Riemannian Geometry IV

Michaelmas 2014

The Epiphany 2015 webpage

Time and place:   Lectures: Tue 11:00, Wed 12:00, CM107
Problems classes:   Th 16:00, CM101, Weeks 4,6,8,10
Instructor: Pavel Tumarkin
e-mail: pavel dot tumarkin at durham dot ac dot uk
Office: CM110; Phone: 334-3085
Office hours: Tue 12:00 -- 13:00 and by appointment

Textbook:

The content of the course can also be found in any standard textbook on Riemannian Geometry, e.g.

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: 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; Hopf-Rinow theorem

    Handouts:

    Homeworks: There will be weekly homework assignments. Selected exercises are to be handed in on weeks 3, 5, 7, and 9