I am reading Agricola & Friedrich's Global Analysis. On page 85 they prove this corollary of the Stokes' Theorem:
Let $\mathcal{M}$ be a compact, connected, oriented manifold without boundary and assume the function $f:\mathcal{M}\to\mathbb{R}$ satisfies at each point $x$ the condition $\Delta(f)(x)\ge 0$, where $\Delta(f)=div(grad(f))$ is the Laplacian of $f$. Then the function $f$ is constant.
According to this question, it also appears as Exercise 3.12 in Do Carmo's Riemannian Geometry.
The proof is easy by using Green's formulas. What puzzles me is that it is called "Hopf's Theorem", and I am unable to find any other reference that attributes this result to Hopf. Is it an obvious (not for me) corollary from one of the more famous Hopf's theorems? Is it known by any other name?
If you replace the compact manifold with a compact domain in $\mathbb R^n$ equipped with a Riemannian metric, the corresponding result is the Hopf Maximum Principle.
I'm not sure if Hopf proved the generalization to compact manifolds. Regardless, the main difficulty in generalizing the proof the maximum principle from the Euclidean Laplacian to the Riemannian Laplacian is dealing with the coefficients of the operator - the topology is not really a problem. Thus it still makes sense (to me, at least) to name this result after Hopf.