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

TL;DR

This paper presents a novel proof of a classical topological result for orientable, compact surfaces using the Bochner technique, linking curvature and tangent vector fields.

Contribution

It introduces a new proof method based on the Bochner formula to establish the torus characterization among 2D surfaces.

Findings

If a 2D Riemannian manifold admits a non-trivial tangent vector field, its Gauss curvature is a divergence.

A continuous, zero-free tangent vector field implies the surface is a torus.

The proof leverages the Bochner formula, Whitney embedding theorem, and approximation techniques.

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.