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Czechoslovak Mathematical Journal

, Volume 56, Issue 2, pp 359–376 | Cite as

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

  • Jan Kubarski
Article
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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.

Keywords

Lie algebroid Euler class index theorem integration over the fibre flat connection with singularitity 

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Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2006

Authors and Affiliations

  • Jan Kubarski
    • 1
  1. 1.Institute of MathematicsTechnical University of ŁódźŁódźPoland

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