Riemannian Geometry IV

Michaelmas 2015

The Epiphany 2016 webpage

Time and place:	Lectures:	Tuesday 15:00, CM107
		Thursday 17:00, CM101
	Problems classes:	Tuesday 16:00, CM107, Weeks 4,6,8,10

Instructor: Anna Felikson
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

Textbooks:

S. Gudmundsson, An Introduction to Riemannian Geometry, Lecture Notes, Lund University (2015).
J.Lee, Riemannian Manifolds, An Introduction to curvature, Springer (1997)
M.P. Do Carmo, Riemannian Geometry, Birkhäuser (1992)

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.

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); 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)!!!

Homeworks:

There will be weekly sets of exercises; stared questions to hand in on Thurdays, weeks 3,5,7,9. -- (+/- notation used for marking)

Week 1: ----s-4 Exercises ---- Hints ----
Week 2: ----s-4 Exercises ---- Hints ----
Week 3: ----s-4 Exercises ---- Hints ----
Week 4: ----s-4 Exercises ---- Hints ----
Week 5: ----s-4 Exercises ---- Hints ----
Week 6: ----s-4 Exercises ---- Hints ----
Week 7: ----s-4 Exercises ---- Hints ----
Week 8: ----s-4 Exercises ---- Hints ----
Week 9: ----s-4 Exercises ---- Hints ----
Week 10: -----.. Exercises ---- Hints ----
Christmas problems ---- ----

Handouts:

Typical exam questions: ---- see here

Fun: ---- Hairy Ball Theorem in 1-munute video

Who is who: ---- Riemann, ---- Hausdorff, ---- Jacobi, ---- Lie, ---- Leibniz, ---- Nash, ---- Levi-Civita, ---- Christoffel, ---- Gauss, ---- Hopf, ---- Rinow. ---- Bianchi. ----