Integrability of nonholonomic differential-geometric structures

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

R. V. Vosilyus

2. l-lntegrability Conditions The nonholonomic differential-geometric structures we consider [I] are involutive distributions b of vertical tangent bundles of extended spaces JIE. In what follows we also denote by the letter b the O,-modules corresponding to these distributions of the sheaf of germs of their local sections. If the smooth function F: JIE ~ R for arbitrary choice of local sections ~: ~E-+b satisfies the equation 2 (~ F = 0 , then it is an invariant of the structure considered. The fact that the distribution b is involutive lets us get local bases of invariants {F~} and express any other invariant F with the help of the equation F - F(F~). Bases of invariants admit arbitrary substitutions P~ =P~(F=) , which must satisfy only the unique obvious requirement

det @F~/o~) # o. Letting F ~ - Ca , where C ~ denotes any constant quantities, we get integral manifolds, fibering the base space J~E of the structure. At each point of the integral manifold the corresponding fibre of the distribution b is its tangent space. Setting --

~

~a~kWa~

we get the equation ~a

k a ~kO~F~+~ O.P=O

of the distribution has relations

5.

Under changes of local fibering charts and bases of invariants one

of the form

~ o o f + ~oAr ~= ~ 1 ~ ( ~

+ ~,~.F~).

We shall use the notation

and the equations b

~.k

g/), ff, A: + / g A r

, "lb

%

-- "~

., bgy,',~ = " ~ b

,

which follow from this. An example of Lie nonholonomic differential-geometric structures is the vertical tangent bundles Yr(~) of nonlinear systems of first-order partial differential equations ~cJiE. Let {F~} denote a basis of the ideal of the sheaf of germs of smooth functions of the space JIE, equal to zero on the subbundle ~. Such bases change according to the rule

c o n t a i n i n g an a r b i t r a r y n o n d e g e n e r a t e m a t r i x

corresponding to the nonholonomic structure

IIAglt. The germs

b=Y'(~) are found from the equation

Vilnius State Pedagogic Institute. Translated from Litovskii Hatematlcheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 27, No. 2, pp. 236-245, April-June, 1987. Original article submitted March 31, 1986. 114

0363-1672/87/2702-0114512.50

01988 Plenum Publishing Corporation

Under changes of local charts one has

However, by virtue of the identities F~ - O, which hold on the subbundle ~ C JIE, from this we get equations of the form indicated above. The differential extension of the structure b is the second-order Lie nonholonomlc structure p#(b)cT~ , which is defined with the help of the following relations (cf. [i]): r~*(b)={~Z:~!D~*~eZ

:* |

=~(~)~b}.

Here D # denotes the reduced Spencer differential operator, which, as already indicated, acts according to the rule ~'a a k D :~ (~, 0,, + ~..~,0~ +

THEOREM I.

a ~.,,, Ook l)_ -

[ @ ## ~a -

. ~.,) ~9.+

(d~# # "~,a -

a k [,~) 0,,] |

dx'.

The components of the germs

of the differential extension p=~(b) satisfy the equations b %k"l-"ab"z

~b

~ + .~ ~) ~ + 0~.~ ~

;"~kl b ."1" ,iv ~ I .,"ab

= O.

Proof. It follows from the definition of the differential extension p#(b) extended structure is generated by those germs

for which one has

~ eb

and the additional requirement -~b=k

Since

~eb

that the

(~,~ ~b -

.~, (~)/~~

~b

.,~) +

-

~.~) = o.

one has b

~k~b%

--v~

@R-differentiating which, we get the relations indicated in the theorem. THEOREM 2. The differential extensions structure p# ~). Proof. equations

d~/= of the invariants F~ are invariants of the

It suffices to rewrite the relations found in Theorem I, in the form of the

.e ( ~ * ) ~ = o ,

.a ( ~ . ) ~ r ~ = o .

We shall consider only normally transitive nonholonomic Lie differential-geometric structures, i.e., nonholonomic structures with surjective canonical projections b-+Yv. Definition. Nonholonomic structures, which have surjective canonical projections p~(b)-+b, are called l-integrable. In this part of the paper we find necessary and sufficient conditions for the l-integrability of formally transitive nonholonomic Lie differential-geometric structures. To this end, setting F=T~/b, we construct the normal bundle F ~ JIE of the given structure b and we denote by the symbol ~: ~ - + F the canonical projection to the quotient. Let J~F be the reduced differential extension [I] of the bundle F with respect to the canonical flat connection of the projective limit J~E. Lifts 7:F ~ J~F of the canonical projection J~F * F are called reduced differential-geometric connections of the space F. Definition.

Differential-geometric connections

generating a commutative diagram T~ r,.,_+ T~'

t are called ~-compatible.

r

l 9

. j,~F,

115

We find the coordinate expression for t h e condition of ~-compatibility. consider an arbitrary vector field

of the bundle T~ and we make use of the local fiberlng chart (x ~, / , ) ~ , Setting

we get a coordinate expression for the canonical projection

To this end, we

y~ of the space F.

.~:T~-+F.

The reduced differential extension J~F is covered by local fibering charts, which have fibre variables of the form (ya, y~), and hence lifts 7:F ~ J~F giving the reduced differential-geometric connection, are given with the help of the equations

y~'= THEOREM 3.

~ff~y~.

The relations Iif~nb

--.,. p i b n c

~Y ~=~+ ~ a ~ -

dr A b ~ i -- O ,

o

r~,,a~ k =

are necessary and sufficient conditions for ~-compatibility. Proof. It suffices to note that the composition of morphisms help of the equations y=

To~

is defined with the

A=kWblA~.b

=.z'* b

~k--r,,,-ab% ,

provided the differential extension p#(~) has the coordinate expression cAb ~k+Ab~, Z*b 'Ji ] "#k "~ " i

"-*b "~ "

THEOREM 4. All rz-connections which one can associate intrinsically with the nonholonomic differential-geometric structure b, satisfy the equation Fk~ - A= + po, in which p~ denotes the intrinsic tensor field of the structure. Proof. the law

We consider an arbitrary intrinsic rl-connection of the structure.

In this case

gjt l~,, ~ r icj c "= Jb.'.~ +fbZ~=>,~A~

of transformation of coefficients of the structure and the law d r # b_ dp c --b g~gt O, f~-f~g~g~ Fpd-r~=

of transformation of coefficients of the connection contain the identical contraction g ~ o . f ~ . Eliminating it, we get the relation "=j--c "=_ = b ~k c A~ F~--Ad-;%gd(A~ Fkb--AD,

by virtue of which the variables p=

a

__ ~=,~ l

~b

r'b

*ka l Aa

=

are the components of an intrinsic tensor field. The zero tensor field is intrinsic for an arbitrary nonholonomic differentlal-geometric structure. Consequently, there exists a class of intrinsic rl-connectlons, satisfying the equation

Connections of this class are characterized invarlantly with the help of the inclusion r1(rg=b by virtue of which they each have a formally transitive nonholonomic differentlal-geometric structure. In particular, they also have tangent structures b=T~(~) of formally transitive differential equations w c JIE. We recall that the differential equation ~ is called form-

116

ally transitive if and only if it has a surjective canonical projection ~ * E. The lifts F:E ~ of these projections are connections of the bundle ~:E * M. In local flberlng charts they are given with the help of the equation

~=r~(x,,/). Setting

rf~= - ~ r L we get the coefficients of t h e of the identities

corresponding Bliznikas rx-connection [I].

However, by virtue

~(x,, r, r;(~,, .~))~ o one has Ab d, r kb+ A , = O ,

and the canonical Bliznikas rz-connections are intrinsic connections of the tangentstructure

Tv(~).

Intrinsic Fx-connections let us use. the decompositions r l ~ = ~'~b ~ ~(/--k) b - r~k ~=,

found in the first part of the paper. V~

Setting

b = i As T Y ~ iAb + ' i p ~ c

-- ~ d ~ b ,

we get the mixed covariant derivatives of the tensor field A~k. THEOREM 5.

The equations

are necessary and sufficient conditions for ~-compatibility of the differential-geometric connections considered. Proof.

It suffices to use Theorem 3, the relation b *kc-- nc

and t o make the corresponding calculations considering the properties of the tensor fields ~b and ~7~b , constructed in the first part of the paper. THEOREM 6. Formally transitive nonholonomic differential-geometric structures are lintegrable if and only if they admit pairs of ~-compatible connections. Proof. It follows from the definition of l-integrability itself that one has this property of a nonholonomic structure if and only if there exists a Fi,2-connection of the form r1.~:b-+pe(bL One now constructs the corresponding reduced connection "f:F-~J~F with the help of a simple diagram chase in the following commutative diagram with exact columns: 0 b

0 .r,,, , p ~ (b) ~p# (0

r f r,. ,_, "ii.

1''e (~)

F ---Z-~--} J~F I

0

0

In fact, if we choose a vector f in the bundle F, then since the canonical projection is surjective, one can find an element ~eT~= with ~-image f. Setting "r

(.~ = (r* (~)or:,, ~) (r 117

we get a unique well-defined map y:F--+J~F, which the above-cited commutative diagram generates. That the construction is well-deflned is proved with the help of the following standard diagram chasing arguments. W e choose another vector ['eY~f , which also has ~-image f. Since ['-[eKcr~ and since the left column of the diagram is exact, one can find a vector ~ eb with i-image ~' - ~ with respect to the natural imbedding i:b-+TL The commutatlvity of the diagram leads to the equation

r,.~ (~') = r~., ( ~ + p* (i) or~., (~), and the exactness of its right column lets us get the identity we need

7'(f) =P* (?) or~. d~') =7(f), By virtue of the relations

the linear map

y:F-+ J~ F is a lift of the canonical protection

.~:J~F--+F

and is a ~-compatible connection.

Now we consider an arbitrary pair of ~-compatible connections, which are generated by the commutative diagram

It suffices for us to extend it to the above-cited commutative diagram with exact columns. To this end we take an arbitrary vector ~ of the structure b and we construct the element :=(P,,~~ of the bundle T[. By virtue of the identity ?oi=0 one has the equation p=(~)(:)=0 and one can find a vector ~ep~(b) , which will have p#(i)-image ~. Since the differential extension p#(i) is an imbedding of the corresponding modules, one can only choose the vector fi in one way. It will also satisfy the relation FI.~(~)= ~ which is needed. Theorem 6 is completely proved.

The intersection

3=bnC*|

U,

i s c a l l e d t h e symbol o f t h e nonholonomic d i f f e r e n t i a l - g e o m e t r i c notation it is generated by the vectors

structure

[1].

In coordinate

= ~ dx k | c),,, whose components satisfy the equation b

~k

--

"." 9

The tensor product ~ : @ ~ is the sheaf of germs of local deformations of intrinsic rl-connections of the form FI:T~'->D. Their infinitesimal operators dr=oo - I ' ~b db are s e c t i o n s

o f t h e module

b,

so i n t h i s

case t h e L i e s t r u c t u r e s

satisfy

[0L dr] ~ b It follows from this that the vertical curvature field R ~ o f the structure by an intrinsic Fl-connection is a section of the sheaf A2~v*|

b , generated

THEOREM 7, The horizontal curvature tensor fields R~:j of intrinsic F1-connections of nonholonomic differential-geometric Lie structures are sections of a sheaf of the form

C* | 118

C* |

Proof, In the first part of the paper we saw that the differential extensions nonholonomlc structures corresponding to rl.connections are generated by germs b

~

b

b

c

p#(r0

of

,,~kl~;a

However, from the inclusion r~ (T9 = b one gets the inclusion

pC (rx) (T") = p# (b), and the extension of the connection structure p~ (b) The equations

Fs=p~(r 0

is an intrinsic connection of the extended

A~ .~b- ~= ;b 0, ~kl+(Ol Ab T . ~ b u l j - . k ~ , , i "~b~, = v

of the extended structure let us get the equations Ay

which

since

~b

b ~o

T ~i

~n'b "~k

T

.rar %i

--

"ab

~kis

.F2(T)cp~(D) , must hold for the variables ,~=

~k ~-- a ?

<

r~)

From this the assertion of the theorem being proved follows

~ f ( ~[ r~ I b+ rdc [irk,d b) = O. THEOREM 8. Nonholonomic Lie differential-geometric structures are l-integrable if and only if for a suitable choice of intrinsic F1-connections one has nb

Proof.

--~ipb~c

9

It suffices to use the relation .~a

-- 0 - 1 DO ijb - - z.. .r~bU

and Theorems 5, 6, and 7. However, one should note that Theorem 8 has a sufficiently complicated algebraic character to occasionally make its practical application more difficult. Hence the following considerably simpler criterion for l-integrability is of interest. THEOREM 9, Nonholonomic Lie differential-structures tions with covariant differentiation

which admit intrinsic F~-connec-

V~:3--> ~* | are l-integrable. Proof. We consider a nonholonomic differential-geometric cited Fl-connection. Then by ~ - d i f f e r e n t i a t i n g the relation

A=k =b _ b ~k--

0

structure which has the above-

,

we get the equation ~b~#A~k 0 kV i b = which is an algebraic consequence of this relation. field k~, letting us use an identity of the form

Consequently,

there exists a tensor

~7 ~ A ~ k

7 ~b = X~, A~ k.

From this, setting

~ = ~,~ - ~,,

dJa-O, ~

_

we get the equation cited in Theorem 8.

One can get another approach to the problem of l-lntegrability of nonholonomlc differential-geometric structures by generalizing differential operator 119

of the form X = D # o Od | F 0 , D # rood ~# (U* | o,~ . THEOREM i0,

One has an exact sequence

|

p* (b) --"~ ~b _ x A ~ . Proof. 5.2 of [2].

|

We only note that the proof of this theorem is analogous to the proof of Theorem Everything necessary for this technique is also developed here.

In the expression of the differential operator X one uses the intrinsic Fl-connection of the nonholonomic structure b , which lets us use the decomposition

b ffi r~ ( ~ ) @ 3. Consequently, we can verify the surjectivity of the canonical projection steps on the first and second summands, respectively. I. Sur~ectivity on the Symbol

of the symbol

~

~ .

p#(b)-+b

in two

We take an arbitrary section

= ~ dx ~ | ~, and we construct the corresponding tensor field ~a b X (~) = ~ v %~ + Fba I, ~~kj) dx l ^ dx k | Oh,

~#(~*|

which we consider modulo the sheaf

Let

denote the covariant derivative obtained with the help of a nonholonomic symmetric affine connection.

By virtue of the equation

X(~) = ~* (V* D, the canonical projectio n considered covers the chosen section ~ if and only if we can construct a tensor field ~b , which satisfies the equations

~r ~= o, ~b (~r ~k + P:~k-- ~tk ~ Now let us assume that the projection 8~-differentiating the relation

.~,k: ~;) = O.

p#(b)"*'b

is surjective on the symbol

~.

Then

8~-differentiating the relation

A~k =b _ b %k--

n v~

we see that the condition ~k

must be a consequence of it.

k

~p

~

~o

bk

c

From this, setting #

~

c

~

b

~

ap

bk_

~

A~k

we get the first condition of Theorem 5. 2. Surjectivity on the Summand FI(TV). We take an arbitrary vector field ~=~a~a and we construct a section FI(~) of the nonholonomic differential-geometric structure b. The tensor field R~t~ ~ is the x-image of this section. By Theorem 7 it belongs to the sheaf ~* | , and by virtue of the identity

is the 6#-differential. Consequently, the canonical projection p#(b) -+b is always surjectire on the module FI(TV). We have obtained another Proof of Theorem 8.

LITERATURE CITED I.

120

R. V. Vosilyus, "Contravariant theory of differential extension in the model of a space with connection, " in: Problems of Geometry [in Russian], Vol. 14, Moscow (1983), pp. 101-176.

2. 3. 4.

R . V . Vosilyus, "Generalized linear structures and the geometry of submanifolds. I-III," Liet. Mat. Rinkinys, 17, No. 4, 31-82 (1977). H. Goldschmidt, "Existence theorems for analytic linear partial differential equations," Ann. Math., 86, 246-270 (1967). H. Goldschmidt, "Integrability criteria for systems of nonlinear partial differential equations," J. Diff. Geometry, !, 269-307 (1967).

GEOMETRIC PARAMETERS OF THE CHORD LENGTH DISTRIBUTION OF A CONVEX DOMAIN E. Geciauskas

UDC 519.212.3

We investigate the distribution function of the length of chord of a convex domain in the plane, when the intersections of the domain with random lines form a chord of random length. We assume that the measure of a set of lines is invariant with respect to translations and rotations. It was known from [i] that the distribution function of the length a of a chord of a convex domain in the plane can be represented by the expression I

(i)

P { . < ~ } = l - - z- f dG, K.G~O

where dG is the density of a set of lines in integral geometry, L is the parameter of the c o w e x domain K. The goal of the present paper is the clarification of the question, which geometric

p a r a m e t e r s o f the domain does the d i s t r i b u t i o n

f u n c t i o n depend on.

We set

L(x)=

.f d6.

(2)

K,G~O

THEOREM. 2n

P{~<x}=I-T

I

r

.[ [H(r

(3) '

0

for x _< d, ~,(x)

P{~<x}=I-

i

9

f [H('b.)+H"(,b)]ces~(x, 5)d5 ,

,

,

+ L1 [H (,.L)sin :~fx, ~)+H'(.,~)cos~(x ,'m*,(~) ,

Y/Jr

(x)

(4)

r (x)

for d _< x _< D, where H(~) is the support function of the convex domain K at the point at one end of the chord, a(x, ~b) is the angle between a chord of length x and the tangent to the contour curve of the domain K at this same point~ H' (~) and H''(#) are the first and second derivatives of the support function, d and D are the smallest and largest diameters of the domain. Proof. Since dG - dpdgo, where p is the distance from the line G to the origin, ~o is the direction of the perpendicular to the line G, one has

L(x)= f h(x, O a % where h ( x , ~) i s t h e d i s t a n c e from t h e o r i g i n i n the d i r e c t i o n

~ to a chord o f l e n g t h x.

Institute of Mathematics and Cybernetics, Academy of Sciences of the Lithuanian SSR. Translated from Litovskii Matematlcheskil Sbornik (Lietuvos Matematikos Rinklnys), Voi. 27, No. 2, pp. 255-257, April-June, 1987. Original article submitted September II, 1986.

0363-1672/87/2702-0121512.50

9

Plenum Publishing Corporation

121