Riemannian Geometry IV
Epiphany 2012
| Textbooks: | The content of the course can be found in any standard textbook on Riemannian Geometry, e.g. |
- M. Do Carmo, Riemannian Geometry.
- S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry.
- J. Lee, Riemannian manifolds: an introduction to curvature.
- P. Petersen, Riemannian Geometry.
Preliminary course content (subject to change):
Geodesics as length minimizing curves, exponential map, the Gauss lemma; geodesically complete Riemannian manifolds and Hopf-Rinow theorem; Riemannian curvature tensor, Ricci curvature, sectional curvature and scalar curvature; manifolds of positive curvature and Bonnet-Myers theorem; manifolds of nonpositive curvature and Hadamard-Cartan theorem.
Schedule:
Week 11: Overview of the first half of the course; variations of curves, length and energy, symmetry lemma
Week 12: The first variational formula of length, geodesics as critical points of length functional; exponential map, Gauss lemma
Week 13: Proof of Gauss lemma, geodesic balls; geodesically complete metric spaces, Hopf-Rinow Theorem
Week 14: Riemann curvature tensor and its properties; sectional curvature
Week 15: Sectional curvature: examples; Ricci and scalar curvature
Week 16: The second variational formula of length, Bonnet-Myers theorem
Week 17: Jacobi fields and conjugate points
Week 18: Orthogonal Jacobi fields; theorem of Hadamard-Cartan
Week 19: The sphere theorem, spaces of constant curvature, comparison triangles, theorem of Alexandrov-Toponogov
Homeworks: There will be weekly homework assignments starting from week 12.