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

• Volume: 56, Issue: 2, page 359-376
• ISSN: 0011-4642

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Abstract

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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 $\mathrm{s}l\left(2,ℝ\right)$ or $so\left(3\right)$ 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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Kubarski, Jan. "The Euler-Poincaré-Hopf theorem for flat connections in some transitive Lie algebroids." Czechoslovak Mathematical Journal 56.2 (2006): 359-376. <http://eudml.org/doc/31034>.

@article{Kubarski2006,
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 $\mathbb \{R\}$ or $\mathop \{\mathrm \{s\}l\}(2,\mathbb \{R\})$ 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.},
author = {Kubarski, Jan},
journal = {Czechoslovak Mathematical Journal},
keywords = {Lie algebroid; Euler class; index theorem; integration over the fibre; flat connection with singularitity; Euler class; index theorem; integration over the fibre; flat connection with singularitity},
language = {eng},
number = {2},
pages = {359-376},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The Euler-Poincaré-Hopf theorem for flat connections in some transitive Lie algebroids},
url = {http://eudml.org/doc/31034},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Kubarski, Jan
TI - The Euler-Poincaré-Hopf theorem for flat connections in some transitive Lie algebroids
JO - Czechoslovak Mathematical Journal
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 2
SP - 359
EP - 376
AB - 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 $\mathop {\mathrm {s}l}(2,\mathbb {R})$ 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.
LA - eng
KW - Lie algebroid; Euler class; index theorem; integration over the fibre; flat connection with singularitity; Euler class; index theorem; integration over the fibre; flat connection with singularitity
UR - http://eudml.org/doc/31034
ER -

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