TITLE: Applications of Gauss-Bonnet theorem
QUESTION [8 upvotes]: In wikipedia,I was pretty amazed to find a proof of fundamental theorem of algebra
using Gauss Bonnet theorem.
I think given how central it is to mathematics with its far reaching generalizations like Riemann-Roch Theorem and more,I am wondering if there are more.I would also be happy to see striking applications of its generalizations.

REPLY [3 votes]: Shameless plug: in this note I showed that a two-dimensional analogue of the positive energy theorem follows essentially trivially from Gauss-Bonnet. 
(Remark: positive energy theorems are two-dimensions is not new. But previously published results assume that spatial sections are diffeomorphic to $\mathbb{R}^2$. The main contribution above is that this topological assumption can be removed as it is a consequence of the other assumptions + Gauss-Bonnet.)