Can anyone point me toward a clever, intuitive proof (or almost-proof) of the Gauss-Bonnet theorem? It seems too cool to not have beautifully simple way of seeing it.
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Gauss-Bonnet theorem: the integral of the Gaussian curvature over a surface depends only on the number of holes in that surface.pic.twitter.com/fk3lI8nuLa
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Kind of magic, right? The full version is \int K dA = 4 \pi (1 - g), where g is the genus.
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There's a solid-state equivalent of this theorem that you derive by thinking about the Berry phase acquired during a closed path in momentum space. I suspect there is a similar route here that involves geometric phase along some path on the surface.
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