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

Title:

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

Creator:

Kubarski, Jan

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:e36e4068-c093-4030-9f12-fd73c370867e
uuid:e36e4068-c093-4030-9f12-fd73c370867e

Subject:

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

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

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. Similar articles:

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

359-376

Source:

Czechoslovak Mathematical Journal | 2006 Volume:56 | Number:2

Harvested from:

CDK

Metadata only:

false