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The Euler-Poincaré-Hopf theorem for flat connections in some transitive Lie algebroids

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Abstract

This paper is a continuation of [19], [21], [22]. We study flat connections with isolated singularities in some transitive Lie algebroids for which either ℝ or sl(2, ℝ) or so(3) are isotropy Lie algebras. Under the assumption that the dimension of the isotropy Lie algebra is equal to n + 1, where n is the dimension of the base manifold, we assign to any such isolated singularity a real number called an index. For ℝ-Lie algebroids, this index cannot be an integer. We prove the index theorem (the Euler-Poincaré-Hopf theorem for flat connections) saying that the index sum is independent of the choice of a connection. Multiplying this index sum by the orientation class of M, we get the Euler class of this Lie algebroid. Some integral formulae for indices are given.

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  1. Institute of Mathematics, Technical University of Łódź, ul. Wólczańska 215, PL-93-005, Łódź, Poland

    Jan Kubarski

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Kubarski, J. The Euler-Poincaré-Hopf theorem for flat connections in some transitive Lie algebroids. Czech Math J 56, 359–376 (2006). https://doi.org/10.1007/s10587-006-0023-7

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  • DOI: https://doi.org/10.1007/s10587-006-0023-7

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