CHAPTER 12

Curvature and Topology: Synge’s Theorem

In Problem 8.3.(7) it was shown that if *M*^{2} is a closed surface in ^{3} then its curvature *K* and its “genus” *g* are related by

(12.1) |

This is the Gauss–Bonnet theorem. In particular, when *M*^{2} 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

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