Paper 4, Section II, F

Geometry
Part IB, 2016

Let α(s)=(f(s),g(s))\alpha(s)=(f(s), g(s)) be a simple curve in R2\mathbb{R}^{2} parameterised by arc length with f(s)>0f(s)>0 for all ss, and consider the surface of revolution SS in R3\mathbb{R}^{3} defined by the parameterisation

σ(u,v)=(f(u)cosv,f(u)sinv,g(u))\sigma(u, v)=(f(u) \cos v, f(u) \sin v, g(u))

(a) Calculate the first and second fundamental forms for SS. Show that the Gaussian curvature of SS is given by

K=f(u)f(u)K=-\frac{f^{\prime \prime}(u)}{f(u)}

(b) Now take f(s)=coss+2,g(s)=sins,0s<2πf(s)=\cos s+2, g(s)=\sin s, 0 \leqslant s<2 \pi. What is the integral of the Gaussian curvature over the surface of revolution SS determined by ff and gg ?

[You may use the Gauss-Bonnet theorem without proof.]

(c) Now suppose SS has constant curvature K1K \equiv 1, and suppose there are two points P1,P2R3P_{1}, P_{2} \in \mathbb{R}^{3} such that S{P1,P2}S \cup\left\{P_{1}, P_{2}\right\} is a smooth closed embedded surface. Show that SS is a unit sphere, minus two antipodal points.

[Do not attempt to integrate an expression of the form 1C2sin2u\sqrt{1-C^{2} \sin ^{2} u} when C1C \neq 1. Study the behaviour of the surface at the largest and smallest possible values of uu.]