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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
    
thanks.but what i want is applications to prove some classical results –  Koushik Feb 6 '13 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 –  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

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