Paper 4, Section II, F
Let be a simple curve in parameterised by arc length with for all , and consider the surface of revolution in defined by the parameterisation
(a) Calculate the first and second fundamental forms for . Show that the Gaussian curvature of is given by
(b) Now take . What is the integral of the Gaussian curvature over the surface of revolution determined by and ?
[You may use the Gauss-Bonnet theorem without proof.]
(c) Now suppose has constant curvature , and suppose there are two points such that is a smooth closed embedded surface. Show that is a unit sphere, minus two antipodal points.
[Do not attempt to integrate an expression of the form when . Study the behaviour of the surface at the largest and smallest possible values of .]