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Riemannian Geometry IV

Michaelmas 2013

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)!!! **
__Homeworks:__

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:__
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Hairy Ball Theorem in 1-munute video