Bott’s vanishing theorem for regular Lie algebroids

Abstract

Abstract. Differential geometry has discovered many objects which determine Lie algebroids playing a role analogous to that of Lie algebras for Lie groups. For example:-differential groupoids, -principal bundles, -vector bundles, -actions of Lie groups on manifolds, -transversally complete foliations, -nonclosed Lie subgroups, -Poisson manifolds, -some complete closed pseudogroups. We carry over the idea of Bott's Vanishing Theorem to regular Lie algebroids (using the Chern-Weil homomorphism of transitive Lie algebroids investigated by the author) and, next, apply it to new situations which are not described by the classical version, for example, to the theory of transversally complete foliations and nonclosed Lie subgroups in order to obtain some topological obstructions for the existence of involutive distributions and Lie subalgebras of some types (respectively).

Anchored vector bundles and Lie algebroids

“…, where X ′ and Y ′ have (f := f k )-decompositions (8) and (9). In the same way as in the case k = 1 it follows that f k is an anchored vector bundle morphism.…”

“…In the same way, if k ∈ IN , p ∈ IN * and X ′ , Y ′ ∈ θ ′(k) , where k = max(deg X ′ , deg Y ′ ) have (f := f k )-decompositions (8) and 9respectively, then…”

“…A morphism of almost Lie structures is a morphism of anchored vector bundles, such that (∀)X ′ , Y ′ ∈ Γ(θ ′ ) which allow the decompositions (8) and…”

Atiyah and Todd classes of regular Lie algebroids

“…known as the Atiyah sequence of the regular Lie algebroid A ( [11]). The differential D ∇ A (3.5) can be simplified if we choose a special linear connection on the regular Lie algebroid A.…”

“…of Lie algebroids over M (cf. [11]). The main purpose of this paper is to investigate whether Atiyah classes of dg manifolds arising from regular Lie algebroids respect the above Atiyah sequence.…”