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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.

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

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.

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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.)

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