Investigations of Laptev-Vasilev seminar at the Moscow University

Journal of Mathematical Sciences, Vol. 78, No. 3, February, 1996

G E O M E T R Y OF O R D I N A R Y D I F F E R E N T I A L E Q U A T I O N S . I N V E S T I G A T I O N S OF L A P T E V - V A S I L ~ E V S E M I N A R AT T H E M O S C O W U N I V E R S I T Y (1980-1992) L. E. E v t u s h i k

UDC 514.763.8

Introduction This review is composed in accordance with the lecture [12], read by the author in 1992 in Kazan' at the International Scientific Conference dedicated to the 200th anniversary of the birth of N. I. Lobachevski. The evolution of the geometric science that started from the ideas of Lobachevski, Gauss, and Riemann, which overturned not only geometry but the whole of mathematical natural science, has led now to the crystallization of the main notions of geometry and the realization of its subject and its place in mathematics. It is universally recognized that geometry studies the invariant intercommunications of different structures on smooth manifolds or submanifolds determined as the fields of geometric-differential objects. Among these structures one can find Lagrangians, differential operators, and differential equations, which we consider as being globally given on a smooth manifold and which are also analyzed in other aspects and by other methods in calculus and in theoretical physics. In spite of the great variety of these and others geometric structures all of t h e m can be determined in terms of smooth manifolds, their smooth maps or jets of these maps with such concrete definitions as a manifold with Lie group structure, or with an action of a Lie group of transformations, or with the structure of a fiber space. This unification of the fundamental notions and, at the same time, the maximum broadening of the bounds of modern geometry, which included in the sphere of its investigations the smooth structures of calculus and mathematical physics, induced the development of a universal invariant method of studying different geometric structures. So far the priority or even the monopoly of it belongs to the moving-frame method and the exterior differential calculus of E. Cartan, which received its modern interpretation in the geometry of fiber spaces in the geometric seminar of the Moscow University, directed successively by S. P. Finikov, G. F. Laptev, and A. M. Vasil'ev. Regarding the monopoly of the Cartan-Laptev method, we mean its uniqueness so far as being a universal method of describing any geometric-differential structures, but, at the same time, its propagation in a mathematical environment leaves much to be desired. Confirming the scientific priority of the native land of Lobachevski and developing the ideas of Cartan the Lobachevski prize winner - the Finikov-Laptev seminar brought a decisive contribution to the modern state of the Cartan m e t h o d by its long-standing activity and gave, at the same time, numerous examples of its applications to the different problems of geometry [3, 4, 11, 16]. The geometry of differential equations became one of the applications of this universal method of geometric investigations. This branch of the seminar's activity was guided by the long-standing works of E. Cartan [24]. The first steps in the geometry of systems of ordinary differential equations were taken by G. F. Laptev [17] and in the geometry of partial differential equations, by A. M. Vasil'ev (we mean here the equations of second order). These investigations were then extended by V. I. Bliznikas and A. K. Rybnikov, and in the geometry of ordinary equations, by L. E. Evtushik, N. V. Stepanov, and their students. The present review gives a description of the works of the last group of members of the seminar on the geometry of ordinary differential equations of higher orders given on smooth manifolds of arbitrary dimension. The review comprises the period from 1980 till 1992 - the year of Nicolai Lobachevski. The investigations of the period before the indicated one have been described in the preceding reviews [21, 22]. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 11, Geometry-2, 1994. 1072-3374/96/7803-0253512.50 9

Plenum Publishing Corporation

253

Let us note the main features of the works being reviewed, which also determine the structure of the present review. There exist two nonequivalent ways of the invariant definition of ordinary differential systems of (p + 1)order on a smooth manifold V.: in terms of the bundle Tp+I(Vn) of (p + 1)-velocities and in terms of its quotient bundle S1)+l(Vn) = T)'+I(V,,)/L~ +1, (L1p+I is a differential.group of the changing of a parameter t on the curve z(t) E V.) of (p + l) dements of tangeney as cross sections of these bundles over the p-order bundles of the same type:

f/)+l:

TP(Vn) _+ Tp+l(vn);

~/)+1: Sp(Vn)

._+ S/)+l(vn).

These two types of systems differ from each other by different powers of the manifolds of integral curves with the only exception being the class of so-called reducible systems, i.e., systems fp+l projectible onto systems of the type qaP+l:

TP(Vn) l'+' ~ Tp+I (Vn)

1

1

A reducible system fp+l and its projection, the reduced system ~/)q-1, have geometrically the same integral curves, but the system fp+l contains one extra equation which determines a family of parametrizations along each solution. The equations of geodesic lines in Riemannian, affiae-connected, and Finsler geometry have precisely this property. The series of works being reviewed contains a solution of the problem of a complete geometric description of systems of the type Tp+l and, separately, of the systems fp+l reducible to q~p+l. One had to find and apply different geometric structures and special methods to solve these problems. The initial part of the present review (Sees. 1, 2) is devoted to a description of these structures, which was found and investigated first by the author of this review [12, 13]. The coordinate structure of the bundles SP(V,,), which is more difficult than the structure of TP(V,~), demands the application of the method of canonization of frames of corresponding order when the systems Wp+l are ex~mlned, while the investigations of the reducible systems fp+l are possible within the general framework and in other geometric terms of the theory of nonlinear stable connections. 1. G e o m e t r l c - D i f f e r e n t i a l M e a n s of t h e I n v a r i a n t D e s e r l p t l o n o f O r d i n a r y Differential S y s t e m s on S m o o t h M a n i f o l d s In this section, we will describe the category of fiber spaces over a smooth manifold and the corresponding differential algebra of structural forms only within the scope of which the construction of the geometry of ordinary differential systems on Vn is possible. The category of bundles associated with the principal frame bundles of different orders over manifolds V,, will now be considered. To each bundle there corresponds a certain system of structural differential 1-forms. The morphisms of this category are described in terms of these forms, and also the instrument of invariant investigations of the main structure is constructed - this or another field of the geometric object determined over some bundle from the specified category. The object generating the field belongs to another bundle. We have to deal with objects of higher orders because of the specific character of the subject. The set of concrete bundles of the geometric objects that we have to deal with will be described first [12, 18, 14]. 1.1. The sequence of principal bundles HP(Vn) of p-frames over Vn with its natural projections (mor-

phisms)

V. += HI(v.)

+-.-.

<--

HP(V.) +- Hp+I(v.)

+---.

is the bar of the category of bundles of the geometric object over V,~. The structure group of the bundle HP(V,) is L~, generated by invertible p-jets of smooth maps from R ') into R" stationary at the point 0 E R". 254

The elements of HP(Vn), p-frames rE, are also invertible p-jets of maps from R" into Vn with origin at the point 0 6 R ", and endpoint x 6 V,, the point of application of the frame. The l o c l coordinates of these jets are x i x~, " ' ' ' x ik l . . . k p " AU other bundles of geometric objects over Vn are determined as associated with the p r i n c i p l bundle Let F be some manifold with an action of the group L~ (it is convenient here to consider the left action in contrast to the right action of L~ on HP(V,)). Then the associated bundle FHP(V,) ~ V,~ is determined as a quotient bundle FHP(V,~) = (HP(V,) x F)/L~ with respect to the L~ action: (r~,Y).l=(r~.l,l-lY),

(r~,Y) 6 H P ( V , ) x F ,

16L~,

x6V,.

The factorization map ~ : HP(V.) • r -+ F H P ( V . ) = (HP(V.) x F)/L~

(I)

assigns to any pair (r~, Y) the element y, = ~(r~, Y) def rpx. y 6 Fz C FHP(V,), where F , is a fiber of the associated bundle over the point x 6 V,~, consisting of the geometric objects y,. In the same way all known objects - vectors, tensors, forms, connections, jets, etc. can be defined. Here the formula of the factorization map =

r

(2)

takes in its coordinate form the sense of the reference of an object y= to the frame r~ at the same point x 6 V,~: the coordinates of Y 6 F are uniquely determined as relative coordinates of the object y= respectively to the frame r~ by means of the n a t u r i l o c l coordinates of the object y= and the frame r E from (2). Any manifold with the trivial action of the group L~ can be taken as a standard fiber F and then FHP(V~) = Vn x F (for example, V,, • RN). The cross sections of such bundles, i.e., the functions on Vn with values in F (for example, AN), can appear during the process of investigation of any geometric structure on Vn as its invariants. Homogeneous bundles of geometric objects (the standard fiber F is a homogeneous space) play an important role in the consideration of the structures on Vn. First of i l , the bundles HP(V~) of p-frames with groups L~ as a standard fiber are homogeneous spaces. All other homogeneous bundles ca~ be defined by factorization of HP(V.) with respect to different closed subgroups G C L{. In order to tie that in with the genera/definition of the associated bundle FHP(V.) = (HP(V,) x F)/L~, being expressed by the factorization formula ~ : y~ = r E 9Y, it is enough to fix here Y = Y0 in any way, for example, coming from the simplicity of the coordinates of the element Y0. Then the formula y ~ = ~0(r~) = ,'~. ro (3) defines a factorization map ~0: HP(V~) -~ FHP(V.) = HP(V~)/G,

(4)

where G is an isotropy subgroup of the element Y0 6 F. Formula (3) expresses the specific adaptation (canonization) of frames r~ to the geometric object y,, when the set of frames r~ 6 ~ol(yz) is tied to every object y= 6 FHP(V,), and object y=, respectively to those frames, i w a y s has the same fixed coordinates, the coordinates of the element Y0. The map ~0 generates the bundle HP(V~) with the base FHP(V,), fibers ~o l(y~), and the structural group G, which becomes the canonical projection of this p r i n c i p l bundle. In the case of nonhomogeneous bundles, partial canonization is applied, when only the m a x i m u m admissible part of coordinates of the element Y is fixed. Let us define morphisms in the category K(HP(V,)) of associated bundles with the aid of morphisms in the category K ( L , ) of manifolds with left action of the group L~ as smooth maps commuting with the L~-action. Each morphism q00: F ~ E generates the corresponding morphism qa: FHP(Vn) -+ EHv(V~) as a mapping i d e n t i c l with respect to V, given by z~ = qa(yz) = qa(r~. Y) = r~ 9qa0(Y); the correctness is obvious. It is obvious that maps of factorizations (1) and (4) are morphisms. A morphism which is a diffeomorphism is an isomorphism. In other cases one can consider morphisms which are directed to bundles of s m i l e r dimension. 255

A morphism written in a coordinate form is in every fiber an algebraic operation of the construction of geometric objects with the aid of other objects invariantly connected with the ones created, and, in this sense, a morphism is the fundamental instrument for geometric constructions. However, all means of differential-geometric constructions do not become exhausted by these operations. It is necessary to add only the operation of differential prolongation, which is reduced to the natural tracing (with the aid of the notion of a jet) of the bundles of larger dimension which are projected to the original bundle by the law of morphisms. A morphism is called a scope, according to the terminology of G. F. Laptev. Since the essence of the Laptev method is contained in the sensible and purposeful employment of the operations of differential prolongation and scope in combination, this method is often called the method of prolongations and scopes. The instrument of exterior differential Cartan calculus, and, more exactly, the instrument of structural differential forms of a bundle based on it, as given by G. F. Laptev, in combination with the idea of a moving frame in its modern universal form, is the algebraic-differential basis of the practical employment of the operations of prolongation and scopes in this method. The search for a necessary scope mapping in an invariant form with respect to the moving frame is essentially addressed by the simple algebraic-differential scope test. Then the description of the instrument of structural forms with its particular realizations is given for purposes of the geometry of ordinary differential equations. The globally defined system of structural 1-forms of p-frame bundles is initial [11]: 9

i

i

i

O.)l , O.)k, Odkl, . . . , O.)kl ...kp ,

they are symmetric by subscripts and satisfy the structure equations .

.

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

.

~ .............

,

d~ k~....kp -- E ot...~l

.

~aj!

o)i

{kl....ke, ^

i . . . . . . . . . . . . . . . . . . . . . . . . . .

....o..

+wmA . . . . . . . . . . . . . .

.

8t

(5)

i

ka+~.... kv}l + t'dl A Odkt...kpl.

The local coordinate representation of the structural forms ovi,wk,wkt,i i ..., ~ *

i

8t .

.

.

(for example,

Wi

=

.

of a p-frame arises from the globally invariant definition of these forms as forms of location but can be restored with the aid of the structural equations (5). The above forms carry all the information about the geometric structure of the frame bundle. The structural forms of the associated bundle F t t v ( V , ~ ) are constructed with the aid of the differential of the factorization map z i d z k, oJi = x~(dx~ - x i , , w " ' ) , ,

~

x'k 9x~ = 51) by the local coordinates x', x}~,x~,,

~: Hv(V~)

"'''

x ik x . . . k p

• F ~ FHV(Vn)

as a completely integrable system of linear differential forms of the following kind: Ay J

:

d y J + ~ . ~ ( Y ) w ~, w',9

i

i

.

,

taji

(0)

globally defined on H v ( V ~ ) • F . The first integrals of this system are the local coordinates (x ~, Y J) of the bundle F H v ( V , ~ ) ; Y J are the globally defined coordinates of a standard fiber F, having, by the condition, an arithmetic space structure or that of its open subset. The formulas of the map ~ : yx = r~ 9Y, written in coordinates yd

=

f J ( i i Xk,~.kl, .

. .

i ) ,igkt...kp, y K ,

where x ~ , x ~ i , 9 ' ' ' x ik l . . . k p are the fiber's coordinates of the frame r~, represent a s~176 integrable system of equations A Y J = d Y v + ~Ja(Y)w c' = O, ovi = O.

256

~ the colpletely

We assume that the standard fiber is transitive if it does not have a unique coordinate system. In this case, it is enough to restrict the map ~, reducing F to any of its coordinate neighborhoods with coordinate YJ and we will come to forms of the type (6) again. Going further and reducing F to the fixed element Y0 E F, we reduce the map ~ to the factorization map ~0 itself of the bundle HP(Vn) - ~0: HP(V,) "+ FHP(V,~), and the structural forms AY J to 9 J = ff~(Y)w ~', w i, that is, to the completely integrable combination of structural forms wi,wk,wkt,i i ..., wik~...4 of the bundle HP(V,) with constant coefficients. The coordinate representation of the morphism

9 ' to) : v "r = fJ ( z L x'k,..., zkl...k,, makes up the first integrals of the system of forms 8J,w i. The practice of geometric investigations shows that homogeneous bundles or bundles with the simple coordinate structure of the standard fiber exhaust the needs of applications. Let us note that the structural forms of frame bundles and all bundles associated with them make up on the corresponding manifold the variable basis of forms setting the sub-bundle of frames of first order on this manifold. The question is the total manifold of the bundle. By virtue of this fact, any differential-geometric structure on a smooth manifold V,,, defined as a fiber map of one associated bundle of geometric objects into another (in a general sense, the field of a geometric object), generates the linear dependence of the structural forms of the second bundle on the structural forms of the first one. These differential relations are the algebraic-differential basis of the invariant investigation of the given structure. D e f i n i t i o n 1. Let FHP(Vn) and EHP(V,) be two given fiber spaces. The identity (with respect to V,~) mapping f: FHP(V,~) -+ EHP(Vn) (7) is called the geometric-differential p-order structure of the type (F, E) on Vn. R e m a r k . The orders of the bundles indicated in the definition may be different; moreover, in many cases, the second bundle has a natural projection (morphism) on the first one and so the map f is assumed to be a cross section. The investigation of a given structure is carried out by the following general scheme. The sequence of jets of the map f : i f , jgf, ... is constructed. Each of these jets, for example, j f, is the map

j f: FHP(V,) --+ J(FHP(V,),EHP(V,,)) into the jet bundle of maps of the type (7), which is also associated with a frame bundle of order greater than one unit. Thus we come to the first differential extension of the initial structure and to all others connected among themselves by a natural projection. We have the possibility of constructing the differential-geometric structures invariantly connected with the original one by finding at each stage of the extension all possible morphisms from the jet-bundles of the corresponding orders into other associated bundles. The general scheme of constructing the geometry of the structure being investigated is contained in the combination of the indicated operations of extensions and scopes. The procedure of adaptation (canonization) is added to this process if it is difficult to realize it in a general frame. For example, if EHP(V,) is a homogeneous bundle with morphisms FHP(V,), and the map f is its cross section over FHP(V,) (FHP(V,) in this case is also a homogeneous bundle), the canonization of p-frames by fixing Z = Z0 E E, the point of the standard fiber, brings us to the sequence of morphisms

HP(V,)

>EHP(Vn)

>FHP(V,~),

and then the cross section f is equivalent to specifying a subbundle HP(V,~) in the bundle FHP(V,~). The above-outlined scheme of investigations by the method of extensions and scopes is realized in terms of the structural forms of the corresponding spaces of geometric objects. Thus, the simple dependence

AY J = --O~------q-6AZ'~of the structural forms of a scoped bundle corresponds to the morphism qa : EHP(Vn) OZ" 257

F H P ( V n ) defined by the scope r : E --+ F of the standard fibers (YJ = ~d(Z a) is the coordinate entry of the scope). This dependence is the scope test. The search for a scope in practice means the search for the J . _,

functions qag(Z) (algebraic), for which the system of forms ~--gzaz appears to be completely integrable, hence making up the structural forms of the desired bundle being scoped. On the other hand, the cross section f : F H n ( V ~ ) ~ E H n ( V n ) implies the existence of the linear depen-. dence AZ a = AZ~w i + AZ~AY J

(s)

of forms AZ ~ on the given cross section under the structural forms AY J, W i of the bundle F H P ( V ~ ) , where Z~, Z~ form the relative coordinates of the cross-section jet j f . Equations (8) are called the structural equations of the map (in particular of the cross section) f. Their exterior differentiation and the application of the Cartan lemma give the structural equations of the prolonged map j f : F H P ( V ~ ) -+ J ( F H P ( V ~ ) , EHP(V~)):

A Z ~ = Z ~ w i + Z ~ A Y J,

A Z ~ = Z~ko: k -1- Z ~ K A Y K, = zz

,

=

A Z ~ = Z~kw k -1- Z ~ I K A Y K, = z

j,

where A Z a, together with AZ~, AZ~, form the structural forms of an object of the first jet. Applying the scope criterion to these forms, we construct the objects being scoped by the object of the first jet. The structural equations j 2 f , ... may be obtained in the same way. If the bundles F H P ( V n ) , E H P ( V n ) are homogeneous and the canonization Y = ]I0, Z = Z0 has been performed, then the structural forms A Y J = g2d(Yo)w a, wi; A Z ~ = r- j ~ , wi of these bundles become completely integrable subsystems of the structural forms of frame bundles and so they participate in the entry of structural equations. We conclude the concise review of the main constructions of the general theory of structures of higher orders on smooth manifolds and of the most important aspects of the method of structural differential forms and its fundamental operations of extension, scope, and canonization. The investigations of the series of works being reviewed in the geometry of ordinary differential systems of higher orders were started and are being continued within the framework of the conception formulated. A preparatory stage was necessary for the realization of this conception. This stage was contained in the selection and description in terms of the structural forms of bundles of the main geometric objects borrowed from the earlier stage of the research. Let us give a description of this stage [10, 11, 15]. 1.2. All bundles of differential-geometric objects of a higher order which we will deal with further, that are associated with a frame bundle of the corresponding order assume nevertheless a straightforward definition in terms of jets of the necessary order. The bundle TP(Vn) of p-velocities is the main bundle which will be the base for all other fiber spaces. It may be defined as a manifold of p-jets of the parametrized curves x ( t ) E V~ (the coordinate representation 9 dx i d'2x i dPx i of p-velocities in the point x ( x i) is x', d ~ ' dt 2 " " ' dtP )" In another series of works, the role of the base is assigned to the quotient space SP(V~) = TP(V~)/L~ of the elements of p-tangency (the coordinates of the elements of SP(Vn) are x i dza d2xa dPxa , dxl, (dxl)2,...,(dxl)p,

a = 2,3,...,p).

The structural forms of the bundle TP(Vn) are obtained by the multiple prolongation of the equations w i = V~dt executed on the curve x(t):

=

258

=

,

=

V~+ld

,

where

av~ = dV? + v&'~, av~ = dv~ + v?~'~ + v?vl~h, AV~ = dv~ + v ? ~ + 3v?v&h + v?vlv:'J~,., AV,~ = dV,~ + V4kwik + (4VskV1t + 3V?V])wi~,

+ 6v,"v;vy,,,h,,, + vtvlvyv&f,,,,~, ,r

~,

a v ; = ~v; + v.L~. s=l

1

2_,

(9)

1

21,v::... ~,~v~:~;'""

"at+...+c,,=p

"

axe the structural forms of the sequence of bundles of p-velocities in which the tangent bundle is the first

member, and V~i, V2i,..., V~ are relative fiber coordinates of the p-velocity with respect to an arbitrary p-frame. The structural forms of the bundle SP(V,~) arising from the awkward formulas of scope TP(V,) --+ SP(V,) are very difficult. Therefore, in works where the bundle SP(Vn) is the base, an adaptation of frames is necessary. This adaptation is connected with the space TP(V,,) by associating with any regular ( ~ r 0) p-velocity t~ the set of p-frames fixing the relative coordinates t~ by the simplest way V1i = ~ , V2' = V3' . . . . = Vd = 0. For the tangent bundle T(Vn) this means the coincidence of the first vector el of z

the 1-frame ei with t , 6 T~(Vn); all vectors collinear with it define an element of S,(Vn), a straight line. Z

This adaptation of frames brings us automatically to zero relative coordinates of the elements of SP(Vn) and to the sequence of morphisms

HP(V.)

>TP(V.)

> SP(V.).

From (9) it follows that the structural forms of TP(V,) turn into the following completely integrable subsystem of structural forms of the bundle HP(V,O i i i Od , W l , a ; l l

, ...

i )Od(p)

"

-~-~dSl...1, P

which includes the completely integrable subsystem wi,w l~,wn,~ . . . , w ~(p) (a = 2, ... , n) of the structural forms of the bundle SP(V,). Since the main problem in the series of works being reviewed is the construction of the geometry of ordinary differential systems, that is, the cross section

TP(V~) ,- - > TP+I(V.)

(

or SP(Vn)

~,+)xSp+I(vn)

)

,

the extended cross sections

j IP+I : TP(V,O --+ J(TP(Vn),Tp+I(v,o),

(j~: s~(v~)-, s(s,(v~),s,+'(v,))) appear and their structural equations axe obtained in the process of prolongation of the original structural equations. The structural equations of the cross section TP+I(V,~) are written in terms of the differential

forms ~', a V?, a V~, ... , a v/, av;+,.' 9

av;~, =dv;'+~ +(v+ ~ ) ! ~ s=l

,V2:... " (~1+".+t~,=p+l

!V2:,,,'~,...k.

"

P

~V;+I.AV; AV:=,P i

=

s

k

(io)

s=O

259

where VJp+lk,.-., 0 , v arc the relative components of j r v+l The structurM equations of the cross section Vv+Ik SP+I(V,,), obtained by the process of canonization, are p

wa(p+l) = , , VVp+lkW , , , o k + X'~,,,a 9 b 2_vvv+lbW(,),

a, b = 2,... ,p.

(11)

s=l

As we see, the original equations of these two types of systems are very different by form, which predetermines the various methods and the application of very different geometric constructions. A separate section will be concerned with the second type of systems9 Now we shall show the first steps of the construction of a geometry of systems fv+l. The first standard operation is the exterior differentiation of equations (10), which is convenient to realize by using the structural equations for the forms AV1/, A V g , . . . , AVv obtained in [14] and their algebraic properties expressed in Lemmas I and 2 in [14]. The further application of the Cartan lemma gives the equations for Y ; i+ l k~' 9 . . ~ Y ; +; l vk AV;,+i k8 = V ;I+ l k87"I

(s,,

= 0,1,..

. ,p).

Let vT(V,,) be the Whitney p-degree of the tangent bundle T(V,) and Jv(TV(V,), Tv+I(V,,)) be a vertical projection of the bundle J(TP(V,,),TV+I(V,~)). A set of p vectors in the tangent bundle is an element in the first bundle. The projection J(TP(V,), Tv+I(V,,)) --+ Jv(TP(V,,), Tv+I(V,)), defined by the maximum i P i s subgroup (V~+lk , .. 9 , i 1 of components V~+lk , s = 0 , . . . ,p, is an element of the second bundle. The following statement [14] sets the morphism:

~1: Jv(TP(V,),TP+I(V,)) •

-=+ vT(V~).

(12)

P r o p o s i t i o n 1. The formulas ~p~ =

1)]

(p +

V.k

v

s!

v k

,Tzl

(13)

8=s

where r are component8 of the set of p vector8 v, = ~ek, define the morphism ~1 and, as acon, equence, its maximum projection ~21 (~ = 2,... ,p): Jv(TP(V=),TV+I(Vn)) XT,(v~) TV+I(V,,)

+'>

1

vT(V~) p--1

l r(v )

The proof of Proposition 1 is based on the application of the scope test, the check of which is realized by the differentiation of formulas (13). During this process, the expression of the differentials of the values entering in the right-hand side of the formulas over the corresponding structural forms leads to the following relations:

,

(p+l)! =

s! AvA

_, -

which realize the scope test. The projection (indicated in Proposition 1) of the morphism is given by the part of (13) corresponding to the values of ~ = 2,... ,p. The map ~31 takes place in the following test of reducibility of the differential system fp+l. 260

T h e o r e m 1. The differential system fp+l: TP(Vn) _+ Tp+l(Wn) is reducible, i.e., it is projected into some differential system ~p+l: SP(V~) __+S~+I(Vn) if and only if

~.(t,) = ~.(t,)~,(e).

t, ~ T~(V,,),

where rP(t p) = t 1 E T(Vn) is a canonical projection from TP(Vn) onto T(Vn) and A,(t p) are some scalar functions on TP(Vn). The test of the reducibility may be written in the explicit coordinate form P

s!

( p + 1)! V/+2_ _ ~ e ! ( p + l __ e)! = e!(s-

~)!

.

xzk "p+ll

.~z' Vs + l - - e

=

A,V~

(14)

with regard to the formulas of scope in Proposition 1. The conditions of reducibility (14), after excluding the factors A~ in the natural frame, turn into a system of partial differential equations of first order on the functions setting the right-hand side of the system fp+l. It is clear that after the vanishing of any factor A, the corresponding set of conditions contains n independent equations. In connection with this fact, a natural restriction of the class of reducible systems arises. Definition 2. The differential system fp+l is said to be strongly reducible if

~1: Jv(TP(V,~),TP+I(V,~)) •

--+ o 6 PT(V,~)

or, in relative coordinates of the vertical projection,

(p+ 1)! -

P y;+2_.

-

.:

s! -

,+1,

"v'. + 1 - .

=

0,

= 1,2,...,p.

(15)

Since the conditions of reducibility (14) are written in relative coordinates of the jet jfp+l in an arbitrary frame, they have an invariant character and immediately may be written in local coordinates. Thus for systems of the second and third orders we have, respectively, one or two groups of conditions for reducibility (or strong reducibility): (1)

~=fi(x,~):fi(x,~.)-2

{ }i (2)

"x'= fi(x,~c,~) : fi(x,

10(x, ~) . ~:k = A(x, ~)~i; 0~:k

l Of'(x,~,~) ~k

~l(x,~,~)~i;

~'---~T'--= x.,. . z) . - ~20fi(x,~,~)~ x"k _ 31afi (x, ~ k k0~,~~)

= )~2(z,~c,~)~ci"

It is easy to obtain from the conditions of reducibility for second-order systems that fi(z, Jc) = fio(x , 5c) + #(x, ~)~i, where f~o(X, :~) are arbitrary homogeneous functions of the second type with respect to ~k. The strong reducibility means ;~ - 0. The second group of conditions of reducibility for the third-order systems represents a special form of entry of functions fi(x, :~, ~), setting the system by its first partial derivatives. This form of entry is analogous to the entry of the homogenity conditions in sense of Euler for second-order systems. The last group of the conditions of reducibility from the p groups of the conditions of reducibility for (p + 1)-order systems fp+l contains higher components V~+1 of the (p + 1)-order velocity and in the natural frame again gives for the (p + 1)-order systems the differential form of entry of the right-hand side of the system, which, together with the previous groups of conditions of reducibility, provides the property of reducibility. The observation valuable for the general theory of strong reducible ordinary systems while constructing the geometry of the systems was made of the third and fourth orders [11, 15]. This observation made it possible to find a very useful geometric structure - the basic differentiation operator. The construction of 261

the basic differentiation operator connected in this case with the ordinary differential system fp+l of the strongly reducible type arises as the result of the conversion of the conditions of reducibility [14]. As is obvious, the conditions of strong reducibility (15) contain components V1/,V2i,..., V~, V)+ 1 of (p+ 1)-velocity linearly (besides their presence as the arguments in components V~+lk), i s forming in this respect a system of the triangular kind. Therefore, formally solving this system on V2/,V3/,..., V), V)+I linearly entering in it over V1/, we obtain the following form of entry of conditions of the strong reducibility V2/

i i = r ~ k Y t , V3/ ~-- F ~ k Y t , . . . , Y~ = Fp_aI, V1k , V;+l = r ; k v l k,

(16)

i , s = 1,... ,p, are defined as a result of solution of a triangular system by the where coefficients the F ~ following recursion method: s+l k -- ~ X=I

y'

s+l p+ll

= C p+~-x: p+IC p+l p+ -x ,

X-Ik,

r'0

(17)

s=0,1,...,p-1.

The values F~k are polynomials in union up to order p with respect to vertical components i

1

of the jet of j f p + l : Fik = P : , (V]+x],... , V~+ir ) .

(18)

The transformation of the complete differential of these polynomials by its expression over the structural forms of the components V~+lZ i s shows, in view of the scope test, that the values F~k combined with V~,, V2k,..., V~ make up the differential-geometric object the structural forms of which are expressed according to the scope test:

oP k

A r / s , = o v;'+ l

v;i+ '

On the other hand, considering the cross section s : Vn --+ TP(V,) and the differential prolongation of its structural equations AV/=71,w' '.- A T i , = f i , , w t, s=l,2,...,p, we can show the coincidence of the structural forms of the bundle J(Vn, TP(V=)) AT~s k with the structural forms AF~k of the object Fsi k" Having fulfilled all the necessary calculations we come to the conclusion that formulas (17), (18) define the morphism r Jv(TP(V,),TP+a(V,)) -~ J(V,,TP(V,)) identical with respect to the base TP(Vn) for these bundles. Since formulas (17), (18) of the morphism are reversible, r is an isomorphism. T h e o r e m 2. There exists an isomorphism

Jv(TP(Vn), T "+a(Vn)) -+ J(V~, T'(V~))

r

between the bundles Jv(TP(Vn), TP+I(Vn)), J(Vn,TP(Vn)) defined by the formulas s+l

ris+l

k

_- - ~_.,ts'~s'}'lVp't'li ~"~ Tri p-s-1+~Ft ~--1

= C v+~-x: p+IC v+l p+I-a ,

262

k,

~ ! k = ~lk'

s=0,1,...,p--1.

(19)

If the differential system fp+l : TP(Vn) ~ Tp+l(Vn) is given, then the isomorphism ~ induced the cross section r : TP(Vn) --+ J(Vn,TP(Vn)) of the bundle J(Vn,TP(Vn)) over the base TP(Vn). The cross section obtained in this theorem defines a very useful structure for the geometry of differential systems which may be given also as an independent structure (the basic differentiation operator). Let us note that the components of the object Fil k , - . . , Fipk arising as coefficients in the converted conditions of strong reducibility (16) may also be determined for an arbitrary system fp+l by the same formulas of scope (17). The theorem is formulated precisely for this general situation. Nevertheless, this object plays an important constructive role in the construction of the geometry just of the strong reducible differential systems. But since the main structure to which the strong reducible differential systems of higher orders are reduced is a nonlinear stable connection in the frame bundle of the higher orders given by the transfer object of the corresponding order, an original description of the bundle of transfer objects and its structural forms should be given first. Any element of the quotient bundle T H P ( V n ) / L ~ is said to be a transfer object [8, 11]. The quotient bundle THP(V,~)/L~ also has an equivalent definition as t h e v e c t o r b u n d l e JP(V.,, T(V,,)) of p-jets of the cross section of the tangent bundle T(Vn). The structural forms of this bundle may be obtained as the left-hand sides of the p-times prolonged structural equations AV1i ---- dV? + VlkW~ = F~w k of the cross section Vn --+ T(V,~) and have the following form: Odi , A V /

Arikt = dr~l + AFiktm =

i

.~ d V / - ] - V?od~, rn " rkl.~..i - r'kmw'[' -

j

i_

"

A r i k -.~ d r ~ q- Plkw I -- r~od t "~ V / W ~ l , " i + rlrn wk., i - rm.;~l i rn + V1rn wkl.,, i r'.~l~' + rkrn wl,,,

"

j

i

i

p

j

"

"

-- Yijwkt

" "

(20)

p--1

AF ik~...~p = d F ik~...k, + E

e' F (kl...ka z. ~ika+l...kp)l -- E cd(p--cr cd(p--ct)[ P! ~ iI(kD.,ka ~dlka+t...kl~ ) "Jl- E ll iCd kl...kpl 9 ot~l otmO

A morphism from the bundle JP(V~, T(Vn)) into T p+l(wn) is very important for the geometry of ordinary differential systems of higher-order singularity of this bundle. P r o p o s i t i o n 2. The following scope formulas, recursively connecting the relative components V i of (p + 1)-velocity with the relative components of the transfer object =

Vi4 .

.

.

.

.

/ = r'

.

v? +

pi l d k l /. i1l ./ m r ~ v ? + 3 r t , v t v / +-k,.~.~ 1 ,

= .

.

.

.

.

.

s=l

.

.

.

.

.

.

.

.

.

.

.

.

.

"al+-.-+Ot,=p

.

.

.

.

.

.

.

kl...k,

.

.

.

.

.

.

.

.

.

.

.

V:: ...

.

.

.

.

.

.

(21)

.

! V2:

set the morphism T(V,,)) -+ This fundamental property of the transfer object develops very easily. Having written in arbitrary local coordinates any cross section of the tangent bundle Vn -+ T(Vn) and the corresponding system vl . . . . = fi(x), one can calculate any derivative of an integral curve at some point x: v~ = ~i = okfi(x).~k = o k f i ( x ) v t , vi3 = "x"i = O k f i ( x ) ~ k + Okotfi(x)~kk t = Okfi(x)Vk2 + OkOtfi(x)vkl v~, . . . , where, by virtue of the arbitrariness of the cross section V,~ --+ T(Vn), we deal with arbitrary values of the natural coordinates of

9

the velocity vl, V 2i,

V3i

and of the transfer object F~ = Okfi(x), F~k! = OkOtfi(x),..

. ,

( O~ = ~x0)~

9

The formulas obtained have the same form by passage to some other field of natural frames. This fact makes it possible to speak about the invariance of these formulas which, in a general frame, coincide with the formulas from Proposition 2. Certainly the scope test can be used for checking the invariance of formulas (21) since the structural forms of bundles TP+x(Vn), Jv(Vn, T(V,~)) are known (10), (20). The fundamental role of the morphism qo2 is uncovered in the close linking of structures of the strong reducible system and nonlinear stable connections of higher orders, the discovery of which without the morphism qo2 would be impossible. 263

2. G e o m e t r y o f t h e R e d u c i b l e Differential S y s t e m s fv+1: -rv(v,) ~ Tp+I(Vn) This section is devoted to a review of [11-15, 18, 19], dealing with the geometry of strong reducible differentia/ systems of third and higher orders. From here on, strong reducible systems will be called reducible systems. The decisive role of the stable nonlinear connections of higher orders which permit one to reduce all invariant properties of a reducible differential system to a purely geometric structure will be shown.

2.1. T h e I n t e r l a c i n g o f S t r u c t u r e s : N o n l i n e a r S t a b l e C o n n e c t i o n s , Basic O p e r a t o r s , a n d R e d u c i b l e D i f f e r e n t i a l S y s t e m s . The three structures indicated in the title are in dose connection when under specific conditions one of these structures of a higher order generates another one. This property plays a crucial role in the construction of the complete reduction of a reducible differentia/ system of higher order to the structure of a nonlinear stable connection of the corresponding order. Here the basic operators of higher orders essentially help in the coordinate-free search of the main morphism (the scope), giving the desired object of the nonlinear stable connection. The general notion of a nonlinear connection of higher order in an arbitrary principal bundle was first given in [8]. In this review, this notion will be transformed in conformity with the principal frame bundles of various orders and defined specifically toward the required direction (the notion of a stable nonlinear connection). D e f i n i t i o n 1. The mapping 7~: Tq(V.) ~ J P ( V . , T ( V . ) ) , identical with respect to T(Vn), is said to be a nonlinear connection of q-order in the p-frame bundld m'(v.). The ordinary connection in the principal bundle H v ( V , ) as the transversal to its fiber L~-invariant distribution is a very special case of 7~, when q = 1 and, for each Tz(Vn), the m a p 7~' linearly acts into the fiber of transference objects. This type of connection has to be excluded from usage when investigating ordinary differential systems of order higher than second. Another specialization of the notion of a nonlinear connection (Definition 2) turns out to be necessary here, but even in this form such a general notion of connection makes it possible to define the parallel translation in any bundle associated with Hv(Vn). First of all, let us write the structural equations of the map 7~, implying all differentia/-geometric information about it. In these equations the structural forms (20) of the bundle J v ( V , , T ( V , ) ) by virtue of the map 7~ linearly depend on the structural forms (10) of

the bundle Tq(V.): Ar'

= evg +

- r', t + v/

9

=

m

i

h =

"

--

Av/,

"

_

m

i

+ 2r(kwt)m

i

-

m

Frnoakl +

-

Vl

m

i Wklm

= r klrn

" 3r'jck, L"

j itxrh,~ = drh~, + rk,~,~

.

~

.

i

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

,

.

.

.

v

"

.

.

.

.

.

.

.

.

.

.

.

p!

.

.

rl

.

.

.

.

.

.

Or)! J" (kl...ka p--1

E

a=O

(Av0 ~ = J ,

.

.

.

.

.

.

.

.

.

.

.

.

.

o

.

0.7i

ka+l .-.kiD)|

_I

"--

OL)!X l ( k l . . . k a

r ikl...kp s! AVst,

264

., _3r~(kW{,,, )

( ik l r n jn A V J ,

= r .

+ 3r jcke i

~ = 0,1,...,q).

k a + t . . . k p ) "3L

l~3kl...kpi

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

(22)

Let E(V~) be any bundle associated with HP(V~) with the structural forms AY J = d Y J + 62S~(Y)w~, ~. The presence of the operation of covariant differentiation in such a general situation is confirmed by the following statement [15]. Proposition 1. The connection 7~ generates a covariant differentiation operator in the form of the

mapping VT: Tq(V,,) •

TE(Vn) --+ TvE(Vn),

identical with respect to E(Vn), defined in a general frame by the formulas V~Y J = hyS(tv) -

r"

=

, r,,

(23)

where TvE(Vn) is a vertical subbundle of TE(V,), AY+(ty) are the values of the forms A Y + for arbitrary ty 6 TE(V~) at the point y 6 E(V~), F ~ are components of the transference object of the connection 7~ depending on the element t q E Tq(V~), and A.rYJ are the relative components of the covariant differential of the object y. This proposition reflects those general situations where the object y E E(V~) is completely independent and therefore the covaxiant differential depends on the vector t u E TyE(V~). But the operator V-~ can have a more concrete content in cases in which it is necessary to restrict it to some submanifold. For example, when the curve x(t) E Vn is given, the covariant differential xT~Y z determines the field of the object V~y(t) E TvE(Vn) for the field of the object y = y(t) (given on the curve) along the holonomic field of the curve velocities tq = jqtx(t). This field of the object has the same meaning as in the classical linear case: V-tY J =__0 means the parallelism of y(t) with respect to the connection ~qP. Moreover, the parallel translation of elements of the bundle E(Vn) along the given path x(t) is uniquely defined with the aid of the integration of the equations V . t Y J 0 by the initial position of the element Yo = y(to). Thus, the isomorphism specific for the given connection 3'~ and for the given path x(t) is established between the fibers It is necessary to consider another restriction of the operator V~, which is useful for the characterization of stable connections of special type. But first let us give, for q -- p, the following definition. D e f i n i t i o n 2. The nonlinear connection 7 p = -y~ is said to be stable if the map 7 p is a cross section of JP(V,, T(V~)) over T'(V~). This definition is correct because, in virtue of the morphism ~02, we have =

E(V,,)[=(,).

JP(V,~,T(V,~)) ~2> Tp+I(V,,) t,5 Tq(V,) and, therefore, JP(V~, T(Vn)) is a bundle over each Tq(V~), q = 1, 2 , . . . , p + 1. The particular significance of this type of nonlinear connection is contained in the fact that by virtue of Definition 2 and the presence of the morphism ~02 (see (21)) the map ~2 oVP: TP(V,) --+ TP+I(Vn) is a cross section over TP(Vn), i.e., it is an ordinary differential system of order p + 1. Thus, the nonlinear stable connection 7 p is naturally projected onto a certain differential system of order p + 1, obtaining a quite geometrical sense. As can be shown by the stability test mentioned below, this property eliminates those of linearity for the connection 7 p. And, moreover, with respect to the given definition, the property of stability of the connection 7 p appears from the second order (p = 2), that is, from the frame bundles of second order. Let us turn to the stability test of the connection 7 p, which wilt be formulated in terms of the operator V~. Let us apply the operator V-~ (see (23)) to the concrete bundle Z(Vn) = TP-I(Vn), having restricted its action from T(Tp-I(V~)) to TP(V~) C T(Tp-I(V,)) and simultaneously having reduced the space TP(V,) xT(v,) TP(Vn) to its diagonal identified with TP(Vn). Thus, the following action is defined:

VT: TP(Vn) =-+Tv(Tn-I(vn)). The comparison of the value Vx(tP), t p E TP(Vn), obtained in this case with formulas (21) of the map ~02 with regard to Definition 2, gives: 265

P r o p o s i t i o n 2. The cor~nection 7 p is stable if and only if VT: Tv(V,,) -+ 0 E Tv(TV-I(vn)), or, in the relative coordinates of the transference object and the p-velocity, v,y?

=

- r

v? =_ o,

=

-

v,v

=

-

- 3r

,vr

- r

,

=_o, V?yl'vy = o,

(24) P

V,yV;_ 1

=

V; - ( p

1)!

-

E i ~1 +".+~a=p--1

.

s=l

_

_

v2,

=0.

...

The differential system fv+~ = :~ o 7P: TP(V~) -~ Tv+I(V,,), defined in this case by the connection 7 v, possesses the following invariant form: 9

v

1

VvY~ = W;l'l - P ! E ~ I I 9 E 1~ik l . . . k o s=l al +'"+o~a =p

~

1 v o,1kl

1 "I/'kl ~,

"" " h ~ . . ' "

(25)

so that its integral curves are those curves x(t) E Vn for which the tangent p-velocity fftx(t) is autoparallel: V~(jv(x)) = 0. Let us note at once that conditions (24) indicate the autoparallelism of the tangent ( p - 1)-velocity along any curve in V,, and represent the functional identities on the components of the transference object. For example, the simplest of them mean, in the natural frame, the identities

'

dt 2

(

dx'

F'k z l, --~-,... , dry ] - - ~ = 0

with respect to the coordinates of a p-velocity. These conditions are consistent and are not so difficult; thus, the differential system defined by the stable connections may be arbitrary in spite of the specific form (25) of its entry. Here the functional arbitrariness of the stable connection 7 p generating the fixed system fv+l is also very great. It is interesting that any differential system of arbitrary order p + 1 obtains meaning similar to the equations of geodesics in the affine-connected space but only on the level of higher-order velocities. Therefore, it turns out that one can set up the problem of the invariant association with the differential system f v + l of the canonically stable connections .yv, generating, by the map q02, the same system fv+l : qo2 o .yV = fv+l. The solution of this problem reduces the system f v + l with all its invariant properties to the geometric structure of the stable nonlinear connection 7 v. All its projections 7 v, q < p, are also defined. In particular, the projection 7~ with the components F/k, V1i is a nonlinear connection of order p in the bundle of linear frames, and the projection 7v2 with the components F~t, F~, V1' is a nonlinear connection of the same order in the bundle of second-order frames. The basic differentiation operator was discovered as an intermediate structure in the investigation of reducible differential systems under the conversion of the conditions of strong reducibility (15), (16) ([15]). But this operator can also exist as an independent structure of higher order [10]. D e f i n i t i o n 3. Any cross section of the bundle J(V~, Tv(V~)) over Tv(V~)

~v : Tv(V~) _+ J(V~, Tv(v~)) is said to be the basic differentiation operator 6v over the space Tv(V,) of p-velocities. The operator 5v generates t h e following operation. Let E(Vn) be any bundle associated with a frame bundle of any o r d e r . If the mapping x : Tv(V,~) -+ E(Vn), identical with respect to Vn, is given, then the operator 5P assigns to its first derivative

j x : Tv(V,,) -+ J ( T v ( V , ) , E ( V , ) ) 266

the more simple m a p p i n g 5P(x): TP(Vn) --+ J(V,,,E(V,~)), 5P(x)(t p) = jx(tP) 95P(tP),

t p E TP(V,,). d

The basic differentiation operator also induces an operator, generalizing the total derivative operator ~ :

~ : TP(Vn) --+ T(TP(Vn)),

gf(t p) = gP(tP) 9t x,

t 1 = prt p E T(V,),

which assigns to the m a p x : TP(Vn) --+ E(Vn) the map (an analog of dz 6f(x): TP(V.) --+ TE(V.), 6[(x)(t p) = 6P(x)(tP) 9t 1. The operator 6p, by virtue of the isomorphism r is equivalent to the cross section

Sp: TP(V.) __+Jv(TP(V.), Tp+x(V.)),

Sp = r

o 6P.

It should be noted that the iterated application of operators 6P(x) and 6~(x) acts from the space TP(Vn) respectively into the spaces J(V,,, J(Vn, E(V~))) of nonholonomic jets of higher orders and into the tangent spaces T(T(E(V,,))) of higher orders. Let us also note that the operator 3~ is related to the operation of d total prolongation ~-~ of the map x : TP(V,~) -+ E(V,~), determined as a restriction of the m a p

dx: T(TP(Vn)) -+ TE(V~) dx

to the subspace TP+I(V,~) C T(TP(Vn)), - ~ : TP+I(V,~) --+ TE(Vn). In the case of setting the system of differential equations fp+l : TP(Vn) ..+ Tp+ l(Vn), the total derivative dx dx d'-T is reduced to this cross section identified with TP(Vn) and is treated as the m a p -~-: TP(Vn) -~ TE(V~). The corresponding operation ~ ( z ) : T~(V,,) --+ TE(V,~) arises as a composition of the maps

TP(V,, ) 6f) T(TP(V,~)) ~,, TE(V,~), where the m a p ~ is a vector field on the space TP(V,~) of p-velocities. Under some conditions the map 6~ turns out to be a differential system fp+l of order p + 1, and the operation ~ ( z ) coincides with the operation ~

P+~ of total differentiation with respect to this differential system. The map ~1 participates

in the characterization of these conditions. Let us compose a composition of the maps:

T'(V.) ~

J(Vn,T'(V.)) ~ ~'~162') T(V.) xv. "-~T(V.) = PT(V.).

The m a p [Tr. ~1 o r o 6p assigns to the p-velocity t p E TP(V,,) a p-vector from "T(V,,); this gives a vector interpretation of the nonlinear bundle TP(V,,). But this is understood only in the ease in which this map is a local diffeomorphism. Another partial case is important. D e f i n i t i o n 4. T h e operator 6P: TP(V~) ~ J(V~, TP(Vn)) and the map ~P corresponding to it are said to be exact if ~01 o (r o ~P) = ~1 o SP : TP(V,) -+ 0 E P-IT(V,).

Proposition 3. The exactness of the operator JP is equivalent to the fact that the map ~ : TP(Vn) -+ T(TP(V~)) is the cross section Tp+I C T(TP(Vn)) (i.e., it is a differential system). Moreover, the operator ~ coincide~ with the total derivative ~ [ p + t with respect to the differential system f~+~ = ~ . The case in which the operator 6P itself is generated by the differential system (by its jet j v f ~+1) is of particular significance from the point of view of the geometry of the differential system f ~ + l set as an original structure. 267

D e f i n i t i o n 5. The operator ~v is said to be differentially generated (generated by the differential system fv+l : Tv(Vn) ..~ Tv+I(Vn)) if ~P = r v+l, where j v f p+I : Tv(Vn) --~ Jy(TV(V,), Tv+I(Vn)) is a vertical jet of the map f l , + l and

r

Jv(TV(V,), T'+I(V,)) ~ J(V,, Tv(V,))

is the isomorphism defined in Sec. 1. Let us recall that in Sec. 1 the possibility was shown of reformulating the conditions of the strong reducibility of a differential system in terms of the components F~ k, F~ k,---, F~ ~ of the object which is an operator 6v = r o j v f p+I generated by this differential system. Precisely in this situation do the operator ~P and the isomorphism r appear for the first time. Now we can formulate a test of the strong reducibility with the aid of the notion of exactness of.the operator ~m. Proposition 4. The differential system fv+~: Tv(Vn) _~ Tv+~(Vn) is strongly reducible if and only if

the operator 6v generated by the differential system fv+l is exact and the differential system ~t defined by it coincides with f n + l . The test of Propositions 3 and 4 is realized by the component entry of the operators 6v and ~tp, which is necessary for the further application of these operators in the geometry of reducible differential systems. Let us take an arbitrary operator ~v: Tn(V,) ~ J(V,, Tn(V,)) whose value 6v(tv) belongs to the bundle consisting of the 1-jets of all possible cross sections v: re', --+ T(V~) : ~ ( t ~) ~ J(V~, Tn(V,)). Therefore, the element 5v(t~) over the point x e V, is set by the system of relative coordinates V~, V~,..., V~; Fio~ = ~, Fli/~,F2ik, 999Fvi ~, where x'," V/ (s = 1,... ,p) are the coordinates of the p-velocity t~, where the jet of i ~v(t~) is directed from the point x(x~), and Fs~ , s = O, 1,... ,p, are, for each s, the quadratic block of the Jacobian matrix of the jet ~v(t~) with respect to an arbitrary p-frame according to the group V~, x ~ of coordinates of the p-velocity. (F~ ~ correspond to the derivatives of the cross section V/ = fi~(x~) by the coordinates x k of the point x ~ V,,s = 1,... ,p.) Let us also note that in the case in which Fist, together with V~ form the components of an arbitrary element of the bundle J(Vn, Tv(Vn)), which is determined as a jet of some cross section v : Tv(V~) (Vi = fi(x~)), F~ ~ are the coefficients of the structural equations of the cross section v: $

(26) r----I "ax+..-+~r=s

where s = 1 , . . . , p . Let us also note that the Caftan differential prolongation of the equations AFis k = Fi kt wI gives the possibility of counting the structural forms AF / k of the bundle J(V,, Tv(v,)) with basic forms AV/, AV0~ = w i. Therefore, the structural equations AFis k

i t = Fskt AV'~'

s = 1 , 2 , . . . ,p r = 0,1,...,p

(27)

correspond to the given operator 6P. The complete entry of these equations will be given before the next section, when the coordinate form of constructions will become the basic one. The operator 6~ corresponding to the basic operator 6v was defined as a composition of 1-jets 6~(t p) = ~v(tv) . t 1, t 1 = prtP; therefore, 6~ is a vector field on Tv(V,) - ~Pt(t p) e Tt,(Tv(V,)). Hence 6f(tv), as a vector of the tangent space T~,(Tv(V,)) at the "point" ff E Tv(Vn) with the relative fiber coordinates V1i, V2/,..., V;, is characterized by the coordinates

(x,,. v ; , . v ; , . . ..,

.

v;l . = v ;. , v;1. =

v;1 =

=

rp i

Let us turn to Proposition 3, where the reformulation of the exactness of the operator gv is given. The coordinate form of the definition of the exactness of 6p is identical to entry (15) of the conditions of strong reducibility in the definition in Sec. 1 for values e = 2 , 3 , . . . ,p. But, by virtue of the isomorphism 268

r Jv(T~(Vn),T~+I(Vn)) -+ J(Vn,T~(V~)) and its coordinate entry (17), relations (15) are transformed into the equivalent relations Y~ = V~I = r l k Yl k, V# = Y#l = ri2 k Y ? , . . . , V~ = V ; _ l l

=

F/~_I~V~.

This means that 6~ associates to the p-velocity with relative coordinates V1/, V2/,..., V~ the (p+ 1)-velocity with coordinates V~/,V21,... , V~, i V~+x i = F~,i ~V1k. Thus, the exactness of the operator 6 p is equivalent to the fact that the corresponding operator 6~ is the cross section 6~: TP(Vn) --~ T~+I(Vn), which is set by the f o r m u l a s W;% 1 = ~ipk(Vsl)W?,where F;~ depend on the p-velocity V ,i, s = 1 , . . . ,p. Let us note that under the conditions of Proposition 3 the operator 6~ does not have to be differentially generated; conversely, in the face of the corresponding operator 6t~, it, by igself, generates some differential system. It is also obvious that for the exact operator 6~' the operator 6~ acts as a total derivative with respect to the system fP+l = 6~ defined by it. Everything mentioned before also confirms Proposition 4, where the exact operator 6~' is differentially generated. It is necessary to consider the coordinate entry of the action of the operators 6P and 6t~ on maps setting the geometric objects on the base of the bundle T~(V~) of p-velocities. Let >r T~(V~) --+ E(V~) be a map setting a field of some geometric objects over T~(V,) with the corresponding structural equations

A Y J = d Y J +r

s = 0,1,...,p,

where YJ are the relative coordinates of the elements y = x(ff) E E(Vn), AYJ, w i and AV~ are the structural forms of the bundles E(V~) and TP(V~), and YJ~ are the relative coordinates of the jet j x . The action of the operator 6P on x, defined by a composition of jets jx(tP) and 6P(ff) E J(V,, TP(V~)) 6p(z)(tp) = j,(tp)&p(tp), results in the jet from J(V,, TP(V~)) the relative coordinates YJIk of which are determined by the coordinates Y J i of the jet jx(tp) and the components P~k = &~, Fit,, s = 1 , . . . ,p, of the jet 6P(ff) being determined according to the formulas

ffkY J

r' YJik = YJ~ + V.LJ sI~.,k.

The symbol 6~, denoting an action Of the operator 6P on the elements V = x(ff) E E(Vn), may be represented in the following form, which is general for any bundle of geometric objects: .

.

.$ !.

()?, . Dl_t'~tlFl

+(

)lp rp1 t`.

(2s)

By the aid of this operator notations we can easily write the operation 6~(x), taking into account that (~f(x)(t p) = ff(x)(ff).t I is also a composition of the 1-jets t 1 = prff e T(V.) and 6P(x)(ff) E J(V., TP(V,,)): =

[()0 + ( , ),r,k] , Vl

*, yJ = , YJV? = Y E v3,

(29)

d to the system fp+ 1 = 6[, which, in the case of exactness of the operator i f , gives a restriction of ~-~ determined by the exact operator d Ip+,

=()F,

~p-1vk

~p~k Lrl

We have described the most important (for the geometry of differential systems) unit fp+l =v 6p =~ 7 p of the cyclic interdependence of the three fundamental structures fp+l, $p, 7p given in the reviewed cycle of works. This unit gives a solution of the problem of the complete reduction of a reducible differential system to the geometry on manifolds V, with the structures of the stable nonlinear connection 7 p invariantly defined by each such system. Concluding this subsection, we shall describe a general scheme of the possible solution of this problem and one of its concrete particular solutions in some class of the reducible systems of arbitrary 269

order. The next subsection deals with a complete description (with explicit coordinate formulation) of the solution of this problem for any reducible systems of third or fourth orders. Let fp+l : TP(V,,) _.+ TI,+I(Vn) be a reducible differential system and $P: TI'(V,,) --+ J(Vn, TP(V,,)) be the basic differentiation operator 8P = r o j v f p+I generated by it. The exactness of t h e operator SP with some additional property of it is equivalent to the reducibility. This property is contained in the fact that the corresponding operator $~, being by virtue of the exactness of $P also a differential system, coincides with fp+l = $p. A simple observation became the item of origin in further constructions - SP projects into the map 7~: TP(V,,) ~ J(Vn, TP(V,~)), which is a nonlinear connection in the bundles of frames of the first order. One can easily observe that 7~(F~k) is the application of ~P to the projection r : TP(V~) -+ T(V~). Therefore, (SP)P(r), the p-times application of 8P to r, is a map into the space J(V,, T(V,)) of the semiholonomic (in general) p-jets of the cross sections T(Vn), which includes the subspace JP(V,, TP(Vn)). This property naturally separated the quite extensive class of systems fp+l elementarily reducible to the nonlinear stable connection of p-order [10]. D e f i n i t i o n 6. The reducible differential system fp+l is said to be regular if (SP)P(r) acts into the space of holonomic p-jets (SP)P(r): TP(Vn) --~ JP(Vn, TP(Vn)) C JP(V~, T(V~)), being thus the nonlinear connection

=

Regular systems are characterized by an initial symmetry of the components

Fik = F~ k, Fit: = Fklt,..., i i Fkt...k, = FiJ,,Ikd...It,p

of the object. Applying the operator ~P a necessary number of times to the conditions of reducibility, one can extract from them the conditions of stability of the invariantly associated connection ~,P = (~/')P(r), and at the last stage it is stated that 7 p is projectible into the original system fp+l = 9~2 o 7 p. T h e o r e m 3. The nonlinear connection 7 p = (jP)P(r): TP(Vn) --+ JP(Vn,T(V,~)) inherent in the regular

differential system fp+l is stable and includes this system as the projection 992 o 7 p. Thus fp+l obtains the following geometric sense: V~ jP,x ( t ) = O, x(t) E V,. This result convinces us that this reduction is possible for arbitrary reducible systems with the aid of the invariant scheme given in Theorem 3. In fact, the papers dealing with the geometry of a reducible differential system from the third to the sixth order confirm this sureness [11, 15, 18, 20]. Unfortunately, the corresponding general construction of the reduction of systems fp+l of the reducible type to the geometry of nonlinear stable connections 7 p has not yet been found. The next subsection deals with the demonstration of a solution of the problem of the reduction of systems of the third and fourth orders in an effective invariant form of the relative coordinates of the objects. It is seen that the general scheme and its complications depending on the increasing order are already observed on these two minimal orders. The general role of the basic differentiation operator and its technical possibilities, without which the tracing of the difficult constructions of the nonlinear stable connection has to be conducted blindly, are guided only by the universal scope test. Let us note that the scope test on the whole plays the control role in the correctness of the choice of scope formulas, although it also contains an element of prompt (everything depends on the complexity of the desired scope formulas). It should be pointed out that in the coordinate constructions of the next section the checking role of the scope test is necessary at the last stage. It is also necessary to know an explicit form of the entry of the 270

structural equations (27) of the operator 6~, which we write in conclusion:

aI'~k

' ' + 2F1* kV1" wt,~ ' + V21w~l " + )'1 " ' * ' 'vl wktm ' - Fatwk

=dr~k+I'~k0:~

Ar' , =

'' + rplwk

(30)

+ pl p

= r ' ~ ~zxv;

s

. . .1 ..Fk: . .k

w'k~...k.... ~.

(~ = 0,~,...,p).

2.2. T h e C o m p l e t e R e d u c t i o n o f t h e R e d u c i b l e S y s t e m s of T h i r d ( F o u r t h ) O r d e r to t h e Stable N o n l i n e a r C o n n e c t i o n o f T h i r d ( F o u r t h ) O r d e r 2.2.1. The third-order differential system f3: T2(Vn) __+ T3(Vn) is described by the structural equations TrkvrlTrm

i

where V3i 0, Vsi ~, Vai ~ are the relative components of j.fa. The exterior differentiation of these equations, together with the application of the Cartan lemma, gives us the structural equations for the components jfa. The right-hand side of these equations contains the components of j2f3 as coefficients: V3i 2kt,2 Vdkt , "2 1 V3i i ~. As is shown in [13], there exists a morphism from the bundle J2(T2(Vn), T3(Vn)) into the bundles J2(V,,, T(V,,)) of the transference objects of second order. The general conclusion of the previous subsection makes it possible to avoid the mentioned procedure of extension and blind research of the desired morphism. However, the result is obtained in an explicit invariant form of the relative components of all objects. First of all a system is required to be reducible, which is provided (according to Definition 1 of Sec. 1) by the identities 9 1 T;~ 2 Tfk . 2 T/~ 2 ~Zk 1

(a) v;-~3k,~ =_o,

(b) V;-~,~-~V~LV~-O,

the second group of which represents a special form of entry of the system f3. Having converted these identities to the form

v~' -- r~ ~y?, ~ = ~v~ ~,,

r~=

v~' = r ~ y ? ,

-sv~L+~v~.v~L

we obtain the components F~ k, F~ k of the basic differential operator 62: T2(Vn) -+ J(Vn,T2(Vn)). Since the first group of components gives us simultaneously the components of the connections 721: T2(V,,) -r J(V,,,T(V,,)) F~ = F~ k, the conditions of stability of the desired connection 7 2 = 722:V2i - F~ V1k - 0 are also obtained. The projection of this connection 7~(F~) has already been determined. Let us apply to these conditions of stability the operator of the total derivative, which, in the case of a strong reducible system, is expressed by (25) according to Sec. 2.1 by the basic differentiation operator (28) 271

__=

It

21

d

d

i

i

dt d 9

i 1

- rlv, i2

=

"

- rly i

2

- - ( k i t ) ."1 k . , , 1 - - O; i

I'~ = r~k = ]v~k, F~a = r(kl0 = 5(~Fk) 1 ['t/-i20

1/-i21 ~m

12"i22 r~rn

I [T]'i20

1 [vi22 Tzrnl

(31)

= -~ [V3(kl) § "3(krnXli) + '3(krn~21)J wil2 xzm2~

212"i22 w m2vr2]

According to the previous subsection, r k = r~ k form the relative components of the connections 721 = g2(r) : Tz(Vn) --+ J(Vn, T(Vn)) (r: T2(Vn) -r T(Vn)). A repeated application of the operator g2 to this map generates the map (52)2(r): T2(V,) --+ J2(Vn, T(Vn)) D J2(Vn, T(Vn)) into the space of semiholonomic 2jets J2(V,,, T(Vn)) with relative components of its elements F/k, F/kl,. Therefore, V1/,r k , r , ,'' = r'(k,0 generate the components of an element in J2(Vn, T(Vn)), that is, a transference object of the nonlinear second-order connection (Fiklt ] form a tensor, as is easy to check with the aid of the structural equations of this object). Thus, the map (52)2(r) = 5252(r), corrected by symmetrization (let us denote it by 5(282)(v)), determines the nonlinear connection 75 = g252)(r): T2(V,) ~ J2(V,, T(V,)) for which the conditions of stability V2/ - F/~V1k = 0 axe already satisfied and the third-order system determined by these connections has the canonical form (25): V.rV2i = V~ - r'~Yr - rt,v,*v; = o. But this entry of the system in the invariant relative coordinates is obtained by the complete prolongation of the identities V2~ - F~V~ -= 0 by the source reducible systems fs and therefore the system induced by the stable connections 72 = 5(252)(r), if it is the modified form of entry of the source system f3. Thus, the problem of reduction of the reducible differential third-order systems to the canonically associated stable nonlinear connection of second order was completely solved and, moreover, in an explicit invariant form. The morphism ~: J2(T2(V,), Ta(V,)) -+ J2(V~, T(V~)), associating with the system fz : T2(Vn) --~ T3(V,) the stable nonlinear connection 75, which induces the same system is determined in the invariant relative coordinates of the 2-jet of the map f2 by the scope formulas (31). These formulas can be rewritten instantly in the local field of natural frames by the second derivatives of the right-hand side of the equations of this system. The case of a regular system (Definition 6, Sec. 2.1) is characterized by the vanishing tensor F~klll and, therefore, by the source symmetry of the object F~l t = Fit. Thus the following is obtained: T h e o r e m 4. There exists a natural morphism ~: J2(T2(Vn),TS(Vn)) -'+ J2(Vn,T(Vn)) defined in the invariant form (31), which associates with the differential system f a : T2(Vn) _.~ Ts(Vn) the nonlinear connection 75 = 6(262)(r) : T2(yn) ---+J2(Vn, T(V,,)). If the system is strongly reducible, then the connection 75 is stable and the system q02o72 : T2(Vn) --+ Ts(Vn) determined by it coincides with r , the integral curves x(t) E V, of which (and only they) have the autoparallel velocity 75 of the second order: V.rj~z(t ) = O. This theorem represents the first successful experience of tracing of the complete reduction of reducible systems to the constructions generalizing the classical affine connections on bundles of higher-order frames. Although the experience was naturally carried out in the simplest case of third-order systems, the general scheme and technology of reaching the purpose known beforehand - the stable nonlinear connection is observed in it quite well, since the structure of the nonlinear connection and the special technology of obtaining it are worked out'for bundles of frames of arbitrary order. The case of the investigations of reducible fourth-order systems in [15] shows that the technical difficulties increase together with the increase in the order of a system. Let us note that, although in the case of an irreducible system the nonlinear connection is associated with the system by the same scope formulas, it undoubtedly loses the property of stability. Therefore, it does not generate the given system, in general, and hence it does not carry all its properties. 272

2.2.2. Coming to a description of the geometry of reducible fourth-order systems, we expect a result similar to the previous theorem. The purpose is to construct the nonlinear stable connection invariantly associated with the set of the fourth-order systems /4: TS(V,) _.+ T4(Vn), the integral curves of which are the generalized geodesic lines of the desired connection. Now it is the connection of the third order .),S: T3(Vn) __~ j3(Vn)T(Vn))) whose components of the object F'~, F}a , F~t,~ have to obey the structural equations (22)

A r~ = ~rk + r ' ~ t - r'N~ + v:~'~, = r'~ I A V;) rn

i

i

m

m

i

i

rn

m

i

= r'~ ~ v 2 ) AF~l., = d F ~ m + F ~ , ~ j

)

(32)

3F(~Wm) j

i j ) +3Fiw~,~)j j + V~w'~,n " " j - 3F/(kw~m ( - F ijw~,~ ~ klmj~...~vs ~AVi .~ ~.

(s = 0, 1) 2) 3),

and the conditions of stability

v#~

= v~ - r ' # ~ - o,

v , v ~ = v~ - r ~ v ~ - r h v ? v / - o

(33)

The source material for the construction of these connections consists of the structural equations of the cross section f4) setting the reducible fourth-order system

=

v 2 ~ ~ + v~l~(~v?

+

v / ~ t ) + v:~(~v~

+

v'~t

+

v/vy~L)

+ v: ~(~v2 + vg~t + 3 v l v y ~ L , + vlvi"v~t~,~)) and of its three-times differential prolongation - the structural equations of the jet j 3 f 4 for the following components V ~"") V~kt) i~r Vi3sr " 4kirn~ ( ~ = 1 , 2 ) 3 ) s) r = 0,1) 2) 3): AV'iS4k

"-~

]Ti~rAlflr4kl~.~rr) A V ~ kliar

=

vi~rq,4klrn"~'qAVrn )

L"~r4klrnAU'i3sr-~ ]/-i3srq, 4klmj~.~

qA]?'j

(34)

the explicit entry of which is too awkward. The role of these equations in the construction of all necessary objects and in the fulfillment of the scope test of the corresponding object is reduced to a minimum since the main load in the construction of connections lies on the basic differentiation operator of the third order 5S) determined by the system f4. The first step in the scheme is the entry of three groups of reducibility conditions of the system f4 in the source form and in the form expressed over the operator 53: 5~ = ()~ + ( ) l+F(;1lk t + ( )l3r3k, t )l2 r2k 1 i :V? = r~i kVlk (a) V;9 - ~V;

( b ) V;9 =_ ~1 V;i ~V2 + ~1 Y;i ~Y~k ~_ ( l _ . v i

1 i ~6 4 ~ + ~v; Iv: ~)v? = r2i

3 i (c) ~' 9 - ~v; ~v? + ~1 ~4f i 2~l f k + ~1

i

ky?)

~v?

(35)

1 i 1 T f i 2TTI 1 i 3T/-/2 -- (~v~ ~ + ~.4 ,,4 ~ + -#; , ,~ ~ + ~ v ; " ~''~,,, ,,,''~)~,4 = r'~v,'.

273

The explicit expression for the components F~ k, P~ ~, F~t of the operator 63 is discovered from this entry. Moreover, the last group of conditions gives the special form of entry of the system f4 in the coordinates of 1 i j f4. F~ ~ = ~V~ ~ = F'~ are the components of the nonlinear connection 7] = 53(r) : Ta(Vn) --+ J(V,~, T(V~,)) and the first group of relations (35) obtains a geometric sense:

v~v/=

(36)

v~' - r ' ~ v ~ = 0,

which composes the first group of conditions of stability (33) of the desired connection 73: Tz(Vn) --~ J3(Vn, T(Vn)) whose projection 71 = 53(v) is defined. Thus, the second step taking the total derivative of identities (36), as was done in the case of the system f3 _ gives

v , v ~ ' = v~' - r ~ v ~ - r ~ , v ? v / =

(37)

0,

where the system of values rtk ~

1 i 1 i3 0 zv~ ~, r t , = 5 ( ,3r ~i) =(~v~ (~o + v; ~1

m m (~rx,) + v ; ,2, , (3 ~~ m o + v~' 3,,, 3(~r3,>) 3

(3s)

forms the relative components of the transference object of second order. But, in this case, identities (37) combined with (36) compose a complete system of conditions of stability of the desired connection 73(r'~, r~,, r~,~) with the projection 7~(F~, F~t) already determined and having the origin 72 = 5(353)(v) identical to the case f3. In system f3, relations (37) compose the final form of entry of this system, whereas for f* they form the concluding group of the conditions of stability of connections 7 3(rk,rm,rktm) i i i with the as yet unknown i leading components rktm, which have to be defined over the coordinate of the jet jar4 so that the structural 9 i equations (32) for components P~tm are satisfied. We need not worry about the values F ik, rkl - structural equations (32) are satisfied since the connection %2( r ik, Fro) i is obtained in a coordinate-free form 732 = 5(353)(r). The third, final and more difficult step directed at determining F~t m consists of using the third degree ( 5 3 ) 3 ( V ) o f action of the operator 53. As was noted in Sec. 2.1, the map (ba)3(r): T3(V,~) -+ J3(Vn,T(Vn)) acts into the semiholonomic jets. The image (ba)a(r)(t 3) is set by the transference object F~ and its first and second basic derivatives: 5 ~ r t k = rgk] I = r t k 0 J l - r t k m 1 F l lm •--I- ..t k i n2, t~2 r1a -~- ~.t i k 3r e .pt 3m1 , 3

i

i

i

0

pi

1pj

Di

2 rJ

ri

(39) 3rJ

5rap kit = F kltlm = F kit m + ~ kit j ' t l m "Jl-J- k[I rnJ- 2rn 3t- J- kl/ j J"3rn"

(40)

If the symmetrization of F~I t gives the object F~t = F(kl0, F~k, which is the transference object of second i order, then dealing with I~kltl m is considerably complicated. The second group of the stability conditions (37) was obtained from the first group of stability conditions (36) by an application of the total derivative operator. Regarding the fourth-order system of f4, the total derivative operator should be applied to the identities (37) taking into account that riktVlkV] = r~ltVkV1t, 3

i 9

"

i

k

I

m

(41)

The form of entry of the system f4 (41) obtained in this way will be converted as follows. Let us find a differentialprolongation of identities (35b) by V~ (in the natural field of frames this means the differentiationof these identitiesby the arguments "~'t):

vi4 ~ - vl ~ ~ v/ + 2 v~ ~ ~ v~' + 3v~ ~,~v~, whence we have

274

Taking this into account, let us make the following change in (41): Yl

= r

i

vl

rn

__ - - ~1( V ~ i 3k

=

I

i3

vx

rn

" ~ - i t -2V~32r'm k r n X l l "~- 3vi433r,,,~zkTpl kmX21}Vl "1 = ~[IkIY?VI1,

where =

k t

,rlt

(42)

form, as can be checked by the structural equations of j2f4, the components of the ordinary object of the affine connection depending upon the elements of the bundle T3(Vn). Thus, entry (41) of the system f4 acquires the final form V~ - r ~ . v ~ - 3 r ~ , v ~ v :

-

-

. , k l r n vr lk v v, ,1z .v,1 J.

-= O~

where 9 " " ' F}a.~ = r'[jl(klT~,~) + r(klo~ ),

(43)

expressed finally by the coordinates of j 3f4, form (combined with F~l , l"/k, V1/ ) the components of the object of the nonlinear stable (in virtue of (36), (37)) connection 73 in the bundle H3(V,,) of third-order frames. The statement is checked by a test of the assumption that the above-mentioned expressions (43) in terms of the coordinates of the jet j3f4 by virtue of its structural equations (34) written in expanded form are subjected to the structural equations (32). The complete analog of Theorem 4 is proved. T h e o r e m 5. There exists a natural morphism

9 : JZ(Ta(V,),Ta(Vn)) -~ Ja(Vn,T(Vn)) determined in the invariant form (see (38), (39), (40), (42), (43)) which associates with the differential system f4: T3(Vn) _.+ T4(V,,) the nonlinear connection 73 = 9 0 j3f4: T3(V,,) _.+ j3(Vn, T(Vn)). If the system f4 is strongly reducible, then the connection 73 is stable and the system q~2 o 73: T3(Vn) -+ T4(Vn) determined by it coincides with f4, the integral curves of which (and only they) have the autoparallel velocity of the third order with respect to 73: VTjZtx(t) = O. The obtained result gives us the following test of regular systems f4: i

i

i

F[kll] = 0, F(kltl,,,) = Fkltlm. Thus, the mentioned part of the review gives us a description of the main results obtained in the series of works presented dealing with the geometry of reducible differential systems of higher orders. The central problem is successfully solved of constructing the morphism ~ reducing the whole geometry of the system to the object F'k, F~I , F~tm of nonlinear stable connection that is stated, for example, for fourth-order systems in the last theorem. Explicit formulas of the morphism ~ for a system of third order are mentioned. They are quite tolerable, which cannot be said about the explicit formulas of the expressions of the components of the object of the connection F~, F~t , F~t~ of the system f4 over the components j3f4 , that is, over the derivatives of the right-hand side of the system up to the third order inclusively. i However, we shall write the simplest part of the formulas of the morphism ~ for the components F ik, Fkt, comparable with the corresponding formulas for fa: 1"~=

F~lt 1

i2

31/-m3

1

i "

1 1

i3

1 i3 ~V~ k, il

m

3[lzrn21zJ3

i F}a' = F(kl0, 1V, 2

m

l rm3irJ

1

i3

3 l,ri3

m

I

jV~

J 31rr3]

r V 4 IJ"

But the formulas for F~t,~ are already immense and contain not 8, but 224 members similar to the formulas for F~It. However, this doesnot mean that it is impossible to manage this monster, since we have the simple expression (43) for F~t~ in terms of the operator 53 and the operation determined by it. As we know, the operator 5 is completely computable by the formulas of the isomorphism r Its action on any object is also easily computable by the familiar universal formulas (28). And since the operator determines by itself the reducible system having generated it all the invariant properties of the system can be expressed in terms of 5 and its repeated actions. These possibilities of the operator 5 appeared in full measure in the construction of the determining geometric structure - the nonlinear stable connection. 275

3. T h e G e o m e t r y o f t h e O r d i n a r y D i f f e r e n t i a l S y s t e m s S~(V'n) --+ ~P+'(Va) The systems of the third order are the source of the construction of the geometry of reduced systems on an arbitrary dimension manifold [14]. The extension of the investigations for the reduced systems of arbitrary order [5, 6, 7, 22, 23] has become a straightforward generation of the constructions appearing on the model of the third order (the second order is an exception to the general scheme). By virtue of this a more detailed a c c o u n t of the geometry of third-order systems is relevant. We shall put it into a separate subsection. Reviewing the results for systems of arbitrary order, we shall restrict ourselves to the statement of the main facts. The final subsection deals with a new result on the geometry of the ordinary equation y(s) f(x, y, y', y", y(a)y(4)) on the plane [1, 2]. 3.1. G e o m e t r y o f t h e R e d u c e d T h i r d O r d e r D i f f e r e n t i a l S y s t e m . A system of third order, determined as a cross section ~03: Su(Vn) --+ Sa(Vn), is described by the following structural equations: =

a,b,c=2,...,n,

0.),,1 b 2 b a = V3a Oodk _~_ V3a ,bOJ1JffV~bOJl,,

i,k=1,2,.

..

(44)

,n.

The exterior differentiation of these equations with the aid of (5) gives its first partial prolongation: Tr e 2

Av~ :

avd I

a

2 c

.~o~,~ + .~1

__~ dV~

~ + V; 'b~c"

2 ,

o --

3(0jr,_

a

1

cO.),,,

V2

1 c cWb

--

2V2 b'O . ) , '

_ 2 V ~ 2e(Wbc' - ~%wn) ,cc ' + 3 ~ t , ' -- 5bWln a '

v o ~ i ~ + v ~ ~,0~,c + V 2 ] 2 The first canonization Va~2 ~ 0, Va~' :=t 0 at the expense of the third order frames reduces equations V.a2 ab , Va~ to the canonical form

~t, = ~ ~, ~ , , + ~1 ( y : ~ 0 ~ + v ; l

c1W ,o + y . ~ 2

o~,,) c

(45)

a - - ~ObOJ11, ,ca , ~Mbl, _~. } (V3a ~ ~oJ k _]_V3a~,cO.)1c "~- V3a ~ 2c W c, , ) ;

(46)

the prolongation of Eqs. (45) gives A Vd 2~o2 : dVd ~ 2c + v ~

., c03b -- V ~ b dO3c "~- V3` 2b 2cUJ1

2 . --

- 3(~oWb + ~ w c ) = V G ~ 2b c , C03d _{_3(0.)~c -- (~cCMb . 1'

av~

b2o,

dV~

-

r b -- V ~ 2 1 ~w~ d

-

--

2r a

1

~(2w~,

-

-

-

(~C03,1)

---- ~ V a a 2b e ,

~ + V / 2b~ , Odd , _ V ~ 2d0 ,0Jbd _ ~ v ~ ob ,0J,,

-{- 3(w~,, - ~bw,11) a ' = VY2 2b,,o (VV~I2_

c

V:220wk c

+ V3a~21 dc d 0'', t

V3a

~22

d dCd, l~

Alia2, ' 2be , A V ~ ~ are similar notations). Whence for the values

V:_- - ~ n,n V a. a 2a2 c, we obtain

Av 2 = ev: + v&l

AV~

, a 2(a ,c), iV~

V 2 - - - ~1 V ; a 2aOl

- V,~wc ' d +~'c, = v v :

..~_ V : 1 o , ) , _~ v l Odwd_~ v : ,dCdld -~- V : 2d03,1, d d a vra22 , = d V ~ + V~cw d - V~cO3db- V ~ = VV~c + yahc 03,,, a cd + 3 ( W t c - 36(bwc),) a ,

A v2 = dV2 - 2 V 2 ~ + ~ : , ,

276

v;a

- (,~ - 1)~,h, = VV2.

(47)

Thus, the following continuation of the process of canonization is possible:

v)~0;

vi~ ::, o;

v2 ~ o,

reducing Eqs. (47) to the form w o1 = v ) o + v : o ~ , + v : l Wt r

a

a = VV) dOJ 1, -~- Vc] 2dO)ll

1

-- 3~(bWc)l = 0 ; a"l l

1 a "~VYbc +

1 l'Ta 2 2

1

~ "3 b cW11,

- ( n - 1)w~n = VV2.

(48)

(49) (50)

The joint solution of system (50), the contraction of Eq. (46); W aa l l

__ l ( n

_

1)Wl111 ---- 13 A'-~l f "a3l a ,

and (46) give 9~n

= Vs o k

, . 1~"1c + v . ~ , ~ no, + .*'~

~11 = v ~ ~ + v 2 ~ [ + v : ~ 7 l .

(51)

The process of prolongation of the initial structural equations is concluded by the two-times prolongation of Eqs. (48) by the form wl: A v : o = ~ v 2 o _ v d o1We,

Av:ool

=

dvclOO_v2OO1 W ed

--

, ~ "~-0Jcl

vv201,

V l O l O ~_ t~7cll 1 = V V c l O O1 1"

(52)

The final step in the canonization V~~ ::~ 0, V~~176 ::~ 0 reduces Eqs. (52) to the form 1 = vvl?, Wcl

1 = vv:O0, 03cll

(53)

whence with regard to (49) we obtain 1 a 2 1 1 a ~oa = ~y~ ~ 1 1 + ~vvio + 3~[~Vy~l.

(54)

Let us summarize the canonization which we have already done and in the process of which, at the expense of the specialization of frames from H s (Vn) the following components of the object scoped by the jet j4~s were reduced to zero: v ; ~ = v~l = v ~ = v : = v y = .loo ,~11 = v , ~ o. (55) By the same token, the subbundle H~3(Vn), characterized by the structural equations (44), (45), (48), (51), (53), (54), W~ = Vgw r Oja :

(02bal, 02tl 1,02tc, 02C, 1 We1 1 ,r 1

1 a }, Wl11,03111

(56)

03~b = {t~l, 0 3a, 0 J 1a ,W11,0J111}, a

is selected in the bundle H a (Vn). The system of forms {wa, w r }, as follows from the structural equations of the bundle Ha(V,,), is completely integrable and contains the integrable subsystem wl,w",wl,w11 , ' * ~ win, including the system wl,w",w~,w~l of the structural forms of the bundle S2(Vn). Hence we obtain the sequence of morphisms of bundles

Hs(Vn) -~ Q(Vn) -+ S2(V.) --+ S2(V,,), where S2(Vn) is a homogeneous bundle determined by the structural forms w q', Q(Vn) is a homogeneous bundle with the struetural forms w a, w~. From this point of view, system (56) is a system of structural equations of the cross section a~,. : S~(V,) ~ Q(V,,). (57) Here H~3(V,,) consists of those frames in Hs(V.) which form the complete preimage of the cross section a~,s (57) with respect to the morphism X. Thus, the prolongation of system ~o3 to j4~oa and the canonization which we have already done result in the following: 277

P r o p o s i t i o n 1. The differential third-order system ~3 invariantIy determines the cross section ar S2(V,) --~ Q(V,) with the structural equations 03~ = V~03~ as a result of canonization (55) of the object scoped by the jet jaq$3. This proposition summarizes the general result of the muir;step process of constructing some sequence of geometric objects invariantly connected with a differential system and the procedure based on this sequence of canonization of the third-order frames. This process proceeds nonidentically and the success of the choice made wiU be seen from the next geometric result of the formally realized procedure. A preliminary analysis of the result obtained permits one to draw the first important conclusion. As is seen from the entry of the following subgroup of structural equations of the frame bundle d03 1 = 031 A w I --~03a A03., i

,~

=

<,,' A 03;1 +

03" ^

03L +

d~" = 03b ^03~ +031 ^03[, d ~ = 03~ ^ ~ +03~ ^03~ + 1

a

03~ A 03,,

^03t, +03" ^03L,

~ , " = 03~A(03t --~b03,)+ = 1 03~ A 0 3., , + 03b A03~,,

(58)

a~', =03~1A (03t - 9~t03])+03~ A (Zo& - ~t03h) +03~ A 03t11 +03x A 03hl, . 1 -~- 03a h 03.,1 1 -~ 031 ^ 5~1 d0311 = 031 A o3;1 + 0311A 03a1 + 203[ A 03al 1 " is completely the system of forms ~'l,~'",~,,~n,~n'....1 of the bundle b2(Vn) completed by forms 03,,03b integrable axld, therefore, makes up the structural forms of some associated bundle P(Vn) having morphisms P(Vn) --+ b2(Vn). Moreover, structural equations (58) of the forms w 1, w" ,~1,~n,,~11,,,,,,03~ '" '" ,' ,1 of the bundle P(V.) contain, besides these forms, only the system of forms w a (see (56)). Therefore, the whole union of the differential 1-forms {031:I 031 , 03t }

1 03cl, , 03b,, . 03bc, . ~ o. {03, ,03 . ,0,~,. ,03,1,0311,031,03b, a , , a 03c,

, 1 a ) 03111,03el,, 0361,

is a completely integrable subsystem of the system of structural forms of the third-order frame bundle g~(v.). This subsystem determines over the total manifold of the bundle P(Vn) the subbundle .ffI(P(Vn)) C H(P(V.)) of its first-order frames (even some G-structure of first order over the manifold P(V.), since the coefficients in (50) are constant). By the same token, the morphism P(Vn) ~ S2(Vn) associates with the cross section ~r~3 from Proposition 1 the cross section ~,s : P(Vr,) -+ HP(Vn) covering it. It is taken into account here that the bundle t'IP(Vn) is a product of bundles P(Vn) XS~(V.) Q(Vn) along the common base S2(Vn) since the structural forms of the bundle H(P(Vn)) are the union of the structural forms of the bundles P(V,), O(V.). Thus, the cross section S2(V,~) --+ Q(Vn) and the c r o s s section P(V,) --+ ffI(P(Vn)) covering it are in bijective correspondence and are characterized essentially by the same system of structural equations 03~ = V~,w 'p (the additional forms 03,1,wb. of the bundle f t ( P ( V . ) ) which are independent axe not contained in these equations). Thus, Proposition 1 admits the following geometrically more capacious reformulation. P r o p o s i t i o n 2. The cross section a~,~ : S2(Vn) -@ Q(Vn) invariantly determined by the differential system qa3: S2(Vn) --~ S3(Vn) with the structural equations 03a = V~/~03~b,

03!b = {031 ,03 a ,031, a 031,,031,}, a ,

03a =

1 03e,, , 03b,, a 03bc, a W,1 a 1,031,,, 1 03e,1 1 } 03e,

is naturally lifted up to the cross section ~r~: P(V.) --+ ft(P(Vn)) with the same system of structural equations and the additional independent forms 03x,03b" ' " The cross section &~a, which is the field of linear frames of the manifold P(Vn) is characterized by the basis of linear differential forms 031, 03., w~, 03[,, w~,, w~, 03~, globally defined on P(Y,), which are dual to the field of frames 5~0~ and subordinate to the structural equations do.)1

=

031 AO31 +~'~1, d031 = 0 3 1 A0311 "~-~-~' dWll = 0 3 1 A0311 + ~ ' ~ 1 ,

do3" = 03b A03~ "~-W1 A03~, d03t -~-03~A03ea + ~ t 0 3 1 A03111 + n ~ , " ~ +031 A 03~'1 + 03~ A0311 + f~l, " d03~' = 03~ A (03~ - ~031) d~', =

278

03~1A (03~ - 2~03[) +03~' A03h + f i b .

(59)

The 2-forms fl are the semi-basic forms of the base S2(Vn) of the bundle P(Vn)

f~ = R ~ 0 3 ~ A 03r The structural equations obtained are the aim of the preceding tracing; its main point is the closed nature, excluding the appearance of new differential forms. But its geometric sense is not found here. First of all, let us define more exactly the construction of the bundle P(Vn) -~ S2(V,,), which becomes the medium of the geometry of the system qo3: S2(Vn) ~ S3(V,,). The product H(V,~) • S2(V,,) inherits from the manifold H(Vn) the structure of the principal bundle with the same group GL(n, R), base S~(V,), and structural forms 031,03at,03, ,031,,0311,03,,tZb a ' ' "~,We" 9 ' As is easily seen, the relations 03c' = V:r 0 = V V : , which are among Eqs. (56), make the hyperplane stretched on the vectors ge of the frame (g,, ~'c) e H(Vn) a function of an element of the bundle S2(Vn), hence a function of the bundle Sz(Vn) ~ S2(Vn). Appearing thusly in the above-mentioned product subbundle in which the plane of vectors g~ is connected with the elements S2(Vn) is t h e b u n d l e P(Vn) C H(Vn) • with the structural group GL(1, R) • G L ( n 1, R) and forms 03' ,03% 03~, 03h, 0311, 031, 03L Considering the structural equations (59) for the fiber forms Wl~, 03~ of the bundle P(V~), one can easily discover, by virtue of the Cartan-Laptev theorem, that the following statement holds: P r o p o s i t i o n 3. The linear differential forms 031,03b; a determined on the principal bundle P(Vn) --~

S~(V,,) and subordinate to the equations do311 = 03' A 03111 -4- ~"~I =

ill,

are by virtue of these equations the forms of some connection on P(V,O, and the semi-basic forms ~2,,"f~2-9 are the 2-forms of the curvature of this connection. It is appropriate to stress that the curvature forms ~ , ~ , as other semibasic forms f~ of the structural equations (59), are expressed by the products of forms 03', 03", 03~, 03,,, ~ 03,,' of the bundle S2(Vn) representing the base of the principal fiber space P(V~):

fii

kI~,~03 ~' ^03~',

~ =-~,~,

=

9

The coefficients of these exterior forms are the functions on P(Vn) expressed by the components of objects entering the Eqs. (56) after its substitution into the exterior equations (58), thus admitting the form (59). As is seen, all other coefficients of Eqs. (59) are constants. The proposition stated above does not give complete information about the role of the structural equations (59), since here only the part of these equations dealing with the forms 03,, ' 03ba is characterized. It was necessary to investigate the "constant" fragment of these equations remaining after the hyperbolically possible vanishing of 2-forms ~2', f~, ~2~1, f/It, f~, f~ in order to obtain the exhausting characteristic of the structural equations (59). The answer to this problem is obtained in the following theorem. T h e o r e m 1. There exists a class of systems among the third-order systems ~s: S2(Vr,) _.+ SS(Vn) for

which and the structural equations (59) have the form d03 1 -- od1 A 031,

d031 = 031 A Call ,

d~" = 03b ^03~ +03' ^03~,

d/-dll = 031 A 0311,

d~t = 03~ ^ 03~ + ~;03' ^ 031,,

d ~ =03~ ^ @t - ~03~) + 03' ^03~ +03~ ^03h, d ~ , =03h ^(03t - 2~t031) + 03~ ^031, 279

of the Maurer-Cartan equations of a Lie group (let us denote it by P~) represented in the local coordinates x 1, x 2 , . . . , z'* by the transformations ~1 = C] x l + cl C1xl "~- C '

Cl 9 C -- CIcI =

i,

(60) (Cl:~ 1 -}- C) 2

dS x a The systems of the indicated class are reduced to the form (dxl)3 = 0, which is invariant under transformations (60) by a special choice of coordinates. This result helped to give a proper geometric interpretation of the structural equations (59) in the general case. The point is that the group P~ includes as a subgroup the structural group GI,,~-a of the principal bundle P ( V , ). In this case, the bundle P(Vn) can be extended to the principal bundle with the structural group P~: P(Vn) = P(V=) x P 2 / G x , n _ l . The forms of some connection of Cartan type in the extended bundle P ( V , ) D P(Vn) can be restored (as is demonstrated for the general situation in [9]) by the forms satisfying the exterior equations (59). The forms'wl,w" ~ , bw ~l , w" l~l", w. a1a , .~ l1, ~ "" by themselves are the restriction of the corresponding forms of the indicated connection to the subbundle P(Vn) C P(Vn). Thus, all preceding constructions invariantly connected with the third order differential system 9~3 bring us to the fundamental result. T h e o r e m 2. The principal fiber manifold P(Vn) = P(Vn) x P~/GI.n-1 over the base S2(V,~) with the structural group P~ and some Cartan connection the restriction of whose forms to the subbundle P(V,) C P ( V , ) are the forms ~ , wa, w[, ~ h , w h , w~, w~ subordinate on P ( V , ) to the structural equations (59), is invariantly associated with every ordinary third-order differential system on the smooth manifold Vn 903: S2(Vn) --~ S3(Vn). The exterior 2-forms f ~ , f~$ are the curvature forms of the connection and f~l, ~ , ~'~1, ~'~]1 are the forms of its torsion. Let us note that since the forms w~, w", w[, w[1 , wll, Wl1, w~' set on the manifold P ( V , ) the field of co-frames, P(V,) is a completely parallelizable manifold. The sense of vanishing of the curvature-torsion forms n I' = n ~ = n ~ l = a l l = a l = aS = 0

is contained in the statement of Theorem 1. And finally, from the geometric results obtained, the main conclusion follows. The Cartan connection constructed and its structural equations (59) imply the complete information about the geometry of the differential system ~3, since the integral curves of these systems and its geometric sense are expressed in terms of the connection forms w 1, w', wt, w[1, Wlx, w~, wt . P r o p o s i t i o n 4. Integral curves of the system 9~3 differentially extended into the bundle S2(Vn) are determined in it by the differential equations w ~ = w~ = w[1 = O, by virtue of which these curves and only these are characterized as geodesic lines, which are the lines along which the second-order tangent elements are parallel with respect to the Cartan connection associated with 903. The generalization of the results of this subsection to the systems of arbitrary order obtained in [5-7] will be described very concisely in the next section. 3.2. T h e F o u n d a t i o n s of t h e G e o m e t r y of R e d u c e d S y s t e m s of a n A r b i t r a r y O r d e r The differential system ~v+x: SP(V~) __+ Sv+~(V~) of order (p + 1) is characterized by the initial structural equations ~.da

= v ; + , O(Mk +

1 b +...+

p b

(02b m ~.dbl...l )

(61)

$

and its Caftan prolongations of the same high order. The conducted prolongation is partial and the choice of the objects being extended is prompted by the experience of the investigation of systems ~3. The technical complications increase not in the arithmetic progression, the choice of the equations being prolonged was dictated by the following arguments. If we start from the fact that by the type of systems ~3 the forms =

280

, O d l l , . . . , 0dp, Odll , r

Odb }

(62)

are candidates for the role of connection forms, then by the method of prolongations and scopes it is neccesary to construct objects after the canonization of which all new differential forms entering the structural equations for forms 03~ = {03:,03",03~,..., wp, a 03:1} 1 would be expressed over the basic forms (62). The entry of structural equations for forms (62) permits us to detect the union of new differential forms 1 1 1 a a a : 03c~03c11~03c1:}03be,03b:~'''}03bp-1~03:::~'''~

031 a p-1~03p-kl"

(63)

The first formal part of the constructions which was successfully carried out in [7] was focused on the search for the objects containing in the equations the enumerated forms, which are expressed after the canonization of these objects over the basic forms. The substitution of the obtained expressions of forms (63) into the structural equations for the forms of 03a (62) subordinates these forms to the following exterior equations: d03 1

=031AW]

"t-~1,

d03]

=03:A0311" t - ~ ] ,

d03111 = 031A 03111 " [ - ~ ] : ,

d~ ~ = 03b A 03~ + 03: A 03~, d ~ = 03g A 03; + ]~031 A 0311 + aS, a 1 + s(p+l-s)03a dod,a = 03,b A (03t -- 3(~b031) : $--1 A03:1 +03:

A 03a,q'1 q- f ~ , dwp. =03p~A(03t_p6t03~)+~03~ 2 ~-: A031:+ ~p~ ( ~ = : , " " P - :),

(64)

which include only forms of the system 03% where the exterior 2-forms : a cja O1 O1 o a ~'~ ~~"~: ~ . . . } ,,,lw ,~1: ~,,~:, a~b

are expressed only by the structural forms w 1, 03a, 03~,..., 03~,,03~1 of the bundle SP(Vn) --+ SP(Vn) covering the bundle SP(Vn) with one-dimensional fibers corresponding to one additional form 031:. It is clear that the forms 03~ in (62), by virtue of (64), make up the basis of forms on the subbundle P ( V , ) of the principal bundle H(V~) Xs(v~) SP(V,,) with the structural forms 031~03~ ~031~. 9 . . ~03p}03:l~: ~ : .1 ~t~b . ~~ cx and the group GL(n,R). The subbundle P(Vn) C H(Vn) x S(v~) SP(Vn) is selected by conditions such that the vectors ge of the frame (eq, e~) E H(V~) at the expense of canonization reducing to the relations 03~ = V ~ w k +V1~03~+... +Vlfw~ are placed into the hyperplane II = F(j~qoP+:(sP)), sp E SP(V~) scoped by the jet jqqpP+: of some order q. Thus, a fiber of the bundle P(Vn) consists of the frames (g:, &) E H(V~) associated with the elements ~ of the base ~ ( V , ) , so t h a t ~o E n = F ( j ~ + I ( ~ , ) ) , ~, = pr~'. The structural group C L ( ~ , R ) of the enveloping bundle is reduced to the subgroup Gi,n-1 E GL(n, R). It is proved that Eqs. (64) under the conditions f/: = f ~ = f ~ = f~p = f~]i = f~ = f/~ -= 0 are identified with the M a n r e r - C a r t a n equations of some group Lie representable by the following transformations: ;~1

C] x l Jr cl --

C1~ 1 -~- C ~

C~ . C - -

CIc1 ~

1~

c$z b + c ~ + c~z 1 + ~.1 Ca(2:l"lp ~,~

(68)

(C:Z: "~- C) 2

In this case, the system

~p+l is reducible to the simplest system

(dxl)p+l

-- 0, which preserves this entry

under the change of coordinates by transformations (65) of the group P~. Since this group includes the subgroup Gl,n-1 C P~, the bundle P(Vn) is expandable to the principal bundle zb(Vn) over the same base SP(Vn) but with the s t r u c t u r a l g r o u p P~. This brings us to the fundamental result for the systems ~0p+l : Sp(V,,) ._+ Sp+I(Vn) of arbitrary high order p + 1, generalizing the corresponding theorem for the systems q0~. T h e o r e m 3. The principal fiber manifold P(V,) D P(V,,) over the base Sv(Vn) with the structural group PPn and 80me c o n n e c t i o n o f C a f t a n with the connection f o r m s 031, 03a 0 3 ~ , . . . , 03~, 0311, 031, 03t, restricted

to P(Vn), and the structural equations (64) is invariantly associated with every ordinary differential system of p+ 1 order on the smooth manifold V, q0p+: : Sp(V,) _~ Sp-I-:(Vn). ~~],~~ are the curvature forms, and 281

are the torsion forms of this connection. The integral curves of the system ~ p + l become geodesic lines on Vn equipped with such a connection, that is, the lines with autoparallel element of tangency of p-order. All invariant properties of the system r p+I and its special classes can be determined in terms of the associated connections. For example, the test of the linearity of the system qap+l with respect to the higher derivatives of the right-hand side of the equations of the system is stated as follows. P r o p o s i t i o n 5. The system ~ P q " iS reducible to a form linear with respect to the higher derivatives of the right-hand side of the system if and only if the part of the components T/a "p-t'l PP be of j2qap+I forming the tensor which is part of the whole curvature tensor of the Cartan connection is equal to zero. As is seen, the classification role of the Caxton connection of the system ~ + ~ and its invaxiant curvature torsion is certain. But it is far from simple to realize it because of the inevitable calculating obstacles. The situation dealing with the system ~ + x setting on the plane (z, y) is easier since, in this case, ~ + x is written only by one equation and thus the "bacchanalia" of indices is essentially diminished, although the obstacles of calculating the local coordinates of invariants and invariant objects obtained in the moving canonical set of frames of different orders are inevitable. The final part of the review deals with new results on the geometry of ordinary differential equations on the plane, obtained for the development of the constructions stated in this section. 3.3. N e w R e s u l t s o n t h e G e o m e t r y of O r d i n a r y E q u a t i o n s on t h e P l a n e The fundamental and fax advanced investigations on the geometry of ordinary differential equations on a plane of arbitrary high order were carried out by N. V. Stepanov and reflected by him in reviews [21, 22]. The basis of these investigations is the above-described Carton connection, its structural equations and invariants, the profound algebraic analysis of which and its scrupulous calculation in the natural field of frames permitted one to obtain an understanding of the problem of classification of equations. However, there is a wealth of problems of more concrete investigations of equations of a certain order. And after the death of Stepanov (1991), work was conducted in this direction and geometrically interesting results in some special class of equations of fifth-order were obtained [1, 2]. The equation of fifth order on the plane (x, y) ~'~I, ~'~' " " " '~'~p, ~ " ~ l l1, a fl~, fl~

yC~)= f(~,y,V,y,,,u<~),yc,))

(66)

by the way of the process, described in general form, of the repeated prolongation of the original structural equation for (66) 3 2 4 2 (67) O d2l l l l l = y l ~ 1 + v ~ 2 + v d ~ + v~1~1 + y~ ~11, + y~ ~,,11 results in the structural equations of the Carton connection, admitting in the case of Eq. (66) the following form:

a~l =~1 ^~I + ~x, a ~ =~1 ^~1~1+ ~], a G = ~1 ^ ~ 1 + ~]1, a j = ~1 ^ ~ + ~ ^ ~ , a ~ = 2~ 1 ^ ~ h + a~, a~l~ = ~ ^ ( ~ _ ~1) + ~1 ^ ~ , + ~ 2 ^ ~ h +a~, a G =~a~, ^ ( ~ - 2 ~ ) + ~ ' ^ ~1~,, + ~ ^ ~ h + ~ , 2 2

2 + 3~1 ^ ~ 1 + i2n1,

2

2

In particular, the prolongation of (67) gives the equation for V4, vr42 2 T,'43 2 rr44 2 dV24 _ V C w l q_ w212 _ 2~,dll = V V 4 m V41~l .~_ V42 W2 q_ V41w12 q_ V22 ~11 q- 1/22 ~d111 -~ V22 ~1111,

reducible by the first canonization V4 ==t 0 to the form ~

282

- ~h

= v~,~' + v ~ ~ + v ~

vr42

2

~r43 2

Tr44 2

+ ~ ~,, + ~ ~,,, + ~ ~ , , , .

(68)

Prolonging this equation by the forms WIII~ 2 W1111 2

~y:~ - y:~(w~- 2w::)- y:r

6 w h ) = vv=?,

we reveal in 1/2424the relative invariant whose vanishing 1/2424-- 0 means the linearity of f by y(4) and, on the other hand, turns the invariant V243 into the relative one, the reduction of which at the expense of canonization to the constant V243 :=$ - 1 gives the relation 2 W22 -- 2W~ = VV243 = V423:to 1 ~-''" ~- Tr434 1/222W1111.

(69)

The partial prolongation of this relation

av:?: vt?r -

av#??

-

awl) = vv:~:, o""~ : : VV4233 , 0V222 COIl -

shows that V242a~is a relative invariant whose vanishing V~2324= 0 turns V24323into an absolute invariant. Let us require this invariant to have the constant value V2423~- - 1 . Thus we have selected in the class of equations of the type ~(~)

= ;:(~,y,y,,y,,,r

+ ;oO,y,,y,,,~(~))

the subclass of equations satisfying the conditions V~ff # 0 (the conducted canonization V2423==~ 0)

V~

- O,

V24d23- - 1 .

(70)

Taking into account the dependence of the form w] over the other forms in (69), we obtain from structural equations (68) the equations for the rest of the independent differential forms

:Win) +

dw 1 = to 1 ^ to,: + 0, 2 ^ ( w h - 2 2

d~ 2 =to: ^to:2 + t o 2 ^ (2to:! - w h )

h l,

+ ~2,

= w I ^ to:~: +w:' ^ ( w : : - W~l) + w 2 ^ ( 2 w h -

w~::)+ a~,

= w: ^ to~ll + w:? ^ ( ~ : : l - 2Whl) + a:~l, dw:2:: = to:::: 2 ^ (:wll2 __ W:)-t'- w:=ll ^ (~:: - W l b + 3W121 ^ will "4- ~"~111'-2 2 2 dWll:l = tolll: A (w121 -- 2w:) ~- 2w1211 A toll ~- ~12111, =

toI ^to:i

(71)

+ w: ^ (to:'i-- :Will) 2 2

i , + :Wllll ^ w2 + ~=~:, I 1 ) 0321 A (W:l ~to 2 2 4-

2 I 2 d~::l=w~Atolll+tollllA(:Wl-

where the forms ~ are also semibasic but differ from the previous forms ~2 by additional members. The forms (], just as the forms ~, can be written by components. Thus, some class of fifth-order equations is invariantly selected. There is a total system of invariants in our hands, permitting one in like manner to obtain theoretically a complete classification. But this problem is apparently especially complicated. In this case, a quite concrete class of equations is selected by some set of invariant conditions and it demands to be characterized. Two basic questions must be answered: what is the set of equations forming the selected class and what is the special geometry induced on the plane (z, y) by the equations belonging to the class considered. T h e answer is contained in the theorems which we suggest for the conclusion of this review. 283

T h e o r e m 4. The class of the fifth-order differential equations

y(5) = f(z, y, y', y ", y(S), y(4)), selected by the invariant conditions V~4 ~ O, V243 =~ - 1 , V4s4 = O, V2~ss - - 1 , is characterized by the structural equations (71) and consists of the following equations: Y(5) = ( 5y,, y O+) ~(x,y,y,) + D(x,y,y',y") ) y(') + B(~,y,~',y",r

(72)

where B, D, ~ are arbitrary smooth functions of their arguments. The way of obtaining structural equations (71) was described above. Then it was necessary to calculate the values V24a, V434, V42323over the derivatives of the right-hand side of the equation up to third order. That is technically not a simple problem, but it was solved and hence conditions (70) were converted to a system of partial differentials the integration of which reduces to a class of equations (72). The geometric sense of structural equations (71) is expressed by the following T h e o r e m 5. Structural equations (71) by a suitable change of the basis of linear forms entering them are reduced to the form dO~ = Oo~ ^ (o== - Oo ~ + Ool ^ o~ + Oo2,

dO~ = Oo~ ^ (el - Oo 0) + oo~ ^ o~ + Oo~, d o 1 = o~ ^ (el - el) + o~ ^ el + og,

dO~ = o~ ^ (o== - e l ) + o

~ ^Oo2 +o12, (7a)

dO~ = o o ^ (Oo~ - o~) + o21 ^ o ~ + o ~ dO~ d ( O ~ - o o) =

el~ ^ (o ~ - el ) + o~ ^ o ~ + o ~ ,

200 ^ Oo = + o~ ^ o~ - Oo~ ^ o~ + o~,

d(el - o~) = 201 ^ o~ + o~ ^ o~ - eo= ^ o~ + o i ,

where the forms O~ are the forms of connection in the principal fiber bundle H(P2, $ ' ( x , y ) ) with the projective structural group 192 on the plane and with the base S4(x, y). Actually, the substitution

Co== ~ ; ,

e a = ~ ~, 1

2

e ~ = - ~ o h + s~'X~l,

e~ = ~ 4 - ~ ; ,

o~ = ~~O J=l l l l ,

e~= = ~ ' h e 2 = 0311,

s~'~.,'-=

01 ---~ 20.) I - - {MI21,

gives structural equations (73) for the new basis. The same dependence connects the forms of the second power 0 and ~. Since under @ = 0 Eqs. (73) are identified with the structural equations of the projective group of the plane, according to the Cartan-Laptev theorem the forms 0~ must play the role of connection forms. The class of equations (72) admits a natural classification on the subclasses on the basis of the four independent relative invaria~ts entering the forms 0~. It is shown that in the case of vanishing of all four invariants the rest of the invariants also vanish, which means G~ - 0, and we come to the group of linear-fractional transformations of the .plane (x, y). This limiting case corresponds to the equation of the class (72), y(S) _ 5Y(s)Y(4) 40(y(Z)) 3

y,,

9 (y,,)'- '

(74)

whose integral curves are all nonsingular second-order curves on the plane (x, y):

b n 2 + 2bmxy + b22y2 + 2blx + 2b2y + b = O. Equation (74) and its connection with the second-order curves on the plane (x, y) have been well known since Sophus Lie. 284

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