Suppose we have a closed, orientable, smooth surface $\Sigma$ immersed smoothly in $\mathbb R^n$ via $f:\Sigma \rightarrow \mathbb R^n$. Impose a Riemannian structure on $\Sigma$ by taking $g_{ij} = \partial_if\cdot\partial_jf$, the metric induced on $\Sigma$ by the immersion $f$. The inner product here is just the usual inner product from $\mathbb R^n$.
The mean curvature vector is $ \vec H = \Delta f, $ where $\Delta$ is the Laplace-Beltrami operator on $(\Sigma,g)$.
Consider the integral of the mean curvature vector over the surface $\Sigma$: $ \int_\Sigma \vec H\ d\mu. $ It seems rather plausible that this ought to be zero in the case where $\Sigma$ is closed, embedded, and has only one codimension. Is this known? Is it easy to prove?
If it is not zero in the generality above, as a surface immersed in $\mathbb R^n$, is it equal to some expression involving topological information of $\Sigma$?