Abstract.
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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Dedicated to the memory of Philip Bell
Research partially supported by NSF grant DMS-9703852.
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Bell, D. The Gauss-Bonnet theorem for vector bundles. J. geom. 85, 15–21 (2006). https://doi.org/10.1007/s00022-006-0037-1
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DOI: https://doi.org/10.1007/s00022-006-0037-1