No title

8. 9. i0. ii.

12.

13. 14.

K. A. Borovkov, "Rate of convergence in the invariance principle for a Hilbert space," Teor. Veroyatn. Primen., 29, No. 3, 532-535 (1984). Yu. L. Daletskii and S. V. Fomin, Measures and Differential Equations in InfiniteDimensional Spaces [in Russian], Nauka, Moscow (1983). V. M. Zolotarev, "Metric distances in spaces of random variables and their distributions," Mat. Sb., 101(143), No. 3, I04-141 (1976). V. M. Zolotarev, "Approximation of distributions of sums of independent random variables with values from infinite-dimensional spaces," Teor. Veroyatn. Primen., 2_1, No. 4, 741-758 (1976). A. I. Sakhanenko, "A method for finding estimates in the invariance principle," in: Fourth International Vilnius Conference on Probability Theory and Mathematical Statistics. Abstracts of Reports [in Russian], Vol. III, Vilnius (1985), pp. If0-111. A. I. $akhanenko, "Estimates of rate of convergence in the invariance principle," Dokl. Akad. Nauk SSSR, 219, No. 5, 1076-1078 (1974). F. G6tze, "On Edgeworth expansions in Banach spaces," Ann. Probab., ~, No. 5, 852-859

(1981). 15. 16.

17.

18.

L. Gross, "Potential theory on Hilbert space," J. Funct. Anal., l, No. 2, 123-181 (1967) o P. Kotelenez and R. F. Curtain, "Local behavior of Hilbert space valued stochastic integrals and the continuity of mild solutions of stochastic evolution equations," Stochastics, 6, 239-257 (1982). A. I. Sakhanenko, "On the rate of convergence in the invariance principle for martingales," in: Fifth Japan-USSR Symposium on Probability Theory. Abstracts of Communications, Kyoto (1986), pp. 40-41. V. Strassen, "The existence of probability measures with given marginals, " Ann. Math. Statist., 36, No. 2, 423-439 (1965).

I-INTEGRABILITY OF NONHOLONOMIC DIFFERENTIAL-GEOMETRIC STRUCTURES UDC 514.763.8+514.763.3

R. V. Vosilyus

3.

Holonomic rt.~-Connections

The theory of nonh01onomic differential-geometric structures is a natural generalization of the geometry of classical infinitesimal structures on smooth manifolds. In particular, the results on the l-integrability of nonholonomic structures of the first order we have found generalize the corresponding chapter of the theory of G-structures, begun by E. Cartan. The connection with nonholonomic structures is achieved with the help of passage to the corresponding Lie differential equations. In this case we must consider holonomic F1. a-connections of tangent bundles T - T(M), which have additional properties, allowing us to develop their general theory in a more convenient form than was done in the first part of the paper. Holonomic Ft.=-connections of the tangent bundle T ~ M are defined as linear lifts F~.=: By virtue of the natural imbeddings JzT JI(JIT) they are ordinary linear connections of the extended vector bundles JIT.

JIT--~J2T of the canonical projections JzT ~ JtT.

Local fibering charts according to the rules

(x~,~

:i9 :i ~i~)

of the differential extensions JzT

which change

r~ = g~ (j'~p -~' + f ~ ~ ) , -- 6i]

%JlP

-..

"..p) .-g, & (ftp, "i.'+f~p :l , ,'k :t

Vilnius State Pedagogic Institute. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 27, No. 3, pp. 435-440, July-September, 1987. Original article submitted April 3, 1986.

0090-4104/87/0213512.50

01988 Plenum Publishing Corporation

213

let one define a Y~. z-connection with the help of relations "2i ,~

=

FrO

.~.~ ~1 .~ -

-

" F~

:,', "

whose coefficients, by virtue of the holonomicity, depend only on the base variables x i. Setting -- ~ijr'ts--

lijr'~

"

we get the law r;r o.,,~ ,-,s (rt f k f u __ . , ~ u 4- Ie k ,~.t ctP ~ u .l- f k ,zt q p ~ u , q v [ u ~,~ 2st.ai oj,~r., l.lv ~ liir --dtOotoi ~ j ' .' t p o r o j "i - 7 " o i j d v ~ r ~

FIpSt,~i o, P c~s ,rt f k oj~rd l

_ _ . , k .d_.,tcx ~ru trt ~'p ,.~ { k o P o't.~, f k OO O,s Crt -- lijr , ;ijpOSe~rdtu dtp~ijo, 9 dtpsoi ojOr

of change of coefficients of the connection considered. The following assertion has no analog in the nonholonomic case (cf. =he first part of the paper). THEOREM I, An affi-ne connection on the base M is intrinsically associated with a holonomic FI. z-connection if and only if its coefficients satisfy the equation rl _ Flp ~..I

U -- - - i j p -- ~ ' q l

in which p~j denotes an arbitrary intrinsic tensor field. Proof. If we are able, in an intrinsic way, to associate an affine connection whose coefficients are denoted in the new system by Y}k, then

l~k,

fkp= F~pf~-- Tk f~f~, glij

='~U

k

l rl .Poq g k - - " l~s6i ~ j ,

by virtue of which one has l~ p ~ k t "u ku l Fo~tgi gff l g , / , = Y q , + ( F t o f l

From this, setting

u

r,

-

k

k s a t 0 4.. f u ~ k ( . / . . o v -Ys:aftf~)(g, gi ~s+g, "~" ,gs~i) o . -~=-,,,uo~

-

Fe

oP ~S)

- p , oi os~"

(*)

and s-mming over these indices, we get the identity we need ~P=,r

oi &fill

(pl~ k~l~

P! ~--~'~" --'{i~.

-- ~ p s J -- "[i]u

Since the zero tensor field is intrinsic for an arbitrary F1, z-connection, the following very important assertion is valid. THEOREM 2.

Holonomic Fz, 2-connections have intrinsic canonical affine connections

which are torsion-free. THEOREM 3.

The quantities *Jr -- - - u r - - ~ i j ~ r

--ri ~j

~rj ~

are the components of an intrinsic tensor field and satisfy the equation w[f~ = 0.

proof. the form

It suffices to use the intrinsic canonical affine connection and rewrite (*) in r ' k u , T',a ~ t k ~u r~k ~ u ' t _ p ~ ; , . t gl gv_.,lt' .2_-~,~1 v[ ~ ,d L p s t -~- L p s Ok - - F tp ~ x ~ t t $ Up) ~;i ~ j ~r.J k J u - - ~ i j 9 , ( i j ~ 9 - - ; r i ~ ] - - ; j r

THEOREM 6-.

The

~'

I 9

quantities

N,jkt"

__Fk_ jt__

i

F;sjkpFt~+FojF~t+Fo~F. i ~ ~ o _F~i Fua.s_O, yaji

are also the components of an intrinsic tensor field. The proof is obtained with the help of elimination of the variables f ~ transformation law for the components Fbk~. of the given F~. e-connection.

and

[}~, from the

The curvature tensor field R~kt of a canonical affine connection is also an intrinsic field of che PL.~-structure considered. In particular, it satisfies the equation

214

THEOREM 5. With holonomic F~.~-connections there are intrinsically associated linear differential equations ~'k v j . + aCikt ~ u -.

~ j~L - -Rjkt) - -- 0

of the second order. Proof.

It suffices to make direct calculations

in local fibering charts.

We note that %-~ here denotes the differential operator of covariant differentiation with respect to the intrinsic affine connection. The differential equation considered can be written in terms of partial derivatives in the form

which lets us formulate the following assertion. THEOREM 6. The structure of a holonomic rLz-connection is defined with the help of a symmetric affine connection on the space M and two of its tensor fields ~r and ~r the first of which satisfies the equation

=%=0. THEOREM 7. T h e linear differential equation associated with a holonomic is a Lie differential equation if and only if one has ~i,

= o,

=r

l~x.2-connection

= o.

In this case it is the Lie equation of infinitesimal motions of the canonical affine connection. Proof._ The differential extension J, ~ is the sheaf of germs of Lie algebras for which the commutator of the sections (~, ~, ~}k) and (~t,N}, ~k) has components T

"hi " ~ k j

__~l_kj.d_

'

. i

If these components satisfy the equation associated with the given FL~-connection, has p i i s~t ( d s F l t , _ F t ~l s S }p _ F u sI" ~ t p +r,71,~,) s=t .., , i$

in which

the

variables

p

~ , r/, ~

io

s j,.

and

iS - ~ p -

~

ip ~s , ~'~xO ~ t r ~'t

FU, ~t "r" - - j l

remain

Setting the coefficient of the product

-

. k

--jr ~k -- ~Jl ~k, "

then one

-

completely

arbitrary.

~.,:'-%,equal to zero, we get the first equation we

need,

If we now also set the coefficient of the commutator suitable transformations we arrive at the relation ~

Setting p

=

9~;'-~" ~ ~ _~

equal to zero,

then after

~i_~/~;Is 8 j~ - ~ r

jls~t

t, and contracting over these indices, we get the final equation

~}u = O. By virtue of these equations the Lie differential equation considered reduces to the form

and is the equation of infinitesimal motions of the intrinsic affine connection. comple~es the proof of the theorem.

This

THEOREM 8. Covariant constant fields of the intrinsic affine connection are covariant FL.~-constant, if and only if they satisfy the equation IH~

- R ~ ) ~ = O.

The assertion is a direct consequence of Theorem 5. THEOREM 9.

The equation

215

holds if and only if the [~z.2-connection decomposes into an affine connection on the base M and a linear connection on the vector bundle T*| Proof.

The change of variables

lets us rewrite the differential equation associated with the given form "~

"

~ j -- ['jk "11 ,p "7- \ ~ j k l

--

Rjkl)

--

----

Fz.~-connection in the

O.

In the ease we are considering it follows from this that

7~ % + ~%, 4 = O, which can be considered as the invariant form of the differential equation corresponding to a linear connection of the vector bundle space T* | T. of the identity

-~+Pkt~,'P' =----0,

In fact, by virtue

r t k p r_ ~j'~ + P~j,~, + F,j ~kp - F,y ~s- 0, .

found with the help of this equation, we arrive at the relation ~p.k ~ k p a G p~k, - _~,, _ F,~ ~,p - F,~ 8 ~s . _

On the other hand, an affine connection on the base M and a linear connection on the vector bundle T* | T let us, with the help of this relation and the equation

~

kl = R ~ k l

get the corresponding Pz.z-connection. If we construct a Fz, ~ tiation

by setting

~1;~,~=0,then we get a covariant differen-

V : 3"*| 3 " ~ T*| 3"*| 3-, which completely coincides with the covariant derivative connection:

V with respect to the affine

in this case the l"z. ~-connec~ion is generated onl~ by an affine connection of the base, which also induces a connection of the vector bundle T | T.

The curvature morphism R : J* T - - ~ A 2 T * |

of a holonomic

Fz.~-connection

is generated by the composition of morphisms

j~ ~ r , . . j.,3- _o_ Y*|

m| ..... Y*|176

described with the help of the Spencer differential operator. be defined with the help of the equations

A~3"*| In coordinate notation it: can

__ ~ i p ";k j _ D i ~! -., rs - - axles "~.p 7- a~lr ~ .~ r

~i

which generate four groups of components of the curvature tensor. holonomicity one has

~ , , = o,

However, by virtue of the

R~,~=o,

while the remaining components can be calculated with the help of the following formulas:

R)~,= O[ r F s,,l j l +r}% ~ ! I F s,,I kt R;kra - - ;h, ~ F ~l i j k + F'/'I, , ,,

216

, Jl, ~- ~,-,[r ~$1

F**] ~ .

Since the differential extension J2T is not a tensor bundle with base M, the components of the curvature morphism

R:JtT'--~" 2T*| are not components of a tensor field.

Only if

~4%~ =

0,

~/~

0,

=

when one has ~i =0 k. t j - - ~

~Pa~kiJT~ka~PlJ~-'lJ~kiJ

~kP|l"~Jl 9

for F-lifts

does one have ~ . . = :~ 7 ~R ~ .

SOLVABILITY OF A MIXED PROBLEM FOR A NONLINEAR SYSTEM OF EQUATIONS OF SCHRODINGER TYPE UDC 517.9

A. Domarkas and F. Ivanauskas

A large class of problems of the theory of waves (for example, the propagation of electromagnetic waves in a nonlinear medium [I], three-frequency [2] and four-frequency [3] interactions of waves, resonance interaction of waves [4], etc.) are described by systems of nonlinear differential equations of Schr6dinger type

au--!~ =ia~Auk+A(u, ~), (x, t)~Q, k = l ,

...

ot

m.

(1)

Here u - (ut(x, t) ..... u ~ ( x , t ) ) i s an u n k n o w n v e c t o r - f u n c t i o n ; Q = ~ ]0, T[; ~ = R " i s a bounded domain with sufficiently smooth boundary J ~ ; i = ~ - - Z ' - i : a ~ , k = l , . . . , m, are real constants; A is the Laplacian; u is the complex-conjugate function of the function u.

The initial and boundary conditions have the form u~ ( x , O) = uo. ( x ) , x a f2, k = 1 . . . .

(2)

, m,

u~ (x, t ) = O, (x, t ) e O ~ x ]0, T[, k = 1 . . . . .

m.

(3)

In many cases the following assumptions hold: I) the functions ~, 0/~/~, 0flk/~, are continuous and satisfy the inequalities

(4) k--I m

(5) m

where

k-- !

p~>O, _ ~ p k = p , k , ] = l

.....

m.

One c a n a l w a y s a s s u m e t h a t

p ~ 1;

(6) k=i

3) there exist real constants ~I .... , a m such that

V. Kapsukas Vilnius State University. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 27, No. 3, pp. 455-465, July-September, 1987. Original article submitted September 30, 1986. 0090-4104/87/0217512.50

9

Plenum Publishing Corporation

217