f f e r e n t i a l - g e o m e t r i c objects r e l a t i v e to h i g h e r - o ro( d e r differential groups GL q (n, p, R) a r e called exobjects o f q - t h o r d e r . . The connection objects of s p a c e L(np) a r e e x a m p l e s of such objects. The 1 - f o r m s (a = 1, 2 . . . . .
p)
a--I
9( )' = {}(~)~-F ~,~(a)~@(~) ~ ( c )k
(1.13)
C~0
define a linear d i f f e r e n t i a l - g e o m e t r i c connection of s p a c e L(P) if and only if the d i f f e r e n t i a l - g e o m e t r i c obn j e e t r ( a ) i h a s the following s t r u c t u r e : ~(b)k P
dp(a), ~-(O)a
,~(a), ~ - ,,(a) l ( b ) k l ~ l, - - J-(c) h ~.~(c) ( b ) ~ --
tqO ) ~ .~(a)
=
(.1.14.)
Z ~(0)~(~)t~ .~(a) ~ ~ .~(c) a
9
c~O
The 1 - f o r m s (m <- p) m
= r
~] ri
(1.15)
a r e 1 - f o r m s o f affine connection if and only if P
l ~ F ~ l(a)k coil ~- Fz](a) k r l drj(a)k
w
Geometry
of Systems
~ l(b)t 0(bCz {a)a__O!0)t ta) k(0)/" = ~ F i(a)k(c)h @'(c)h* c=O
of Ordinary
Differential
(1.16)
Equations
1. G e o m e t r y of S y s t e m s of S e c o n d - O r d e r O r d i n a r y Differential Equations. The s y s t e m of secondo r d e r o r d i n a r y differential equations +/t' dt~
= 0 \
(2.1)
' dt )
~(2)
can be t r e a t e d as the finite equations of a 2 n - d i m e n s i o n a l s u r f a c e in the space - n ' i.e., as the finite equations of an i m m e r s e d manifold in the s e n s e of Laptev [61]. Laptev [62] e x a m i n e s the g e o m e t r y of d i f f e r e n t i m equation s y s t e m s f r o m this point of view. The g e o m e t r y of the s u r f a c e s mentioned is also called the affine g e o m e t r y of n o r m a l s y s t e m s of s e c o n d - o r d e r differential equations or the g e o m e t r y of paths. Since under a change of local c o o r d i n a t e s of the b a s e V,~ of s p a c e 4 2 ) the s y s t e m of functions H i, which is defined on space It~ ' , is t r a n s f o r m e d by the t r a n s i t i v e law n~,=
o,~" / 4 ' - o " ~ ' Ox i
c)xk Oxh
. d,:___~ ~ " ~_~h dt
dt
(2.u) '
the g e o m e t r y of differential equation s y s t e m (2.1) is equivalent to that g e o m e t r y of s p a c e ~ n ) which is e s tablished in it by m e a n s of the d i f f e r e n t i a l - g e o m e t r i c object H i called the fundamental d i f f e r e n t i a l - g e o m e t r i c
598
2 (1) 9 The differential equations of the d i f f e r e n t i a l - g e o m e t r i c object H i have the f o r m object of s p a c e ~n dH ~ -1- H k ( o k~ -l-, ~ ~ = H ~ko k - -
,H~t~t'),~.
(2.3)
By prolonging this s y s t e m we obtain a fundamental s y s t e m of fundamental d i f f e r e n t i a l - g e o m e t r i c objects of the s p a c e being examined: $(') (H) C ~(2) ( H ) C . . . C f f m ( H ) c .- -,
(2.4)
where
5(')(n) = { n ,
=
nL
nh, 'nh,
Is f
....
It t u r n s out that the subobjects of fundamental d i f f e r e n t i a l - g e o m e t r i c object $(2)(H) c o v e r the object of linear differential-g_eemetric connection F~ = 1/2THe., and also the object of ( t o r s i o n - f r e e ) affine connection i 9 i ~ ~ r j k = 1/2"Hjk of space L~~p (see Bliznikas [3-10]). The d i f f e r e n t i a l - g e o m e t r i c object
~(3)(H) c o v e r s the object of p r o j e c t i v e connection (Thomas p a r a m e -
ters)
II~
=
r~
__~1 n§
+
~,PM~
f(1)i -
n+l
"g~kh.
(2.5)
vi If Vk is the symbol for the nonholonomic b a s e d e r i v a t i v e r e l a t i v e to object Fj (see Bliznikas [7]), then the quantities II~h and l~ij , w h e r e
ILi ~
~ II,-k k Hi~ a ), n +I I (% r[~
(2.6)
define an object of c e n t r o p r o j e c t i v e connection (see Bliznikas [10]). Thus, the fundamental d i f f e r e n t i a l - g e o m e t r i c object ~ ( 4 ) ( H ) o q q v e r s a sheaf of d i f f e r e n t i a l - g e o m e t r i c (i) 9 objects each of which g e n e r a t e s a p r o j e c t i v e connection of space L n 0t depends upon the choice of the c o v e c t o r field, which is intrinsic). The existence of the a b o v e - m e n t i o n e d s p a c e s was f i r s t noticed by L a p tev (in a plenary r e p o r t at the F i r s t All-Union G e o m e t r i c Conference, Kiev, 1962). Stepanov [95-97, 99] investigates the local p r o p e r t i e s of differential equations.' I n v a r i a n t s of the equations being investigated a r e found and t h e i r g e o m e t r i c i n t e r p r e t a t i o n is given, An invariant c l a s s i f i c a t i o n of a n u m b e r of t y p e s of s e c o n d - o r d e r o r d i n a r y differential equations is m a d e and c e r t a i n special c a s e s a r e investigated. The study of the g e o m e t r y of differential equation s y s t e m s is reduced, as in the p a p e r s of Ken, Kuz'mina, and o t h e r s , to the study of a manifold of specific dimension i m m e r s e d in the r e p r e s e n t a t i o n s p a c e of an infinite t r a n s f o r m a t i o n pseudogroup. The investigations w e r e c a r r i e d in a canonized f r a m e . In all the c a s e s c o n s i d e r e d a fully-defined f u n d a m e n t a l - g r o u p connection is a s s o c i a t e d in invariant m a n n e r with the equations being studied. The investigation of the g e o m e t r y of differential equations is r e d u c e d to the study of this connection. In p a r t i c u l a r , the connection is found, explicit e x p r e s s i o n s for the invariants a r e found, and a c l a s s i f i c a t i o n is made of a s y s t e m of two s e c o n d - o r d e r o r d i n a r y differential equations (see [97]) r e l a t i v e to the pseudogroup of the m o s t g e n e r a l analytic t r a n s f o r m a t i o n s of the v a r i a b l e s , All p a i r s of equations admitting of finite t r a n s i t i v e Lie groups a r e found. L u p e i k i s ' p a p e r s [66-69] a r e devoted to the g e o m e t r y of c e r t a i n c l a s s e s of quasilinear s y s t e m s of differential equations of the f o r m a? ( xk' -~-I l -d3 d2 xr(i,] . . . . .
hC~( x~' dx~ ]l =
(2.7)
1,2 . . . . . n; a , ~ . . . . . 1,2,...,m).
By the g e o m e t r y of differential equation s y s t e m (2.7) we m e a n the g e o m e t r y of the s p a c e f~(l) with fundamental d i f f e r e n t i a l - g e o m e t r i c object { aT, h a}. n
599
F o r m = n t h e r e w e r e found the objects of linear d i f f e r e n t i a l - g e o m e t r i c connection and of (torsionfree) affine connection of s p a c e s l) in the g e n e r a l c a s e , i.e., when a ~ = a ~ (x, dx/dt), and in a s p e c i a l c a s e , i.e., when a .a~ = a ~ (x), i n v a r i a n t r e l a t i v e to the r e n o r m a l i z a t i o n of the components of the line element. The linear d i f f e r e n t i a l - g e o m e t r i c connection c o v e r s the f i r s t , while the ( t o r s i o n - f r e e ) affine connection c o v e r s the second differential prolongation of the fundamental d i f f e r e n t i a l - g e o m e t r i c object { a ~ , h~}. It t u r n s out that in the s p e c i a l c a s e the differential equation s y s t e m (2,7) will be equivalent to the s y s t e m of differential equations.of the geodesic c u r v e s of the a s s o c i a t e d space ( L ) , F.1) if and only if the second C a f t a n c u r v a t u r e t e n s o r a jkp of t r u n c a t e d affine connection equals z e r o . In the c a s e given, by the a s s o c i a t e d space ( ), ~]) we m e a n s p a c e L(nl) with l i n e a r d i f f e r e n t i a l - g e o m e t r i c connection. o
In c a s e m = n + 1 a s h e a f of F i n s l e r s p a c e s (L(n1), F) (the p a r a m e t e r s a r e s c a l a r functions A(x)) is a s sociated with s p a c e L{nl). In the g e n e r a l and the s p e c i a l c a s e s t h e r e e x i s t objects of ( t o r s i o n - f r e e ) affine connection, c o m m o n to all the F i n s l e r s p a c e s of this s h e a f and invariant r e l a t i v e to r e n o r m a l i z a t i o n of the line e l e m e n t ' s components. (') (a= 1, 2) When m = 4, n = 2, (a)with the s p a c e ~ ( l ) t h e r e is a s s o c i a t e d a s h e a f of F i n s l e r s p a c e s (L(n~), Fq) with the m e t r i c t e n s o r Fq, having a g e n e r a l affine connection invariant r e l a t i v e to r e n o r m a l i z a t i o n of the line e l e m e n t ' s components. ~
~ (1)
The objects of affine connection of the space L n w e r e found also for m = 1 ; n = 2 ; m = 3 , 6;n=3, m=2.
n=3;m=
2. G e o m e t r y of N o r m a l S y s t e m s of H i g h e r - O r d e r O r d i n a r y Differential Equations. The g e o m e t r y of this c l a s s of differential equation s y s t e m s h a s not been investigated much. By the g e o m e t r y of n o r m a l s y s t e m s of differential equations of o r d e r p + 1 we m e a n the g e o m e t r y of the space L(np) with a specified field (p)
of a d i f f e r e n t i a l - g e o m e t r i c object (see Blizuikas [8-10]) Hi: (P)
(P!
.
(P)
.~P (P)
, i = ~ .H(~)~ i ~(~)~, dH + H ~ o)k--~
(2.8)
c=O
where p+1 (P) ~ k. 1 f(,Dkl.. " f(pa)k~ q)' = (P -k l)! ~ 1a! ~ k,.. ~ p,!...p,! a=~ (Pl+" +pa=P+1) In t h i s case the f i r s t fundamental d i f f e r e n t i a l - g e o m e t r i c object
Cp)
(P)
(2.9)
(P)
~'(1)(/-/) = {Hi, H~)~} c o v e r s the object
of linear d i f f e r e n t i a l - g e o m e t r i c connection of the s p a c e L(np) (see Bliznikas [10]): (o + ,), (, + b - a ) ,
(2.1 0)
(b-.~
while the second fundamental d i f f e r e n t i a l - g e o m e t r i c object connection (truncation o r d e r p = 1):
5~(2)(H) c o v e r s the object of truncated affine
F~(~)a= p + 1 H(p)k(a+l)a.
(2.11)
Only s p e c i a l c a s e s of such s y s t e m s have been c o n s i d e r e d in detail (see Bompiani [i17], Stepanov [98, 103, 104], Homann [172]). Stepanov studies the g e o m e t r y of o r d i n a r y differential equations examined r e l a t i v e t o a g e n e r a l analytic pseudogroup of t r a n s f o r m a t i o n s of the v a r i a b l e s . The b a s i c connection on the s e t of s e c o n d - o r d e r tangent e l e m e n t s with fundamental group g2,6 was found. A n u m b e r of s p e c i a l c a s e s w e r e c o n s i d e r e d and a classification m a d e of the equations with the aid of the r e l a t i v e i n v a r i a n t s which w e r e c o n s t r u c t e d f r o m the c o m p o n e n t s of the c u r v a t u r e t e n s o r of the b a s i c connection. The s t r u c t u r e of the s p a c e of i n t e g r a l c u r v e s of a differential equation was c o n s i d e r e d . It t u r n s out that this space has an affine s t r u c t u r e ; m o r e o v e r , a c u r v e of special type (see [104]) is fixed in an ideal
600
plane. The s t r u c t u r e of a local s p a c e of b a s i c connection of the equation
y(") ~- f (x,y,y', .... y(~-')) was investigated in [103]. It was proved that in this c a s e t h e r e exists a s t r i c t l y - b o u n d e d collection of fundamental groups for spaces with connection: g2,6; g2,13; g3,3; g3,5; g3,6; g3,7; g5,4; g5,5; g6,2; g6,1. Representations, subgToups~ and various geometric interpretations were found for these groups. The problem of r e storing the canonic equation on the corresponding E-group was solved. If we examine n + 1)-th order ordinary differential equations (see Udalov [105-107], Mirodan [213]), then the curves of a projective space Pn are in a one-to-one association with them. Therefore, the objects, definitions, and properties invariantly connected with a curve c a r r y over to the associated differential equation. Since the curves of a projective space Pn fall into three types (0 <- q -< n - 1 is the class of a curve, i.e., an arithmetic invariant): q < n - 2 , q = n - 2 , q = n - 1 , the associated differential equations also fall into three distinct classes. It turns outthatanordinary linear differential equation of class q < n - 1 is reduced by a change of variables and a linear change of functions to an equation with constant coefficients if and only if the projective curvatures of the corresponding curve are constant. If q = n - 1 , the equation reduces to the equationdn+ly/dxn+i = 0.
w
Geometry
of Systems
of Partial
Differential
Equations
1. G e o m e t r y of S y s t e m s of F i r s t - O r d e r Differential Equations. system ~
We c o n s i d e r the differential equation
(3.1)
H~ (u, x).
(i,] ....... 1,2 ..... n; a,~ . . . . . 1,2 ..... m). By the g e o m e t r y of this differential equation s y s t e m we m e a n the collection of invariants and of inv a r i a n t o p e r a t i o n s r e l a t i v e to the pseudogroup of t r a n s f o r m a t i o n s
0u 4:0, detl[ ox" det o--Yl
=/=O,
(3.2)
a s s o c i a t e d with s y s t e m (3.1), i.e., we understand it as the g e o m e t r y of a differentiable manifold Vn • V m 1. with a field of a fundamental d i f f e r e n t i a l - g e o m e t r i c object Ho~. VH~
= H sik o k _~_ i rlsf~ 0 ~ 9
(3.3)
If as pseudogroup (3.2) we take the m o r e g e n e r a l pseudogroup
us = us, (s), x' = x" (x', 0x'l 0us' =~ det ~
0,
(3.2~
det ~ = 0 ,
then the g e o m e t r y of differential equation s y s t e m (3.1), t r e a t a b l e as the g e o m e t r y of the fibered s p a c e Vn (Vm) with a fundamental d i f f e r e n t i a l - g e o m e t r i c object, is equivalent to the g e o m e t r y of the fibered s p a c e (of the c o m p o s i t e manifold in the s e n s e of Vagner) Vn (Vm) with a given object of l i n e a r d i f f e r e n t i a l - g e o m e t r i c connection (see Vagner [19-21], G a r d n e r [145], Kan [47], Kil'p [49-55], Klemona [190], Kuz'mina [56-59], and others). In this c a s e the fundamental d i f f e r e n t i a l - g e o m e t r i c object s a t i s f i e s the following differential equation s y s t e m : i
i
i
k
Vile--cos = H~ko) -~ H ~ 0 ~.
(3.4)
601
H e r e an object of truncatdd affine connection is c o v e r e d by the f i r s t differential prolongation of the fundam e n t a l d i f f e r e n t i a l - g e o m e t r i c object Hi , while the object of c o m p l e t e affine connection, by the second. Of p a r t i c u l a r i n t e r e s t is the study of the g e o m e t r y of a f i b e r e d s p a c e in the c a s e when the b a s e s p a c e is a s o - c a l l e d affine f i b e r e d s p a c e (see Bliznikas [15, 16]). In this c a s e the object of truncated affine connection is c o v e r e d by the f i r s t differential prolongation of the fundamental d i f f e r e n t i a l - g e o m e t r i c object, while the object of c o m p l e t e affine connection, by the fourth. Let us consider in m o r e detail the g e o m e t r y of differential equation s y s t e m (3.1) r e l a t i v e to pseudogroup (3.2). By prolonging the differential equation s y s t e m (3.3), we obtain a fundamental sequence of fundamental d i f f e r e n t i a l - g e o m e t r i c objects of the differentiable manifold Vn x V m with a given field of the fundamental d i f f e r e n t i a l - g e o m e t r i c object H ~1 , .~(1) (n, m, H ) C s(2) (n, m, H ) ~ .
. . C ~'(p) (rt, fit, H ) C . .
9
(3.5)
,
where g ~ H' I s (2)(n, m, H) H~,
~ H~kh, H ~~ ,
i H~vl,
{g$,
... 9
It turns out that the p-th o r d e r fundamental d i f f e r e n t i a l - g e o m e t r i c object tains two s e r i e s of s u b o b j e c t s , f0) (n, m, H) C f ( ~
~-(p)(n, m, H) always con-
(3.6)
m, H ) C [ (2) (n, m, H ) C 9 9 9c f (pl(n, m, H) C . . . ,
(3.7)
,[(1) (n, m, H)g'f(2)(n, m, H) C " " C'[(V)(n, m, H ) C . . . , where
{~
n)=
{n~},
'f'
f)(n,m,H) = =
Hi~} ....
{HL H~,} .....{")(n,m,n) ={HL
{
'f<~)(n,m, H) = H~, H'CZl%'
" "
H~k, .....n~, '
H ~*I
"~p+l
}
,o}.....
....
When c o n s t r u c t i n g the t h e o r y of the i n v a r i a n t s and invariant operations of any g e o m e t r y a b a s i c r o l e is played by the connection objects c o v e r e d by the differential prolongations of the fundamental d i f f e r e n t i a l g e o m e t r i c object. Only a f t e r this c a n t h e c o r r e s p o n d i n g t h e o r y of c u r v a t u r e be worked out and an invariant c l a s s i f i c a t i o n of s p a c e s with connection be m a d e in a c c o r d a n c e with the p r o p e r t i e s of the c u r v a t u r e t e n s o r s . Thus, the f i r s t main t a s k in the g e o m e t r y of differential equation s y s t e m s (3.1) is this: s t a r t i n g f r o m the differential prolongation of s o m e (finite) o r d e r of the field of the fundamental d i f f e r e n t i a l - g e o m e t r i c object H 1a , i.e., of the fundamental d i f f e r e n t i a l - g e o m e t r i c object Y(p)(n, m, H) (p is finite), to s e e k the obj e c t s of v a r i o u s connections. Next, another t a s k is posed and subject to r e s o l u t i o n , n a m e l y , the t a s k of seeking the c o m p l e t e s y s t e m of differential i n v a r i a n t s and of solving the p r o b l e m of the equivalence of two differential equation s y s t e m s of f o r m (3.1). T h e s e t a s k s have been solved fully only for c e r t a i n p a r t i c u l a r c a s e s of s y s t e m s of f o r m (3.1) (for e x a m p l e , see H e r m a n n [166], D e d e c k e r [132], Klemona [190], K u z ' m i n a [58], H a r r i s o n and E s t a b r o o k [163], S h v a r t s b u r d [108], and others). Objects of affine connections of s p a c e s Vn and V m a r e called intrinsic objects of affine connections, adjoined to a differential equation s y s t e m of f o r m (3.1), if they a r e a l g e b r a i c c o v e r s (algebraic c o n c o m i tants) of the fundamental d i f f e r e n t i a l - g e o m e t r i c object ..~'(p)(n, m, H) or a l g e b r a i c c o v e r s of the d i f f e r e n t i a l g e o m e t r i c objects f ( P ) (n, m, H) and If(P) (n, m, H). The e x i s t e n c e of such connections has been p r o v e d only f o r c e r t a i n c a s e s (see Bliznikas [15]), i.e., when m = n, detlIHi~ I[ ~ 0; m = n - l , r a n k l I H i l I = m (m is even), and the nonzero r e l a t i v e invariant I
g~=
602
m--~. ~il]1
'
"
" k:'Bis]2 ' k~" " "Sim] m. km " K h ~ 2 " im g'~'ili~'" ]m...g'Cklks "" km
,
(3.8)
where ~ i l i 2 " " i m _~_ 8~
"CkrnS~l'"~m~, il al~l
Kim ~
. .~fm CZm~m, "
D
= V ( ~ H ~~ )
and e ~ 1"'~ m is an m - v e c t o r , eit ...in is an n - c o v e c t o r . If m is odd, then ~ ~ 0 . invariant can be c o n s t r u c t e d in a c c o r d a n c e with another f o r m u l a .
F o r odd m the r e l a t i v e
It has b e e n proved that when m < n the p r o b l e m of finding the intrinsic affine connections --I~B~/is equivalent to the p r o b l e m of the intrinsic invariant rigging of m - d i m e n s i o n a l s u r f a c e s of multidimensional affine s p a c e s (this follows f r o m the fact that the s t r u c t u r e of the sequence of d i f f e r e n t i a l - g e o m e t r i c objects (3.7) is the s a m e as the s t r u c t u r e of the fundamental sequence of d i f f e r e n t i a l - g e o m e t r i c objects of an a r b i t r a r i l y p a r a m e t r i z e d m - d i m e n s i o n a l s u r f a c e of a m u l t i d i m e n s i o n a l affine space) (see Bliznikas [15]). T h e r e f o r e , when seeking the intrinsic objects of affine connections, adjoined to the differential equation s y s t e m (3.1) (for m < n), it is e s s e n t i a l to use the t h e o r y of r e l a t i v e i n v a r i a n t s , developed in the r e s e a r c h e s of Laptev, Vaguer, Ostianu, Shveikin, L i b e r , Weise, Klingenberg, and o t h e r s . However, it should be noted that the r e l a t i v e i n v a r i a n t s , with whose aid we actually c o n s t r u c t the objects of intrinsic affine connections, adjoined to the differential equation s y s t e m (3.1), as also the differential r e l a t i v e i n v a r i a n t s , a r e c o v e r e d by fundamental objects of higher o r d e r than the c o r r e s p o n d i n g r e l a t i v e differential i n v a r i a n t s defining the int r i n s i c rigging o f a n m - d i m e n s i o n a l s u r f a c e in an affine space. Thus, the p r o b l e m s in the g e o m e t r y of differential equation s y s t e m s a r e not exhausted by the b a s i c p r o b l e m s in the t h e o r y of s u r f a c e s . 2. G e o m e t r y of Q u a s i l i n e a r S y s t e m s of F i r s t - O r d e r Differential Equations. The f i r s t investigations on the g e o m e t r y of q u a s i l i n e a r s y s t e m s of f i r s t - o r d e r p a r t i a l differential equations w e r e by V a s i l ' e v [22] who investigated a s y s t e m of t h r e e f i r s t - o r d e r partial differential equations with three,, unknown functions and two independent v a r i a b l e s , SII,I. It t u r n s out that the g e o m e t r y of the s y s t e m S~:~ of q u a s i l i n e a r differential equations, t r e a t a b l e as the g e o m e t r y of an i m m e r s e d manifold, is equivalent to the g e o m e t r y of an 8dimensional submanifold of an l l - d i m e n s i o n a l manifold M5,2, whose local c o o r d i n a t e s can be looked upon as the f i r s t i n t e g r a l s of the f u l l y - i n t e g r a b l e s y s t e m co~=0,(0~=0
(i----1,2,3,4,5;)~1,2;
a=3,4,5).
The c l a s s i f i c a t i o n of s y s t e m s S~I,~ is equivalent to the c l a s s i f i c a t i o n of c e r t a i n r u l e d submanifolds of a m u l t i d i m e n s i o n a l p r o j e c t i v e s p a c e (see V a s i l ' e v [22]). F o r e x a m p l e , the c l a s s i f i c a t i o n of the s y s t e m of equations ors s . Ovt O---Z= h t ( x ' v ) ~ +[~(x'Y'vr)
(s,r = 3,4 ..... m; t = 1,2 ..... m)
is equivalent to the g e o m e t r i c c o n s t r u c t i o n of t w o - d i m e n s i o n a l s u b s p a c e s of a tangent s p a c e of a differentiable manifold M m in a neighborhood of the point (x, y, v r) (see Kil'p [50]). Abutting V a s i l ' e v ' s investigations a r e the p a p e r s of Ionov [45], Kan (Shvartsburd) [47], K u z ' m i n a [56], Orlova [88], Petrushkevichyute., [90], S h v a r t s b u r d [108], and o t h e r s . Bilchev [1-3] obtained a c l a s s i f i c a t i o n of the s y s t e m s S(~,2 and found c e r t a i n groups r e l a t i v e to which a s y s t e m $2~:~'~"is invariant. It turns out that the p r o b l e m of i m m e r s i n g a t w o - d i m e n s i o n a l R i e m a n n s p a c e with constant c u r v a t u r e into a t h r e e - d i m e n sional Euclidean s p a c e r e d u c e s to the g e o m e t r y of c e r t a i n s y s t e m s S~:2) (see Bilchev [1]). The study of c o n s e r v a t i o n laws occupies an i m p o r t a n t place in the i n v e s t i g a t i o n of the g e o m e t r y of diff e r e n t i a l equation s y s t e m s (see V a s i l ' e v [22], Bilchev [1-3], Kan [47], Kil'p [49-55], and others), Let us consider the differential equation s y s t e m //a_____h~ (x, u) 0x~ + ha (x, u) = 0 ou~
(i,],k .....
1,2 ..... n;u,~ . . . . .
(3.9)
1,2 ..... m; a , b .... == 1,2 ..... r-~
603
and the nonsingular t r a n s f o r m a t i o n s x ~' - x ~' (x'), u ~' = u ~' (u~ ), A~' H ~ = O , detllA~' II~=0. The g e o m e t r y of differential equation s y s t e m (3.9), a n a l y z e d with r e s p e c t to the t r a n s f o r m a t i o n s mentioned, is equivalent to the g e o m e t r y of the differentiable manifold Vn • V m with a given field of the fundamental d i f f e r e n t i a l g e o m e t r i c object {hai a , ha~: Vhac~ = hacz o)k ~- ha~ 0 O, ,
,k
(3.10)
Vh~ = h~ ~k + h~ 0~" The g e o m e t r y of s y s t e m s of f o r m (3.9), t r e a t a b l e as the g e o m e t r y of s o m e r i g g e d manifold (manifold with a fundamental d i f f e r e n t i a l - g e o m e t r i c object), h a s much in c o m m o n with the g e o m e t r y of s y s t e m s S! t) (as a g e o m e t r y of i m m e r s e d manifolds). T h e s e connections between two g e o m e t r i e s of differential wff~,m equation s y s t e m s a r e analogous to the connections which exist between the t h e o r i e s of holonomic and nonholonomic s u r f a c e s in Klein s p a c e s . In other words, that which in one g e o m e t r i c t h e o r y holds for a conc r e t e differential equation s y s t e m of a given f o r m is, in the other theory, c h a r a c t e r i s t i c of s o m e family of differential equation s y s t e m of a given c l a s s . If m = nr (or m < nr), then the f i r s t differential prolongations of the fundamental d i f f e r e n t i a l - g e o m e t r i c object {h? c~ , h a } c o v e r the objects of affine connections F]k and I~fi,y (see t31iznikas [13, 16]). If r = n, then the object of affine connection 1~3. is c o v e r e d by the f i r s t differential prolongation of the d i f f e r e n t i a l - g e o m e t r i c object {h ac~, h a } , while ~heY object of affine connection F~k is c o v e r e d by the second differential prolongation of the s a m e object. The s t r u c t u r e equations of a fibered s p a c e (composite manifold in the s e n s e of Vagner) Vn (Vm) have the f o r m DO ~
= 0 ~ A 0~,
(3.11)
Do)~= o)k A ~ + 0= A ~ . By p a r t i a l l y prolonging t h e s e s t r u c t u r e equations, we obtain Do)~ = o)k A o)[ + o)k A o)~ -~ 0~ A o)~
(3.12) where i
--
(0~i,
i
--
"
By the g e o m e t r y of differential equation s y s t e m (3.9), e x a m i n e d r e l a t i v e to the pseudogroup of t r a n s formations x"=xi'(x',u~),
u~'=u~'(u~
A~'H"=O,
det It~Our IIq=O, det
Ox" ~ []=/=0, we m e a n the g e o m e t r y of the
f i b e r e d s p a c e Vn (Vm) with a given field of the d i f f e r e n t i a l - g e o m e t r i c object: VhaJz : hau ik o~k J+ ha~z i~ of~ P v h~, - - h,,~ k ok~ = h~ o~ + h~ O~.
(3.13)
If r = n m and the m a t r i x l]h~ ~ 11 is nonsingular, then the f i r s t differential prolongation of the fundam e n t a l d i f f e r e n t i a l - g e o m e t r i c object of s p a c e Vn (Vm! c o v e r s the object of linear d i f f e r e n t i a l - g e o m e t r i c connection F~ and the objects of affine connections F]k and l~fi~. i If r < n m and the s p a c e Vn (V m) is an affine f i b e r e d space, then the object F a also is c o v e r e d by the f i r s t differential prolongation of the fundamental d i f f e r e n t i a l - g e o m e t r i c object (see Bliznikas [16]). An a r b i t r a r y s y s t e m of f i r s t - o r d e r partial differential equations (3.14) ( a = 1,2 ..... r),
604
e x a m i n e d r e l a t i v e to n o n s i n g u l a r t r a n s f o r m a t i o n s of the t y p e s c o n s i d e r e d a b o v e and r e l a t i v e to the n o n s i n g u l a r t r a n s f o r m a t i o n s x r = x ~' (x~, uS), u ~' = u s' (x~, u=) (i, ] . . . . . 1, 2 ..... n; cr ~. . . . . nq-1 ..... n-q-m), is i n v e s t i g a t e d in [74]. In each c a s e , by the g e o m e t r y of d i f f e r e n t i a l equation s y s t e m (3.14) we m e a n , r e s p e c t i v e l y , the g e o m e t r y of d i f f e r e n t i a b l e m a n i f o l d j l (Vm, Vn), the g e o m e t r y of the f i b e r e d s p a c e ( c o m p o s i t e m a n i f o l d in the s e n s e of Vagner) Vn+m+nm, or the g e o m e t r y of a s p e c i a l f i b e r e d s.p~tce L N (N = 1, 2 . . . . . n, n + 1 . . . . . n + m, n + m + 1 . . . . . n + m + nm) w h o s e local c o o r d i n a t e s (x i, us , X~) a r e the f i r s t i n t e g r a l s of the fullyi n t e g r a b l e d i f f e r e n t i a l equation s y s t e m (o ~ 0 , VS--~
S
~ S
o)a : O ,
k ~
1
~
0
with a c o r r e s p o n d i n g given f u n d a m e n t a l d i f f e r e n t i a l - g e o m e t r i c object. Any differential equation s y s t e m of f o r m (3.14) can be r e d u c e d to the f o r m Atacz OC~ + B .ao
tz
+
C ai o)i
=0,
a and C ai ' b e i n g functions of the c o e f f i c i e n t s of s y s t e m (3.14), f o r m the fundaw h e r e the q u a n t i t i e s A ao~ i , Bee, m e n t a l d i f f e r e n t i a l - g e o m e t r i c o b j e c t of one of the a b o v e - m e n t i o n e d s p a c e s and have a s t r i c t l y - d e f i n e d s t r u c t u r e (depending on the n o n s i n g u l a r t r a n s f o r m a t i o n s r e l a t i v e to which s y s t e m (3.14) is examined). If s y s t e m (3.14) is e x a m i n e d r e l a t i v e to the t r a n s f o r m a t i o n p s e u d o g r o u p (3.2), then the o b j e c t s of l i n e a r d i f f e r e n t i a l - g e o m e t r i c and of affine c o n n e c t i o n s of s p a c e j i (Vm, Vn ) h e v e b e e n d e t e r m i n e d for r = 1, m = n; r = mn; n = r m and the o r d e r of t h e i r c o v e r by the f u n d a m e n t a l d i f f e r e n t i a l - g e o m e t r i c o b j e c t {Ad e , B a, C a } h a s b e e n e s t a b l i s h e d . C e r t a i n s u b c a s e s of the g e n e r a l c a s e h a v e b e e n studied individually, i.e., when the t r a n s f o r m a t i o n group for the independent v a r i a b l e s is l i n e a r (r = 1, n = m2; r = n, m is any n u m b e r ; r = m = n), as well as u). in a n u m b e r of s u b c a s e s of the p a r t i c u l a r c a s e , i.e., when A ai s = A aS(x, i T h e g e o m e t r y of s y s t e m (3.14), e x a m i n e d r e l a t i v e to the n o n s i n g u l a r t r a n s f o r m a t i o n s (3.2 t) h a s b e e n studied in a n a l o g o u s s u b c a s e s , i.e., the o b j e c t s of affine and of l i n e a r d i f f e r e n t i a l - g e o m e t r i c c o n n e c t i o n s of the s p a c e Vn+m+nm have b e e n obtained and the o r d e r of t h e i r c o v e r by the f u n d a m e n t a l d i f f e r e n t i a l geometric object has been established. The o b j e c t s of affine c o n n e c t i o n of the s p e c i a l f i b e r e d s p a c e L N w e r e i n t r o d u c e d in [74] and it w a s p r o v e d that if s y s t e m (3.14) is e x a m i n e d r e l a t i v e to the m o s t g e n e r a l t r a n s f o r m a t i o n s of the unknown f u n c tions and of the independent v a r i a b l e s , then the o b j e c t s of affine c o n n e c t i o n of s p a c e L ~ a r e c o v e r e d by a d i f f e r e n t i a l p r o l o n g a t i o n of the f u n d a m e n t a l d i f f e r e n t i a l - g e o m e t r i c o b j e c t { A i a s , t3a, ~ '1a } (the o r d e r of the c o v e r is not h i g h e r than third). 3. G e o m e t r y of N o r m a l S y s t e m s of S e c o n d - O r d e r P a r t i a l D i f f e r e n t i a l E q u a t i o n s . L e t us c o n s i d e r the d i f f e r e n t i a l equation s y s t e m
ou~du~
~ ' '0u]
(3.15)
to the beginnings of w h o s e g e o m e t r y ' w e r e devoted the r e s e a r c h e s of D o u g l a s [135], Su [232-240], and o t h e r s . The g e o m e t r y of d i f f e r e n t i a l equation s y s t e m (3.15) is u n d e r s t o o d as the g e o m e t r y of the f i b e r e d jet s p a c e j1 (Vm, Vn ) r i g g e d with the field of the f u n d a m e n t a l d i f f e r e n t i a l - g e o m e t r i c object (3.16) where
--
kh/~
"13
c~ /./"
(3.17)
605
If differential equations (3.16) a r e invariant r e l a t i v e to the linear t r a n s f o r m a t i o n s ua = A~u" + Aa , then the g e o m e t r y of the c o r r e s p o n d i n g differential equation s y s t e m is called the g e o m e t r y of K - s p r e a d s (in the terminology of Douglas). The g e o m e t r y of K - s p r e a d spaces was examined by ]3ieberbach [115], ]31iznikas [7, 10]~ Bortolotti [118], Ku [198-200], Ishihara and Fukami [180], Suguri [241], T a r i n a and Artin [246], T e o d o r e s c u and Rusu [247], Hu [177], Hua [178], Yano and H i r a m a t s u [274, 275], and others. We r e m a r k that the space of K - s p r e a d s is n e c e s s a r i l y p r o j e c t i v e l y flat when an a b s o l u t e l y - p a r a l l e l v e c t o r field exists for a given surface element (see [178]). 13y prolonging s y s t e m (3.16), we obtain a sequence of fundamental d i f f e r e n t i a l - g e o m e t r i c objects of the rigged fibered space j l (Vm, Vn):
(3.18)
H u> (n, rn)c H 2(n, rn)c ... C H (p~(n, m ) C . . . , where H (') (n, m) = {Hg,
,
H
n
H~vk, H a~f M~e h J t
' ....
The second fundamental d i f f e r e n t i a l - g e o m e t r i c object H (2) (n, m) always c o v e r s the objects of linear d i f f e r e n t i a l - g e o m e t r i c connections of the rigged fiber space j1 (Vm, Vn), as well as the objects of affine and of t e n s o r connections of space j l (Vm, Vn ) (see Bliznikas [12]). If the rigged fiber space j1 (Vm, VII) is a space of K - s p r e a d s in the sense of Douglas, then the third fundamental d i f f e r e n t i a l - g e o m e t r i c object H (a) (n, m) c o v e r s the object of projective connection, defined by T h o m a s r p a r a m e t e r s . This connection is invariant r e l a t i v e to p r o j e c t i v e t r a n s f o r m a t i o n s of object Hia/% i.e., t r a n s f o r m a t i o n s --I i i e H ~ : H~ q- f~T~,
where ~ f l
(3.19)
is a t e n s o r field p o s s e s s i n g the property(IIAflll is the m a t r i x inverse to m a t r i x IIA~ If):
It turns out that the fundamental differential=geometric objeqt H (4) (n, m) of the space of K - s p r e a d s c o v e r s the object of c e n t r o p r o j e c t i v e connection II~u, II~, where II}~. a r e the intrinsic Thomas p a r a m e t e r s o i ~ and IIij IS the algebram cover of the d l f f e r e n t l a l - g e o m e t r m object (II~k , Viii]k) (V i is the symbol of nonholonomic base derivative r e l a t i v e to an intrinsic object of linear d i f f e r e n t i a l - g e o m e t r i c connection). The diff e r e n t i a l - g e o m e t r i c object H N) (n, m) c o v e r s also the object of Cartan projective connection (see 131iznikas .
.
.
.
J~
~'J
9
9
J~
.
o
.
[12]). Thus, with the fourth differential neighborhood of a space of K - s p r e a d s t h e r e a r e always connected c e n t r o p r o j e c t i v e l y - c o n n e c t e d spaces (whose base is the fibered space j l firm , Vn)) and the sheaf of spaces of projective connection in the sense of Caftan, depending upon the choice of the two fields of differentialg e o m e t r i c objects (of the c o v e c t o r field and of the contracted object of affine connection). If differential equations (3.16) are invariant r e l a t i v e to the p r o j e c t i v e t r a n s f o r m a t i o n s ~
A~ u~ -~ A ~ AV u~ q- A
then the g e o m e t r y of the rigged space j1 firm, Vn) is called the projective g e o m e t r y of a space of K - s p r e a d s (see Hu [177]). In this c a s e too t h e r e exist projective connections adjoined to s y s t e m (3.16), but th.ey a r e c o v e r e d by the higher differential prolongations of the fundamental d i f f e r e n t i a l - g e o m e t r i c object H~fl. We r e m a r k that if differential equation (3.15) is examined r e l a t i v e to pseudogroup (3.2'), then the study of its g e o m e t r y r e d u c e s to the study of the g e o m e t r y of a h i g h e r - o r d e r fibered space (composite manifold in the sense of Vagner), i.e., of second o r d e r if the fibered space Vn (Vm) examined e a r l i e r is
606
reckoned to be a f i r s t - o r d e r fibered space with a given field of the fundamental d i f f e r e n t i a l - g e o m e t r i c object
H~: v,,
~
~ ~
,~ ,~ ~ t
~z~ - - "v s v
~ st = H ~
r ~ + H~
0 v.
In this case, depending upon the relations between the d i m e n s i o n s rn and n, we can find the objects of linear d i f f e r e n t i a l - g e o m e t r i c and of affine connections. These connections have a s t r i c t l y - d e f i n e d o r d e r of i cover by a given fundamental d i f f e r e n t i a l - g e o m e t r i c object Ho~fi, which depends upon the r e l a t i o n between the dimensions m and n. 4. G e o m e t r y of L i n e a r and Quasilinear Systems of Second-Order P a r t i a l Differential Equations. It is v e r y well known that many p r o b l e m s of an applied nature often lead to a differential equation of the following f o r m : g ~ (u)
~
Ous Ou~
+ a ~ (u) ~
Ou~
+ a (u) x = 0,
(3.20)
defined on a C2-manifold Vm. If det]]g~fi 11 ~ 0, then manifold Vm can be t r e a t e d as a Riemann manifold whose m e t r i c is defined by the t e n s o r g~/3 i n v e r s e to t e n s o r gC~fi. The differential prolongations of the diff e r e n t i a l - g e o m e t r i c object {gO~fi, a % a'} c o v e r the Riemann connection F~fi and the Well connection II~fl (see Ibragimov [32-42], Ovsyannikov [75-87]): 1
n~, = r ~ + y (Qs 6~ + Q~6~-- Q~g~),
(3.21)
where Q ~
2
n-- 2 (a~ + g~ r~), q~ = g~ Q~. o
Then the differential equation is reduced to the invariant f o r m (Va is the symbol of covariant derivative r e l a t i v e to the Well connection): o
o
ga~ Vs V~ u @ ax = O.
(3.22)
Two equations of f o r m (3.20) are equivalent if and only if (see Ovsyannikov [79, 84]): K~ = K~, H = H, where (K~/3 is a t e n s o r , H is a s c a l a r , Va is the symbol of covariant derivative r e l a t i v e to the Riemann connection) Ks~ = V~b - - V~ b~, H - - - 2a + V~ a~ + + a~ as -{- n 2 2 (n-- 1) R, -
-
b~ = a ~ + gV~I'~, as = g ~ a~, bs = gs~ b~' and R is the s c a l a r c u r v a t u r e of space Vn. The m o r e general c a s e of Eq. (3.20), i.e., when the free t e r m has the f o r m ~I'(x, u), was examined by I b r a g i m o v [32-42] who c o n s i d e r e d the p r o b l e m of seeking the continuous t r a n s f o r m a t i o n groups (in the sense of Lie) a d m i s s i b l e by Eq. (3.20) (free t e r m ~I,), defined on Riemann space Vn. If this equation admits of a motion group of Riemann space Vn of maximal o r d e r n (n + 1)/2 (space of constant c u r v a t u r e ) , then the equation being analyzed is equivalent (relative to the t r a n s f o r m a t i o n s x ~ = x i ( x ) , F[u] =e--iF[uel], F [ u ] = O(x)F[u], F[u] is the left-hand side of the equation) to the following equation CR ~ 0): n+2
n--2
Au + un---~q9(u(1 + r)-T) = 0,
(3.23)
607
where tl
n
r,=
02
e, = • 1, A = Z
If R = 0, this same equation is equivalent to the equation (cp is an a r b i t r a r y function) Au + ~ (u) = o.
When the equation admits of a group of c o n f o r m a l t r a n s f o r m a t i o n s of a Riemann space of maximal o r d e r (n + 1) (n + 2)/2, then it is equivalent to the equation (a is an a r b i t r a r y constant) n+2 AU ~ - Gf~Un - 2 ~- O,
In addition, in e v e r y Riemann space Vn the equation n--___.~2 R x = 0
(3.24)
is invariant r e l a t i v e to the group of conformal t r a n s f o r m a t i o n s of space Vn (Ibragimov [32-42]). In general, Ovsyannikov [75-87] and I b r a g i m o v [32-42] and his students pay m o r e attention to questions on the group p r o p e r t i e s of the equations (group classification, invariant or p a r t i a l l y - i n v a r i a n t solutions, etc.). In o r d e r to underline the difficulties of finding the group classifications of differential equation s y s t e m s of the f o r m defined, we mention a few, as far as we know, as yet unsolved or incompletely-solved p r o b l e m s . Apparently, t h e r e still does not exist a group classification of the equations of m a g n e t o h y d r o dynamics (in the t h r e e - d i m e n s i o n a l case) and the group admitted by the Einstein equations (the general t h e o r y of relativity) is unknown, etc. In o r d e r to emphasize the analytic subtlety and the g e o m e t r i c profundity of these p r o b l e m s we presen% as a solved example, a p r o b l e m from the t h e o r y of nonlinear heat conductivity. Certain problems of nonlinear heat conductivity r e d u c e to an equation of the f o r m (after a p p r o p r i a t e manipulations)
0 Q(U)~x ) Ox
Ou at
(3.25)
The fundamental group of this equation for f (u) = const was f i r s t found by Lie. The main point of the p r o b l e m on group classification of equations is to find the fundamental group for an a r b i t r a r y function f(u) and a f t e r w a r d s to establish those p a r t i c u l a r f o r m s of f (u) for which the fundamental group of the given equation (or, in the g e n e r a l c a s e , of the given system) is wider in c o m p a r i s o n with that group which c o r r e sponds to an a r b i t r a r y f ( u ) . F o r Eq. (3.25) the g e n e r a l Lie group is t h r e e - p a r a m e t r i c : X~ -- -~-, X~ = ~ x ' Xa = 2t
+ x 7x'
o ) " If f ( u ) = u 2m (m is an a r b i t r a r y constant), X 4 = - - t ~ -o + 7u
but when f(u) = e u, it is f o u r - p a r a m e t r i c
mxo + u
the fundamental group is f o u r - p a r a m e t r i c also
. However, for the exelusive ease when
m = - 2 / 3 , this group is f i v e - p a r a m e t r i c {X~ = - - x ~ ~ 4-3xu o ) ~,
o~
~'~"
Thus, many facts on the motion groups of Riemann spaces or of affinely-connected spaces, as well as a s p e c t s of the invariant classification of Riemann of F i n s l e r spaces by the algebraic and analytic p r o p e r t i e s of the c u r v a t u r e t e n s o r , have c o n c r e t e applications in the g e o m e t r y of differential equations, not only on the theoretical side, but also, p a r t i c u l a r l y , in the applied sense. Let us consider the general ease of Eq. (3.20), i.e., the quasilinear partial differential equation (i, j . . . .
608
= 1, 2 , . . . ,
n; a , fl . . . .
= 1, 2 . . . . .
m)
h7 ~ (x, u)
O~x~ (x, u Ox) Ou~zOu-------~ + h ' Ou] = O,
(3.26)
i.e., this equation coincides with Eq. (3.20) when n = 1. In this c a s e the quantities h~ fl and h f o r m a fundamental d i f f e r e n t i a l - g e o m e t r i c object of the s p a c e
K(~) n,m" For the cases m < n, n = m (m + 1)/2, and m < n < m (m + 1)/2 it has been proved that the first differential prolongation of the fundamental differential-geometric object {hi~8, h} covers the objects of linear
differential-geome ic connection of space
while the objects of affine connections r}k and
(see
Lupeikis [68, 70]) are covered by the second differential prolongation of this same object. In [68, 70-73] Lupeikis investigates the geometry of a differential equation system of the form
(3~ (a,b . . . . .
1,2 ..... r),
examined r e l a t i v e to the nonsingular t r a n s f o r m a t i o n s (3.2). In c a s e r = m, m < n it was proved that the objects of affine connections ~ and Flk of s p a c e K(l) a r e c o v e r e d by, r e s p e c t i v e l y , the f i r s t and second differential prolongations of i~e~ fundamental d i f f e r n ' t m l g e o m e t r i c object {h a ~ f i , h a ) , while the objects of linear d i f f e r e n t i a l - g e o m e t r i c connections Mia~ and N i k (of s p a c e K~1!m) a r e c o v e r e d by, r e s p e c t i v e l y , the s a m e differential prolongations of the fundamental diff e r e n t i a l - g e o m e t r i c object. F o r n = r m (m.+ 1)/2, r n = m (m + 1)/2, and r = n m (m + 1)/2 the objects of linear d i f f e r e n t i a l - g e o m e t r i c connection N}.. and M}~ a r e c o v e r e d by, r e s p e c t i v e l y , the f i r s t and second differential prolongations of the fundamental dTf~erenti~-~geometric subobject {h a~fi ) and induce the affine connections F]k and FT~fl with c o v e r o r d e r higher by unity.
r~k
If ~-ihac~fl = h ~ f l (x, u), then in the c a s e nr = m (m + 1)/2 the objects of affine connection and F ~ T a r e c o v e r e d by, r e s p e c t i v e l y , the f i r s t differential prolongation of the d i f f e r e n t i a l - g e o m e t r i c subobject {h a ~ f l ) and the third differential prolongation of the d.ifferential-geometric object {hiaafl, ha}, while th.e objects of linear d i f f e r e n t i a l - g e o m e t r i c connections N~j and M~fi have the s a m e c o v e r o r d e r s as for F jk 1 and Y~T" When r = (rim (m + 1)/2) + 1, with s p a c e t ~ l ) t h e r e is a s s o c i a t e d a s p a c e (g(nl!m, F ) c a l l e d t h e g e n e r a l i z e d a r e a l s p a c e (where F is a function of the coeffic'ien-ts of s y s t e m (3.37)). We can ot~t~iia th.e sheaf of such a r e a l s p a c e s , defined by a s c a l a r function X = X (x, u). The object of affine connection A]k (see Lupeikis [71]), c o m m o n to the sheaf of t h e s e g e n e r a l i z e d a r e a l s p a c e s , is c o v e r e d by the t h i r d differential p r o l o n g a tion of the d i f f e r e n t i a l - g e o m e t r i c object {11a n t i , h a } . C o v e r s of objects of affine and of linear d i f f e r e n t i a l - g e o m e t r i c connections of s p a c e K~l)~ have been found when r = m. In the c a s e of differential equation pendent variables,__the e x i s t e n c e has been tion of space K~n~,m ~ when r = n, m < n; m m e n t a l d i f f e r e n t i a l - g e o m e t r i c object h a s
s y s t e m (3.27) with a linear group of t r a h s f o r m a t i o n s of the indep r o v e d of a linear d i f f e r e n t i a l - g e o m e t r i c and of an affine connec<- nr; r = n m (m + 1)/2 and the o r d e r of their c o v e r by the fundabeen established.
C e r t a i n c l a s s e s of special differential equation s y s t e m s of f o r m (3.27) have b e e n investigated, i.e., systems h aaft (x, u)
O~x~ Ox~ Ox] -4- caa (x, it" Oxi o~---j~~ + bT?~ (x, u) ~ o S -- ' ) ~ + ~'~ (x, u) = o.
(3.28)
in this c a s e the fundamental d i f f e r e n t i a l - g e o m e t r i c object {h~, vht ~I ~ , /~] h a s the following subobjects: {h~], /~} ' ~ . {h~,. b.,] ~ } . {hi~. ~., } .{h~, . b~, ~, I The e x t"s t e n c e of ob j9e c t s of linear d i f f e r e n t i a l g e o m e t r i c and of affine connections of s p a c e K(l!m has been p r o v e d and the o r d e r of their c o v e r by the fundamental d i f f e r e n t i a l - g e o m e t r i c object or by a subobject of it has been d e t e r m i n e d (see [70]). ,
609
By the g e o m e t r y of k q u a s i l i n e a r differential equation s y s t e m (3.27), examined r e l a t i v e to t r a n s f o r mation pseudogroup (3.2'), we m e a n the g e o m e t r y of a s e c o n d - o r d e r f i b e r e d s p a c e (composite manifold in the s e n s e of Vagner) with a given fundamental d i f f e r e n t i a l - g e o m e t r i c object {h i , h } (see B h z n i k a s [10], Lupeikis [72, 73]). The g e o m e t r y of s y s t e m (3.27) has been investigated when r = 1.'n = m (m + 1)/2; n = r m (m + 1)/2; r n = m (m + 1)/3; r = nm (m + 1)/2 for the g e n e r a l (when h a n t i = h a ~ (x, u, 8 x/8 u)) and the p a r t i c u l a r (when h a i t i = hlqUfl (x, u)) c a s e s . It h a s been proved that the objects of affine connection of the s p a c e being e x a m i n e d a r e c o v e r e d in the g e n e r a l c a s e by the second, while in the p a r t i c u l a r c a s e by the fourth differential prolongation of the fundamental d i f f e r e n t i a l - g e o m e t r i c object. We r e m a r k that the existence of an object of affine connection i m p l i e s the existence of an object of l i n e a r differential g e o m e t r i c connection of s p a c e K~(9m with the s a m e o r d e r of c o v e r by the given fundamental d i f f e r e n t i a l - g e o m e t r i c object or by a subobject o[ it as the object of affine connection. The object of affine connection, obtained as a r e s u l t of a partial differential prolongation of an object of linear d i f f e r e n t i a l - g e o m e t r i c connection of s p a c e K~!m, h a s an o r d e r of c o v e r by the fundamental diff e r e n t i a l - g e o m e t r i c object g r e a t e r by unity than the obj'ect of linear d i f f e r e n t i a l - g e o m e t r i c connection. It t u r n s out that objects of affine connection, obtained as a r e s u l t of a p a r t i a l differential prolongation of the components of a linear d i f f e r e n t i a l - g e o m e t r i c object, differ f r o m the objects of affine connection, obtained d i r e c t l y a s a r e s u l t of p a r t i a l differential prolongations of the given fundamental d i f f e r e n t i a l - g e o m e t r i c object, in c o v e r o r d e r and in construction. Special c a s e s w e r e i n t r o d u c e d in [70], when t h e s e objects of affine connections do coincide. F o r e x a m p l e , when r = 1, n = m (m + 1)/2 the object of affine connection, obtained as a~result of a p a r t i a l differential prolongation of the given fundamental d i f f e r e n t i a l - g e o m e t r i c subobject {h~P}, is c o v e r e d by its f i r s t differential prolongation and is contained in the object of affine connection, obtained as a r e s u l t of a single partial differential prolongation of the object of linear d i f f e r e n t i a l - g e o m e t r i c connection, c o v e r a b l e by the second differential3arolongation of the s a m e fundamental d i f f e r e n t i a l - g e o m e t r i c subobject. T h e y coincide only in c a s e h~ fl = h~h'(x, u). The g e o m e t r y of linear and q u a s i l i n e a r s y s t e m s of s e c o n d - o r d e r p a r t i a l differential equations was studied also by Vranceanu [267-269], D o b r e s c u [134], Aeppli [110], J u r c h e s c u [182], Sauer [226], I n g r a h a m [179], I z r a i l e v i c h [43], K r i s z t e n [197], H a i m o v i c i [153-157], H a i m o v i c i [158-160], and others. 5, G e o m e t r y of Norm_al S y s t e m s of H i g h e r - O r d e r P a r t i a l Differential Equations. differential equation s y s t e m (m < n): OP+lx
(XC~,u~OXk OPxk)
-.1-/-./~al
Ou% . . .O~p+ 1
%+1
Ou~
.....
- . . . .
Ou~l . . .Ou~p
O.
Let us c o n s i d e r the
(3.29)
The g e o m e t r i c a s p e c t s of such a s y s t e m (p is a r b i t r a r y , p >- 3) w e r e taken up in the p a p e r s by I z r a i l e v i c h [44], Kawaguchi and Hombu [185], Mirodan [213], Tonooka [251], and o t h e r s , in which objects of v a r i o u s connections w e r e adjoined to s y s t e m (3.29) (it is a s s u m e d that the t e n s o r field gaff is given). Kawaguchi and Hombu noticed that the c o n s t r u c t i o n of an object of affine connection F f l y , c o v e r e d only by the d i f f e r e n tial prolongations of the d i f f e r e n t i a l - g e o m e t r i c object H i~ , + ~ )=_/ - / i =1...%+1". P
~ H~I*""O~p-~-I
=I.*""=p'~-I
co~+H =1 ~ "=p"I-1TM0'1~ + ~~~H=1" ~'1~1" P'= OV o "~p~-I~r ~ l ' " P a ' =1 'O~p~-l/~
(3.30)
a~l
where p+l
~,=, "=.+.= ~] ' (o+o!s: ~'kl""ks ~, 1 f~l( ,=~ (;~+ +j,=,+~ is' -'%'"=i, " " " p@l
'' " f~]I"~''"J-]S--I-I-I'"i~p"{-I - Z
s~2
(p + 1)r 0]= "~ ,~..(p's'-~ 1')! ( z'"%/%+z'"% +z)?'
p o s e s a v e r y difficult p r o b l e m in the g e o m e t r y of differential equation s y s t e m s (3.29) (with a r b i t r a r y p), which is called the Kawaguchi p r o b l e m . In a l m o s t all the g e o m e t r i c r e s e a r c h e s on the g e o m e t r y of diff e r e n t i a l equation s y s t e m s (3.29), by the g e o m e t r y of t h e s e s y s t e m s is m e a n t the g e o m e t r y of the f i b e r e d
610
jet s p a c e s JP (Vm, Vn) rigged by the d i f f e r e n t i a l - g e o m e t r i c object H~(~.,). By prolonging the differential equation s y s t e m (3,30), we obtain a sequence of fundamental d i f f e r e n t i a l - g e o m e t r i c objects of the rigged f i b e r e d s p a c e JP (Vm, Vn):
H (') (n, m, p ) g H (2) (n, rn, p ) C " . C H (~) (n, m, P ) C ' " The d i f f e r e n t i a l - g e o m e t r i c object H 2 (n, m , p) h a s a subobject ized t e n s o r ,
H :'~(m ~(P) which is always a g e n e r a l Cc(p_]_l)kl~
If L~ is a c h a r a c t e r i s t i c o p e r a t o r defined by the equalities p--1
then the quantities Ng Cr
~ L
i [1~Jt]e]CC(P)
f o r m a t e n s o r c o v e r e d by the f i r s t differential prolongation of the fundamental d i f f e r e n t i a l - g e o m e t r i c obi ject H~ If p = 2, 3, 4, then a t e n s o r a~fi exists which is an a l g e b r a i c c o v e r of the t e n s o r s N~(m~ 8 ~(P+D "
and ~i ~(mv(p) (see Bliznikas [17, 18]). JJ C~(p~_l)kh If p > 2, t h e r e e x i s t s a t e n s o r a a l - " ~ q c o v e r e d by those s a m e t e n s o r s (q = 2 ( p - 2 ) ) . With the aid of t e n s o r a ~ t...~q and of o p e r a t o r L T in the g e n e r a l c a s e we can c o n s t r u c t an object of affine connection --I~BT (it is c o v e r e d by the t h i r d differential prolongation of the fundamental d i f f e r e n t i a l - g e o m e t r i c object). This p r o b l e m still h a s not been solved for m > n and a r b i t r a r y p. w
On t h e
Global
Geometry
of Differential
Equation
Systems
In connection with the development of a global t h e o r y of fibered s p a c e s and of the t h e o r y of differentiable mappings on differentiable manifolds t h e r e e m e r g e s a s e r i e s of new questions and p r o b l e m s lying at the juncture of differential topology, a l g e b r a i c g e o m e t r y , functional a n a l y s i s , and the theory of differential equations. T h e s e m u l t i f a c e t e d connections a r e b r i e f l y explained by the c i r c u m s t a n c e that in the a n a l y s i s of differentiable mappings an i m p o r t a n t r o l e is played by the t h e o r y of s i n g u l a r i t i e s , in which t h r e e c o m ponent p a r t s can be delineated: differential, topological, and homological. The f i r s t p a r t is m o s t closely r e l a t e d with g e o m e t r i c p r o b l e m s not only in the t h e o r y of jets and fibered s p a c e s , but also in the g e o m e t r y of differential equation s y s t e m s . It is evident that all t h r e e p a r t s have c o n c r e t e applications also for the intrinsic theory of differential equation s y s t e m s (the w o r k s of Shwarz, Atiyah, Singer, and others) and for M o r s e theory. In v a r i o u s a r e a s of g e o m e t r y , in p a r t i c u l a r in differential g e o m e t r y , we encounter questions on the e x i s t e n c e of the solutions of the s y s t e m s obtained and often it is n e c e s s a r y to know as well the a r b i t r a r i n e s s of the solution. When solving such p r o b l e m s in the local s e n s e we e s s e n t i a l l y use C a r t a n ' s t h e o r y of f u l l y - i n t e g r a b l e differential equation s y s t e m s and, p a r t i c u l a r l y , C a r t a n ' s t h e o r y of s y s t e m s of differential equations in involution. T h e r e f o r e , when solving c e r t a i n g e o m e t r i c p r o b l e m s in .the global s e n s e we need the global t h e o r y of differential equations and e s p e c i a l l y those sections which have d i r e c t g e o m e t r i c applications. In this direction the first, most systematic, investigations are due to Kuranishi (see [202-205]), Kodaira [191], Spencer [192,231], and others. These investigations were based on the local results of other authors, on the theory of higher-order jets (see Ehresmann [138-144]), and on the theory of jets of local cross sections of a fibered space (in the local sense these spaces are Vagner's manifolds or are fibered
s p a c e s in the s e n s e of Vagner). Let N, N' be differentiable manifolds, (N', N, ~r) be a fibered space (v is a s u r j e c t i v e mapping, i.e., 7r: N' ~ N is a projection), JP ~- J P ( N ' , N, v) is the s p a c e o f p - t h o r d e r jets of local c r o s s sections for the f i b e r e d space (N' N, 7r), i.e., f :x EN---~y=f(x)E n-I(x). The s p a c e JP (N', N, ~r) is a fibered s p a c e as well (see Kuranishi [202-205]) and the c o o r d i n a t e s of the points of this s p a c e (in a s p e c i a l l y - s e l e c t e d a t l a s a r e the v a r i a b l e s x i, y a ~ v f / O x i l . . . ~ x i V (v = 1, 2 . . . . . p; i, j = 1, 2 . . . . . n; o~, fi = 1, 2 . . . . . m), w h e r e f is a
611
local c r o s s section. If U'is an open subset of s e t JP and A~ is the g e r m s h e a f of functions on U C l P , then a p-th o r d e r p a r t i a l differential equation on the f i b e r e d s p a c e (N T, N, ~r) is defined by the giving of a certain UCIP and of a s u b s h e a f Z of ideals of s h e a f A ~ locally g e n e r a t e d by s h e a f A ~ i.e., in s o m e a t l a s f o r m a b l e by 1 - f o r m s (v - p - l ) : d y ~"- - p~ dx t .... fOix...t~, ~ ~ d p l~1. .-i v ~ p tz 1 . .i v k d x k.
A c r o s s section f is a solution of Z if E: jP ( f ) --~ 0 for e v e r y JP the differential d (go pp+l), w h e r e (pp+l is the n a t u r a l projection)
zEN.
F o r any function g given on
~+1 : jp+1 -)- jn,
g e n e r a t e s a 1 - f o r m on jp+l. If we apply this operation to equation ~, we obtain a t r a n s i t i o n Z to a (P + D - s t o r d e r differential equation P "(Z) which is called the s t a n d a r d prolongation for ~. It t u r n s out that f is a solution for ~ if and only if it is a solution for P (Z). L e t T be a tangent bundle o v e r N and F (N f) be a bundle o v e r N', c o n s i s t i n g of v e c t o r s tangent to the f i b e r s . A f i b e r e d submanifold RPCJP is called a p - t h o r d e r partial differential equation on N ' (as on the bundle space). Then a solution of equation R p will be the c r o s s section whose j e t s lie in R p. We set Rp+q=]
q ( R p) (-] ] p + q ( N " ) .
The f a m i l y of s u b s p a c e s gp = F (Rp) ('] ~-1SP.T*@n~-I F (N')
(4.1)
in the bundle n-ISpT * | ~-t F(N') o v e r R p is called the symbol of equation R p, w h e r e sPT * is the s y m m e t r i c t e n s o r product (p-fold) of the cotangent s p a c e s T* and ~r0 ~ P~. If V and W a r e f i n i t e - d i m e n s i o n a l v e c t o r s p a c e s and, m o r e o v e r , (vl, . . . . Vn) is a b a s i s in V, while (v 1. . . . . v n) is a c o b a s i s in V*, then the h o m o m o r p h i s m 6(6 = 6p+l,j) 5: W@S 0+1V*@ A/V~-*-W|
p V* | Ai+1 V"
(4.2)
is defined by the r e l a t i o n n
6 ([
where
[EW|
|
..
. A v ' i ) = ~]~o, [ |
v " A . . "A o'i,
*, AIV * is the e x t e r i o r d e g r e e of s p a c e V* and 5v : W | S p+I V~
W | S p V'.
We a l w a y s h a v e 62-~-0,
i.e., ~p,j+l(Sp+l,3-~-0,
while the sequence 0_...~ ~7 @ SPV* ~ 1~7 @Sp-I V" | V* ~-~ W |174
A ~Y "~ -*-...---~ W|
SP-~V" |
| Am*-+ 0 is an e x a c t sequence ( s r v * = 0 for r < 0). If p :" k, then for the sequence {gP} of s u b s p a c e s g~ C W | SP V ~
612
(4.3)
we can define the sequence
8
6
6
0---~ gp_+gp-1 | V*-,.gp-~ | 6 gk | -~
6 V*-~W
|
9
k-1 V* | A p-k+1 V*
(4.4)
Sequences (4.3)and (4.4) a r e connected with differential equations in a natural way, b e c a u s e the family of subspaces (4~ looked upon as a sequence of spaces, is determined, f r o m the algebraic point of view, with the aid of two vector spaces. The groups of Spencer holonomies or the Spencer cohomologies H p - j ' j (see Kodaira and Spencer [192], Goldschmidt [146, 147], Guillemin and Kuranishi [15217 a r e determined with the aid of the equalities (HP-J,J = HP-J,J (gk), p > k):
Hv-I'l = Ker 6p_l,i/Im ~p__1_}_1,1.1.
(4.5)
The sequence of subspaces (4.1), generated by the prolongation of a given differential equation, is called involutive if sequence (4.4) is exact (it is always exact only in two t e r m s ) . The sequence of subspaces (4.1) is said to be q - c y c l i c if Hp,J=O
(p>~k, O<~]<~q).
The differential equation RP is said to be f o r m a l l y integrable if gp+q+l is a vector bundle over R p and 7rP+q :RP+q+t .-* RP +/ is s u r j e c t i v e for all q ~- 0 (see Goldschmidt [147]). A subspace
UCV* is said to be c h a r a c t e r i s t i c for gcW|
if
gN w | The o r d e r e d linearly-independent s y s t e m of elements ~ ..... ~ ~ W is called q u a s i r e g u l a r for the subspace ACV| .r~(A)=mindim ((VQW1)NA), where W1 r a n g e s over all ( d i m W - j ) - d i m e n s i o n a l subspaces in W, if ~(A) : dim ((V | Fn_k) n A) and Fn_ k is the linear hull of the v e c t o r s ~k+l . . . . .
(]--1 ~ k ~ n - -
1)
~n (n = dimW).
It t u r n s out that subspace A is involutive if and only if a q u a s i r e g u l a r basis exists in W (see Guillemin and Kuranishi [152]). On the other hand: if subspace g is involutive, then the subspace UCV* will be c h a r a c t e r i s t i c if and only if U contains a one-dimensional c h a r a c t e r i s t i c subspace (see Gufllemin [15117. Differential equation RP is f o r m a l l y integrable if 7r •: RP +i --* R p is surjective, gp+l is a vector bundle over R p, and H m,2 = 0 (for m >- p). In c e r t a i n c a s e s a f o r m a l l y - i n t e g r a b l e equation R p always admits of an p+/ (analytic) local c r o s s section having a given (p + l ) - j e t f r o m R at a given point (see Goldschmidt [14717. When a given differential equation is involutive, e v e r y q u a s i r e g u l a r s y s t e m can be prolonged up to a q u a s i r e g u l a r b a s i s (see Guillemin and Kuranishi [152]). F o r the g e r m sheaf of solutions of differential equation R p we can c o n s t r u c t the Spencer complex (see Kodaira and Spencer [192], Goldschmidt [146]). When the equation is elliptic, the c o r r e s p o n d i n g Spencer complex is elliptic as well (see Goldschmidt [14617. F o r c e r t a i n c l a s s e s of differential equation s y s t e m s we can in a c o r r e c t and invariant manner introduce the notion of global Cartan c h a r a c t e r s and obtain global c h a r a c t e r i s t i c s of the integrability conditions (both f o r m a l integrability as well as full integrability), as well a s the global p r o p e r t i e s of the solutions. Investigations on the global t h e o r y of differential equations, having an immediate b e a r i n g on g e o m e t r y and on the global t h e o r y of Lie groups or of Cartan subgroups, a r e r e f l e c t e d in the r e s e a r c h e s of Auslander [111], Auslander and Markus [112], ]3reuer [119], Guillemin [151], G u i l l e m i n a n d K u r a n i s h i [192],Goldschmidt [146-150], Johnson [181], Quillen [223], Kodaira [191], Kodaira and Spencer [192], Rauch [224], ~vec [243], -Uchiyama [254], and others.
613
LITERATURE 1,
2. 3.
4o
5. 6. 7. 8. 9. 10. 11.
12. 13. 14.
15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.
614
CITED
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27. 28. 29. 30.
31.
32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.
46.
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