Riemannian Geometry IV
The Epiphany 2016 webpage
Instructor: Anna Felikson
| Time and place:
||Lectures: || Tuesday 15:00, CM107|
| || ||Thursday 17:00, CM101|
|Problems classes:   || Tuesday 16:00, CM107, Weeks 4,6,8,10|
e-mail: anna dot felikson at durham dot ac dot uk
Office: CM124; Phone: 334-4158
Office hours: Monday 11:00 - 12:00 and by appointment
|The content of the course can also be found in any standard textbook on Riemannian Geometry, e.g.
- 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.
- 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); differential
- Week 4: Tangent bundle, vector fields, Lie bracket
- Week 5: Riemannian metric, models of a hyperbolic space. Isometries of Riemannian manifolds.
- Week 6: Lengths of curves, arc-length parametrization; Riemannian manifolds as metric spaces. Levi-Civita connection.
- Week 7: Christoffel symbols. Parallel transport.
- Week 8: Geodesics as solutions of ODE. Geodesics as distance minimizing curves, first variation formula of length.
- Week 9: Proof of first variation formula of length. Exponential map; Gauss Lemma.
- Week 10: Some corollaries of Gauss Lemma. Hopf-Rinow theorem.
If you have any questions you are very welcome to ask (during the lectures, after a lecture, during office hours, in any other convenient time or via e-mail)!!!
There will be weekly sets of exercises; stared questions to hand in on Thurdays, weeks 3,5,7,9. -- (+/- notation used for marking)
Typical exam questions: ---- see here
Hairy Ball Theorem in 1-munute video
Who is who: