# Applications of Gauss-Bonnet theorem

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.

• en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem - but I don't know if MO is the place to be asking this. Feb 6, 2013 at 5:18
• thanks.but what i want is applications to prove some classical results Feb 6, 2013 at 8:14
• i don't think it is unsuitable to ask questions here specially when many questions have already been asked in a similiar vien Feb 6, 2013 at 10:06
• I don't think it makes sense to call Atiyah-Singer an application of Gauss-Bonnet (a generalization, sure). But if applications of Atiyah-Singer to "classical results" would be answers to this question, then the question is way too broad. If applications of only Gauss-Bonnet (and perhaps Gauss-Bonnet-Chern) are allowed, then there are probably a reasonable number of decent answers. For instance, two surfaces with the same constant curvature and the same genus necessarily have the same area. Feb 6, 2013 at 13:11
• Not an application but an operator theoretical interpretation: arxiv.org/abs/1302.0001 Feb 3, 2016 at 9:39

In particular, a hyperbolic manifold with $$\mid\chi(M)\mid=1$$ necessarily is the hyperbolic manifold of minimal volume in its dimension. Ratcliffe, Tschantz and Everitt have used this to find the hyperbolic manifolds of minimal volume in dimensions 4 and 6.
(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.)