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
email: anna dot felikson at durham dot ac dot uk
Office: CM124; Phone: 3344158
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, LeviCivita 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, arclength parametrization; Riemannian manifolds as metric spaces. LeviCivita 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. HopfRinow 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 email)!!!
Homeworks:
There will be weekly sets of exercises; stared questions to hand in on Thurdays, weeks 3,5,7,9.  (+/ notation used for marking)
Handouts:
Typical exam questions:  see here
Fun: 
Hairy Ball Theorem in 1munute video
Who is who:

Riemann,

Hausdorff,

Jacobi,

Lie,

Leibniz,

Nash,

LeviCivita, 
Christoffel, 
Gauss, 
Hopf, 
Rinow. 
Bianchi. 