The fundamental form on a Lie groupoid of diffeomorphisms
by
Ivan Belko
Belarusian State Economic University

The Ehresmann method is an effective method of study of geometrical structures on differentiable manifolds. It is based on the theory of jets and Lie groupoids and it is the development of E. Cartan ideas. The Lie groupoid \Pi^k(B) of k-jets of local diffeomorphisms of differentiable manifold B is essential in the Ehresmann method. This Lie groupoid admits itself many special structures. We mark the following structures among the last ones:

(i) the canonical morphisms of Lie groupoids

\pikk-1:\Pik(B) --> \Pik-1(B);

(ii) the representation of Lie algebroid A\Pi^k(B) as a Lie algebroid J^kTB of k-jets of vector fields on B;

(iii) the truncated bracket A\Pi^k(B) \LambdaA\Pi^k(B) --> A\Pi^k-1(B), which is a vector bundle morphism;

(iv) the representation of Lie groupoid \Pi^k(B) as a Lie groupoid of a vector bundle isomorphisms preserving the truncated bracket;

(v) the fundamental form on \Pi^k(B) with values in the Lie algebroid A\Pi^k-1(B).

The called structures are generalizations of the similar structures on the frame bundles of higher order, which has been studying by V.Guillemin and S. Sternberg, P. Liebermann, P. Molino, Ngo van Que, D. Alekseevsky and P.Michor and others.

Our goal is an exposition of the basic theory of G-structures with an emphasis on Ehresmann method. The Lie groupoid \Pi^k(B) consists of k-jets of local diffeomorphism of B. The Lie groupoid \Pi¹(B) is a Lie groupoid of linear isomorphisms of the tangent bundle TB. The maps \alpha and \beta are the source and the target maps. For any x in B \alpha-leaf \alpha^-1(x) is the principal bundle \Pi^k(B)_x(B, \beta, G^x_x) of higher order frames. The right translation of \Pi^k(B) is an isomorphism of principale bundles. For any diffeomorphism j of B the map j^kj defines an admissible section for \alpha-projection. This section defines itself a left translation j^k of Lie groupoid \Pi^k(B). Let A\Pi^k(B) be a Lie algebroid of \Pi^k(B). This Lie algebroid is isomorphic to Lie algebroid J^kTB of k-jets of vector fields. The section j^kX, X in \GammaTB defines a vector field X^(k) in \GammaA\Pi^k(B) with a current

exptX(k)(jkx\psi)=jkx(exptX o \psi).

The truncated bracket on J^kTB is defined in such a manner. The Lie bracket of Lie algebroid

\GammaJk-1TB\Lambda\GammaJk-1TB --> \GammaJk-1TB

is on operator differential when any of its arguments is fixed. That's why it defines a vector bundles morphism

JkTB\Lambda JkTB --> Jk-1TB.

This morphism becomes the truncated bracket of Lie algebroid A\Pi^k(B). The truncated bracket is very important for the next theorem. The Lie groupoid \Pi^k(B) is isomorphed to Lie groupoid of vector bundle isomorphisms of A\Pi^k-1(B), which conserve the truncated bracket. There fore any element u in \Pi^k(B)^y_x can be considered as linear isomorphism

u:A\Pi(k-1)(B)x --> A\Pi(k-1)(B)y.

The fundamental form \vartheta on \Pi^k(B) is an \alpha-vertical form with values in the Lie algebroid A\Pi^(k-1)(B). Let p in T^\alpha_u\Pi^k(B) is an \alpha-vertical tangent vector. Then

\varthetau(p)=u-1 o (\pikk-1)*(p).

The form \vartheta characterizes the continuations of basic diffeomorphisms. Let \psi be a local diffeomorphism of \Pi^k(B), which conserves the \alpha-leaves and the fundamental form. Then \psi coincides with a local left translation of \Pi^k(B).

Date received: April 26, 2003