Curvature and Topology: Synge’s Theorem
In Problem 8.3.(7) it was shown that if M2 is a closed surface in 3 then its curvature K and its “genus” g are related by
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 ...