Riemannian Geometry IV
The Epiphany term webpage
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)
Typical problems: ---- see here
Hairy Ball Theorem in 1-munute video