course |
course_year |
question_number |
tags |
title |
year |
Geometry |
IB |
24 |
|
Paper 3, Section I, E |
2021 |
State the local Gauss-Bonnet theorem for geodesic triangles on a surface. Deduce the Gauss-Bonnet theorem for closed surfaces. [Existence of a geodesic triangulation can be assumed.]
Let $S_{r} \subset \mathbb{R}^{3}$ denote the sphere with radius $r$ centred at the origin. Show that the Gauss curvature of $S_{r}$ is $1 / r^{2}$. An octant is any of the eight regions in $S_{r}$ bounded by arcs of great circles arising from the planes $x=0, y=0, z=0$. Verify directly that the local Gauss-Bonnet theorem holds for an octant. [You may assume that the great circles on $S_{r}$ are geodesics.]