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arXiv · 1911.08821

A new proof of a classical result on the topology of orientable connected and compact surfaces by means of the Bochner technique

J. M. Almira·A. Romero
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Abstract

As an application of the Bochner formula, we prove that if a $2$-dimensional Riemannian manifold admits a non-trivial smooth tangent vector field $X$ then its Gauss curvature is the divergence of a tangent vector field, constructed from $X$, defined on the open subset out the zeroes of $X$. Thanks to the Whitney embedding theorem and a standard approximation procedure, as a consequence, we give a new proof of the following well-known fact: if on an orientable, connected and compact $2$-dimensional smooth manifold there exists a continuous tangent vector field with no zeroes, then the manifold is diffeomorphic (or equivalently homeomorphic) to a torus.

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J. M. Almira, A. Romero. 2019-11-20. A new proof of a classical result on the topology of orientable connected and compact surfaces by means of the Bochner technique. https://arxiv.org/abs/1911.08821

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