THE STRUCTURE INVARIANTS OF A TRANSVERSE G-STRUCTURE

IVAN BELKO

Abstract

The problems of equivalence and integrability of G-structure were considered by many authors. Ch.Ehresmann, E.Cartan, S.Chern, V.Guillemin, D.Bernard and many others studied this problems. The solution of the equivalence problem is based on construction of structure invariants and cohomology classes correspondent. It is possible to pick out two main approaches to construction of the structure tensor, wich are different in their methods. We can call one of them geometrical, it uses the fundamental form and the forsion of the connection on principal bundle of the frames. Another approach is caracterised by use of differential operators and their Spencer cohomologies. The Ngo van Que paper [1] is an example of this.

Our goal is the construction of a structure invariants for a kth order foliated transversal structure on foliated manifold. Geometrical construction of such tensor for the first- order transversal structure is given in R.Wolak paper [2].

The particularity of our approach is in using of foliated Lie algebgoid anddevelopment of Spencer cohomology method to transversal differential operators. For a foliated manifold (B, F) the transversal k-jets of local diffeomorfisms forms the foliated Lie groupoid Π^{k}(B). His Lie algebroid can be identified with the foliated Lie algebroid Ĵ^{k}(T B) of transversal k-jets of foliated vector filds. In this Lie algebroid the partial plate connection

τ : T F → Ĵ^{k}(T B) is defined by canonical way. The natural projection by Ker τ determines the exact sequance of foliated vector bundles

0 → Ĵ_{o}^{k}(T B) → Ĵ^{k}(T rB) → T rB → 0

The adjoint Lie algebra bundle Ĵ_{o}^{k}(T B) = Ĵ_{o}^{k}(T rB) plays an important role

in construction of a structure invariants.

A regular infinitesimal transversal kth order structure on a foliated manifold is a regular section of the bundle associated to Lie groupoid Π^{k}(B). Such a bundle is foliated and permits to distinguish the foliated sections and structures.

A regular structure S defines a Lie subgroupoid Π^{k}(S) in Π^{k}(B). His Lie subalgebroid is a vector bundle and can be considered as a system of linear differential equations

E^{k}(S) ⊂ Ĵ^{k}(T B).

On the base of the exacte sequance of vector bundles is built the transversal Spencer operator

D̂ : Γ(Ĵ^{k}T B) → Γ(Ĵ^{k−1}T B ⊗ T r^{∗}B).

This operator defines a transversal cohomologycal sequance

δ : T rB ⊗ S^{k+1}(T r^{∗}B) ⊗ Λ^{p}(T r^{∗}B) → T rB ⊗ S^{k}(T r^{∗}B) ⊗ Λ^{p+1}(T r^{∗}B).

In the terms of these cohomologies is defined the type and the degree of a structure S. For a connection in Lie algebroid Ĵ^{k}(T B) also in its Lie subalgebroid E^{k}(S) can be defined the torsion. The class of transversal δcohmology correspondent is an obstacle to the integrability of a foliated firstorder G-structure S.

REFERENCES

[1 ] Ngo van QUE , Du prolongement des espaces fibrés et des structures infinitésimales, Ann. Inst. Fourier (Grenoble), 17, 1 (1967), 157-223.

[2 ] R.A.Wolak, The structure tensor of a transverse G-structure on a foliated manifold, Bollettino U.M.I. (7) 4-A (1990), 1-15.

Belarusian State Economics University High Mathem. Department 26 Partizansky avenue, 220070 Minsk BELARUS e-mail: hmd@bseu.by, niipulm@bcsmi.minsk.by