Riemannian Geometry IV
Michaelmas 2019
The Epiphany 2020 webpage
| The content of the course can also be found in any standard textbook on Riemannian Geometry, e.g. |
- J. Lee, Riemannian manifolds: an introduction to curvature.
- M.P. Do Carmo, Riemannian Geometry.
- F. Morgan, Riemannian Geometry.
- T. Sakai, Riemannian Geometry.
- S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry.
- P. Petersen, Riemannian Geometry.
Preliminary course content (subject to change):
smooth manifolds, tangent spaces, vector fields, Riemannian metric, examples of Riemannian manifolds, Levi-Civita connection, parallelism, geodesics.
Schedule:
Week 1: Smooth manifolds: definition and examples
Week 2: Smooth manifolds via Implicit Function Theorem; tangent space and tangent vectors (derivations, directional derivatives)
Week 3: Tangent space and tangent vectors (equivalence of definitions, examples)
Week 4: Differential as a linear map of tangent spaces; tangent bundle, vector fields
Week 5: Lie bracket of vector fields; Riemannian metric, models of a hyperbolic space
Week 6: Isometries of Riemannian manifolds; lengths of curves, arc-length parametrization; Riemannian manifolds as metric spaces
Week 7: Levi-Civita connection; Christoffel symbols
Week 8: Parallel transport; geodesics as solutions of ODE
Week 9: Geodesics as distance minimizing curves,
first variation formula of length; exponential map
Week 10: Exponential map; Gauss Lemma, corollaries
Handouts:
Homeworks: There will be weekly homework assignments. Selected exercises are to be handed in on weeks 3, 5, 7, and 9
Who is who:
Riemann,
Hausdorff,
Jacobi,
Lie,
Leibniz,
Levi-Civita,
Christoffel,
Gauss,
Hopf,
Rinow,
Bianchi.