  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.
