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.

  • $\begingroup$ en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem - but I don't know if MO is the place to be asking this. $\endgroup$
    – David Roberts
    Feb 6, 2013 at 5:18
  • 1
    $\begingroup$ thanks.but what i want is applications to prove some classical results $\endgroup$
    – Koushik
    Feb 6, 2013 at 8:14
  • 2
    $\begingroup$ i don't think it is unsuitable to ask questions here specially when many questions have already been asked in a similiar vien $\endgroup$
    – Koushik
    Feb 6, 2013 at 10:06
  • 10
    $\begingroup$ 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. $\endgroup$ Feb 6, 2013 at 13:11
  • $\begingroup$ Not an application but an operator theoretical interpretation: arxiv.org/abs/1302.0001 $\endgroup$ Feb 3, 2016 at 9:39

2 Answers 2


Chern-Gauß-Bonnet implies that the volume of a hyperbolic manifold is a constant multiple of its Euler characteristics, with the constant factor depending on dimension only.

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.


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.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.