No title

I-INTEGRABILITY OF NONHOLONOMIC DIFFERENTIAL-GEOMETRIC STRUCTURES

UDC 514.763.8+514.763.3

R. Vosylius

i. r'L 2 -Connection The reader will find the general definition of nonholonomic differentlal-geometrlc structures, including nonholonomic connections in the author's survey [2]. The theory of F lconnections, which are used essentially in what follows is recounted in his joint paper with V. I. Bliznikas [I]. The necessary concepts and methods of D@-technique, recounted in [2], will be given at the appropriate points of the following text also. Letting i,j, k, I, p, t..... I,!2 ..... dimM, a, 5, c, d, e, u..... I, 2, .... d i m E - d i m M , we shall consider the category of C~-smooth locally triyial bundles x:E ~ M, covered by local fibering charts (x~,/), generated by the local charts (x I) of the base M. In the domains of intersection of the charts considered one has

x'=gJ(~'),

~'=f'(xS, ;'=f~

JP),

y~=gb(s

y,).

The differential extensions JkE have local fibering charts (x', / ,

y~,, Yr,, . . . . . .

Y~, .... ,),

which let us use the differential operators

,),.... t, =,~'lc)x~...~)x',, 0~"" z,=~I 0 y,,,...~,, "

c), = ~)1o~,,,

a ..... ,%=0~

.. *a.,,

k--I

Z

'",

p--O

..... and the corresponding partial derivatives

fifo., ip Oij.... ,~,f', =

~...~,~ .... b , = ~ .... i,*,~b,...~/a. In the new fiberlng charts

there are new differential operators

A,

-

...

~-~

a, . . . . . . .

,

and new partial derivatives l

l

g~,...~,=~)i .... l , g ,

a

~

gi .... ipb .... b, = i.... ipo-Ob.... h

ga.

We note that by virtue of the relations i

Vilnius State Pedagogic Institute. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 27, No. i, pp. 28-37, January-March, 1987. Original article submitted March 14, 1986. 0363-1672/87/2701-0019512.50

9 1987 Plenum Publishing Corporation

19

~, =f, (g~(~)), Y"=f" (~'(~'), s" (~', y")), one has the identities

g~hk + g~A~= o. The local fiberlng charts of the second differential extension JzE change according to the rules

.~' = f , (xO, y " = f ~ ( x ' , Yg,

-a _

t

@

a

l

:~

a

YO--g~ ~' f +g~ gSdk *f ,

O, =fa~O,~,

by virtue of which on the bundle space E one has one has the relations Oa

on its differential extension JIE

I =f~b -c)e+ g,k Ok# f~c -Oc,

while on the extension J2E one should use the formulas

Ok f~a c)~+ (oiS C)k f~ a--~lJa

+ g~ g i Oktf[,) 0~,

\olkJa ~ ~l~put wa "~-opelUt JaYUb

a --.~l.SkJa

Moreover

on all the differential extensions, one has the identity

~ =2/ k.

We shall consider the vertical tangent bundles

~=ro(J~e),

T~=r,(E),

having local bases decomposition

~=T"(J,E),

{O. }, {do, di.}, and also {O,,, O~, O~t}, respectively.

The components of the

of an arbitrary vector field of the space T~ change according to the following transitive laws:

-b_

~bc

ba

~, - ~ ~k 2. ~ + ~f'. ~k, ~U + (g,,ft + ~ gl, O, f~. + gp g, tlt f t ) ~, + 9 p :~ b ~,'# b a. +(gtk3; f t + g f gkO;,ft)~

Nonholonomlc dlfferentlal-geometrlc structures are distributions of vertical tangent bundles of the e x t e n ~ d spaces JkE. Nonholonomlc differentlal-geometrle connections are given by dlstr~utions of special form. We shall use Fa and ra,z-connectlons, which mean lifts r 1 : ~ , FL~:~, canonical projections ~ E ~ E and J z E ~ E generated by maps ~ , ~ . The corresponding nonholonomic structures in this case are defined with the help of the relations b1=F1(Y ~ and bx.s=F~,s(~). In local fibering charts one has

which are generated by the coefficients F~ of the connection considered. fibering charts, we arrive at the formulas --e

20

k

d

Changing the

which generalize the classical transformation law for the coefficients of an affine connection. Letting

we get differential operators and the equations [0~,3~]=-O[~l~l,0~, which lead to the invariant form associated with the tensor field R~==0~F~]t. In [i] this field is called the vertical curvature tensor of the connection considered. The horizontal curvature field has components a

# i~k]

I~kl

,.

It generates a linear map T~ ~T*| which is called the curvature morphism of the F1-connection. If Ot denotes the sheaf of germs of smooth functions of the space JkE, ~ the sheaf of germs of local sections of the vector bundle

T~,

then the curvature

morphism

is

generated by the composition of maps

Uo r.. U[

. U* |

U,

|

U~

A

|

Here T* is the dual tangent bundle of the base, which is always considered as its (~)-*image, generated with the help of the canonical projection =~:~E-->M. The differential operator

V~:U~-+U*| of the form V # = D # o F a is the covariant derivative with respect to the F~-connection. local fibering charts it is given with the help of the equation

In

~:~ (~. ~a c),)-_ (c)k # .~ ~a + Fkb a ~) dx k | c),. In the case of the tensor field ~=~d~|

the corresponding convolution

is no longer invariant, and under change of local fibering charts leads to the relation

b? ~ + P~, ~ =A Sg~ gf (0~ ~ + r ~ ~ ) + gfkfg ~ . Let r}k denote the coefficients of an arbitrary affine connection of the base M. virtue of the classical law

By

r~,=g~ g'kfl Vf,, + glkL' we get the equation

Ffk ~p--f]~ gt gk I'I, =t =b ~ t + ~..~ l k J b~",~s~ from which it follows that the quantities

are components of a tensor field. pair of connections.

This is mixed covariant differentiation with the help of a

However, we will come to this construction more naturally in the following way. With the help of the formal derivative ~# one can get canonical lifts %~:T-+T(J~E) of the tangent bundle T of the base. In local fibering charts they are defined with the help of the equations ~(~3t)=~i0~. One can extend the lifted bundles T~=%~(T) in the reduced differential way and get the corresponding reduced differential-geometric connections [2]. Since in the case considered the reduced differential extension is realized with the help of 0~-differentiation, the extended bundle J~(T~) has fibred components (~, ~}), which change according to the rule

a

P

i

~

9

Letting

21

we get the coefficients ~ of the reduced differentlal-geometric connection, which change exactly like the coefficients of an ordinary affine connection of the base. The only difference is that now they depend on all the components of the differential extension JkE and generate a nonholonomlc affine connection. They will also be used in the construction of the operations of mixed covarlant derivatives.

A F1,g-connection is a lift of the form wa

and have coefficients

F~b and

=a

~

FI.2:Tr-+I~2. They have coordinate expressions

a

'~a

k

t

b

"~c

kl

rc

P~b, which change according to the following transitive laws:

- - bi

~ p

J

b

cs

9

t

i ~ b L-g~ ~n~d, f~,

--b

c--bi ~ i j b c # b p @ b rlkdJ~a+g ~, d~# f~rik c gigk(f~ rija--c)ijf~)--glkdp f~a.

9

which satisfies the equations L e t n - d i m M a n d m - d i m g -- d i m M. A t e n s o r f i e l d Plkb, ~ Ptpa=0, will be called symmetric and irreducible. THEOREM i,

The quantities " =Fl**+-~--

~

60Fk)p~--I'lka

Fk)Pa

n+l

are the components of a symmetric and irreducible tensor field. Proof, We shall use only the first equation in the law of transformation of the components of a Fl,=-connection. First we can get from this the relation --bp

and then

the

equation 1

Now i t

only

remains

8~,Fk~p

i p i

to eliminate

b rrjdes _ ~ 1 from this

I

8(pIj)~Q , +-ct s

equation,

terms not containing components of the connection, theorem.

l'~--lt

with

~li g~)S pf ~r _ ~ f ~ i

the help

the

convolution

and we get the assertion formulated in the

THEOREM 2. With an arbitrary pair consisting of a F I and a associate an affine nonholonomlc connection in an intrinsic way. Proof,

of

~b._.:,.~/6a"

Fi, 2-conne=tion,

one can

The relation t ~ b b d p gigged, f~=f~ gag, r gr ~ - r t--b

a n d the first equation in the law of change of the components of a write the identity lku__~.kaul

by virtue

of which

the

--b

i__

i

p

j

b

r

Fi, 2-connection let us

Fpdgj_Fydg~)_glkf~gba,

quantities

r,

1

r,),-r,,,)

are the coefficients of a nonholonomic afflne connection. THEOREM 3, With a nonholonomic affine connection and a trinsically associated a rl-connectlon.

l~1,z-connection, there is in-

Proof. We only note that the coefficients of the intrinsic rt-connection are determined by the formulas

rid _-

22

i

ck

THEQREM 4, If we are given a nonholonomlc afflne connection, a F1-connection, and a r~,2-connection, then the quantities -- *tka

T --qk) va -- u(t

r~).

are the components of a tensor field. Proof.

It suffices to use the equations 9

=

g(k go/; r;,,

r(kO--

t

#

b

I

p

b

i --b

and from the first equation in the law of change of components of a eliminate all free terms. THEOREM 5,

F, and

r,.~-connection to

r,,2-connections generate a tensor field = r ~: --r,~o ~b. 9 ~ = r ~ - Ok# F,,b+ r~=

Proof,

O~-dlfferentlating the equation

we can get the relation f~d(rlkd--b --Ok@--b

~ #

~--hi

i--b

c # d ~ b t ~ b rd (r,~a--@, raa)--(gtkf~ +gtp gkOt f,~) p,. The following identify indicates the next step in the proof: --b ~';b -bl --c -b --c c d c d -bl p t b d f~'d (F~,. d - O~ r , ~ - -F,~: r,~ + -r,o rk~) = g,t gf~f~b (r,po3,~ P~o + F,a r..o) - g~fjc -r,~o rk.d - ( gp, ~ + g(,gk, o.~ f~) F,..

~ by We note that the tensor field we have constructed satisfies the equation .~ltklb--Rb~,, virtue of which the horizontal curvature tensor field is its ~-differential. In the classical case of spaces with affine connection the analogous result is a consequence of Bianchi' s identity.

Again we consider r~ and r~,~-connections. Setting rs=r~or~, we get a lift r~:T~-+T_~, which now means a nonholonomic differential-geometric connection of second order. In coordinates an arbitrary Fz-lift is given by means of the equations - -

containing the coefficients of a ~ r~ =r~.,or, one has the relations

and

"t k b "~ ~ a - - ~ k l b ", ~ a ,

F~t~-connection.

For connections of the form

la _ a rk~ Pkg. -

A F1-connection , considered as a nonholonomic differential-geometric structure, has a differentiable extension P@(rl)- The extended structure is generated by the fields ~.__

a

a

b

k

a

l

whose D#-differentials satisfy D # ~r

|

F, (~').

By virtue of the equations

D~ ~ = [0,~ ~. + r~, ~) 0 . - O,~ r h ~ + r ~ o,~ ~ + ~k) ~1 | dx' one has

~ = (r~ r ~ , - # r~,) ~ and the extended structure p#(Fx) has germs with the following coordinate expression: ~,=~o[a_r~_O,~

~ r.,) c 0bk,] r ~b - re,

Consequently, this is a nonholonomic F2-connection. The following assertion clarifies the geometric meaning of the tensor field ~b. THEOREM 6, Proof,

The equations ~ b = 0

and

p#~,)=F,,,oFx are equivalent.

The assertion is a simple consequence of what was said above.

23

THEOREM 7, We consider an arbitrary affine nonholonomic connection, a rl-connection , and a symmetric tensor field M~Ib- If R~blkdenotes the horizontal curvature tensor field of the rx-connection, and the field ~,# satisfies the equation ~ ] b = R ~ , then setting

we get the coefficients of a I'~,~-connection and the identity llka_~ ,,~bi

~_

|

!

dl

l

b

The ~roof is a simple consequence of the assertions obtained above. In what follows all nonholonomlc affine connections are assumed symmetric. A differential-geometric object Px of the form r~ will be called a contracted object. We shall say that I'~.~-connections obtained with the help of Theorem 7 are decomposable. It remains for us to show that in this way one can get any r10s-connection. THEOREM 8, All Px, z-connections are decomposable. contracted geometric object.

Each decomposition is generated by a

Proof, We consider a given r~.e-connection and we write the equation M#~#--p~k~. following decomposition is possible for this connection

If the

~ k i __ .ai .J_ ~I I'~a r~l ~ b b --.l-'lkb 7 - u ( ! ~ k ) b - - Xll~ Ua,

t h e n one h a s t h e i d e n t i t i e s b ffi ~ r~.

r~= ~

I

(r,~. + r~, ~),

, - rkid).

From this one gets the relation

(n+ 1) rd, ffimrf, + r~f, and the final formulas

The formulas found also let us solve the question of decomposition of an arbitrary r1~s-connection. In fact, we use the contracted differentlal-geometrlc object Pl and we write the equations rtk~,

m(n+l) 1

By v i r t u e o f t h e f o r m u l a s f o u n d a b o v e , t h e s e e q u a t i o n s d e t e r m i n e t h e c o e f f i c i e n t s o f a nonholonomic affine connection and a certain rx-connection, and by virtue of the relations

=--.,

- ~

e. + - - ~ -

~,~ .

= ~&, -p~,

one gets the required decomposition

~t. = ~ , + ~, r~, , - r ~ ~. To conclude the proof of the theorem it remains for us to use Theorem 5. The decompositions of rl,l-connections obtained are not unique and depend on the choice of the nonholonomic affine connections and rl-connections. Hence, setting 24

we get the equation "bt

However / ~ identities

and

P~

I

i

are irreducible tensor fields (cf. Theorem i), from which one gets the

T~k =__~. I

~=

b

I -~-~-~.

This leads us to the formulas

containing the contraction

pt=T~, and

to the final equation al

__ _ a i

kb --Fjkb,

by virtue of which the tensor field the equation

Pf~ has intrinsic character.

From this it follows that

kb "~Fjkb,

which we use constantly in what follows, is independent of the choice of intrinsic connections.

LITERATURE CITED I. 2.

V. I. Bllznikas and R. V. Vosilyus, "Nonholonomic connections," Liet. Mat. Rinkinys, 27, No. I, 15-27 (1986). R. V. Vosilyus, "Contravariant theory of differential extension in a model of a space with connection," Probl. Geometrii, Itogi Nauki i Tekh. VINITI Akad. Nauk SSSR, 14, 101176 (1983).

NECESSARY CONDITIONS FOR AN EXTREMUM IN VARIATIONAL PROBLEMS WITH FREE BOUNDARIES

UDC 517.972.2

A. Grigelionis

Introductio n We consider the problem of minimization of the functional

in the set of so-called multlphase media (~, ~), where ~cR", n>2, is a bounded domain, and ~, still called the boundary of the multiphase medium, is the union of a family of (n - l)dimensional surfaces, belonging to the closure of the domain ~. In particular, ~ consists of two parts: F being free, i.e., variable, and 7 being fixed. The function u minimizes the functional

f~(x,{~w},=,..l~+fg(x,{o.w},.,.._,)~

g[n, .](w)ffi

(2)

m

Institute of Mathematics and Cybernetics, Academy of Sciences of the Lithuanian SSR. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 27, No. I, pp. 38-47, January-March, 1987. Original article submitted March 21, 1986. 0363-1672/87/2701-0025512.50

9 1987 Plenum Publishing Corporation

25