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.

waytoo 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