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Keywords:
Lie algebroid; Euler class; index theorem; integration over the fibre; flat connection with singularitity
Summary:
This paper is a continuation of [19], [21], [22]. We study flat connections with isolated singularities in some transitive Lie algebroids for which either $\mathbb{R}$ or $\mathop {\mathrm sl}(2,\mathbb{R})$ or $\operatorname{so} (3)$ 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 $\mathbb{R}$-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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