Riemannian Geometry IV

Michaelmas 2013

The Epiphany term webpage

Time and place: Tuesday and Friday, 12:00 CM107
Instructor: Anna Felikson
e-mail: anna dot felikson at durham dot ac dot uk
Office: CM 124; Phone: 334-4158
Office hours: Friday 13:00-14:00


Further reading:

  • J.Lee, Riemannian Manifolds, An Introduction to curvature, Springer (1997)
  • M.P. Do Carmo, Riemannian Geometry, Birkhäuser (1992)
  • F.Morgan, Riemannian Geometry, Jones and Bartlett Publishers, (1998)
  • T. Sakai, Riemannian Geometry, Translations of Mathematical Monographs 149, AMS (1996).
  • S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, Springer (2004),
  • Preliminary course content (subject to change): smooth manifolds, tangent spaces, vector fields, Riemannian metric, examples of Riemannian manifolds, Levi-Civita connection, parallelism, geodesics.

    Schedule: --------- red: most important notions, blue: most important statements

  • Week 1: Smooth manifolds: definition and examples; smooth manifolds via Implicit Function Theorem.
  • Week 2: Tangent space (derivations, directional derivatives, equivalence of definitions). Examples of tangent space.
  • Week 3: Differential as a map of tangent spaces. Tangent bundle, vector fields .
  • Week 4: Vector fields: Lie bracket . The Hairy Ball Theorem (without proof). Problems class.
  • Week 5: Riemannian metric, length of a curve. Examples: three models of hyperbolic space. Isometries of Riemannian manifolds. Arc-length parametrization of curves. Geodesics.
  • Week 6: Levi-Civita connection. Christoffel symbols.
  • Week 7: Parallel transport. Problems class.
  • Week 8: Geodesics: Geodesics as solutions to ODE, Geodesics as distance-minimizing curves, First Variation Formula of Length.
  • Week 9: Exponential map, Gauss Lemma, some corollaries.
  • Week 10: Hopf-Rinow theorem. Integration on Riemannian manifolds.

  • If you have any questions you are very welcome to ask (during the lectures, after a lecture, during office hours, in any other convinient time or via e-mail)!!!


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

    Typical problems: ---- see here

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