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As an application of the Bochner formula, we prove that if a $2$-dimensionalRiemannian manifold admits a non-trivial smooth tangent vector field $X$ thenits Gauss curvature is the divergence of a tangent vector field, constructedfrom $X$, defined on the open subset out the zeroes of $X$. Thanks to theWhitney embedding theorem and a standard approximation procedure, as aconsequence, we give a new proof of the following well-known fact: if on anorientable, connected and compact $2$-dimensional smooth manifold there existsa continuous tangent vector field with no zeroes, then the manifold isdiffeomorphic (or equivalently homeomorphic) to a torus.

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