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Fibre integral in regular Lie algebroids. (English) Zbl 0959.58026

Szenthe, J. (ed.), New developments in differential geometry, Budapest 1996. Proceedings of the conference, Budapest, Hungary, July 27-30, 1996. Dordrecht: Kluwer Academic Publishers. 173-202 (1999).
The operator of fibre integral \(\int\) in oriented bundles is used, for instance, to define the Euler class of a sphere bundle and the index of a vector field at an isolated singularity.
In the paper under review, the author adapts the idea of this fibre integral to regular Lie algebroids, defining an operator of integration over the adjoint bundle of Lie algebras. It is based on the result expressing the fibre integral of right-invariant differential forms on a principal bundle via some substitution operator.
Using this operator and its properties, the study of some directions of the theory of real cohomologies of regular Lie algebroids is initiated. The author investigates the commutation of the integral operator with exterior derivatives, substitution operators and Lie derivatives, and the stability of the kernel of \(\int_A\). Transitive (resp. nontransitive) examples of Lie algebroids of rank \(1\) which are invariantly oriented and non-invariantly oriented respectively are given.
In a forthcoming paper, as applications of the introduced operator, the author will study the problem of Poincaré duality.
For the entire collection see [Zbl 0903.00045].

MSC:

58H05 Pseudogroups and differentiable groupoids
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