Mengxue Yang, Department of Pure Mathematics, University of Waterloo
“The Curvature of Higher Dimensional Manifolds, Part I”
How does a change of coordinates affect the Riemannian metric? When are two Riemannian manifolds locally isometric? Riemann has a famous counting argument that suggests the metric is determined by n(n−1) functions. In this talk we will see why this is true. In particular, we will 2 see that for any 2-dimensional subspace of the tangent space, we can find an invariant quadratic function on the space in any normal coordinate system. In an n-dimensional tangent space there are n 2-dimensional subspaces and hence we have Riemann’s claim. In preparation for 2 this goal, we will introduce the exponential map and the Riemann normal coordinates.
MC 5417