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The Geometry of Physics by Theodore Frankel

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CHAPTER 12

Curvature and Topology: Synge’s Theorem

image

 

In Problem 8.3.(7) it was shown that if M2 is a closed surface in image3 then its curvature K and its “genus” g are related by

image

(12.1)

This is the Gauss–Bonnet theorem. In particular, when M2 is a (perhaps) distorted torus (i.e., a surface of genus 1), then (2π)–1 M KdS = 0. Thus it is not possible to embed the torus in 3 in such a way that its Gauss curvature is everywhere positive. This ...

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