Analysis seminar: Reginald Anderson (CMC)

Title: Review of differential geometry

Abstract: 1. Given the embedding of a sphere of radius rho centered at the origin of \R^3 from spherical coordinates, what is the pullback of the flat metric in \R^3? i.e., what is the “round metric” on the 2-sphere of radius rho? 2. If we impose a complex structure on S^2 via identifying charts in \R^2 with \C, what does the round metric on the unit sphere look like in a complex coordinate?

If time allows, we can also cover the following: 3. Christoffel symbols describe the (torsion free, Levi-Civita) connection on the (tangent bundle of a) Riemannian manifold. These are described in terms of the metric via raising and lowering indices. 4. Geodesics record the notion of “straight lines” in a Riemannian manifold. What does it mean for velocity vectors to be transported in parallel to a path, in terms of its parametrization and Christoffel symbols?