The Gauss-Bonnet theorem for vector bundles

Details The Gauss-Bonnet theorem for vector bundles The Gauss-Bonnet theorem for vector bundles

2007-02-06

arxiv.org/abs/math/0702162v1

Journal of Geometry 85, no. 1-2, 15-21, 2006

Mathematics

Differential Geometry

Scientific paper

We give a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle $E$ of even rank over a closed compact orientable manifold $M$. This theorem reduces to the classical Gauss-Bonnet-Chern theorem in the special case when $M$ is a Riemannian manifold and $E$ is the tangent bundle of $M$ endowed with the Levi-Civita connection. The proof is based on an explicit geometric construction of the Thom class for 2-plane bundles.

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