Riemannian Geometry IV
Epiphany 2013
| 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.
| Further (recommended) reading: |
Preliminary course content (subject to change):
Riemannian curvature tensor, Ricci curvature, sectional curvature and scalar curvature; manifolds of positive curvature and Bonnet-Myers theorem; Jacobi fields and conjugate points, manifolds of nonpositive curvature and Cartan-Hadamard theorem; comparison theorems.
Schedule:
Week 11: Overview of the first half of the course (manifolds, tangent vectors, tangent bundles, vector fields, metric)
Week 12: Overview of the first half of the course (affine connections, Levi-Civita connection, covariant derivative, geodesics, Hopf-Rinow theorem, Riemann curvature tensor, sectional curvature)
Week 13: Ricci and scalar curvature. Variations of curves, the second variational formula of length, Bonnet-Myers theorem
Week 14: Proof of Bonnet-Myers theorem. Problems class
Week 15: Applications of Bonnet-Myers theorem. Symmetry Lemma, proof of the second variational formula of length
Week 16: Proof of the second variational formula of length. Jacobi fields
Week 17: Jacobi fields and conjugate points; orthogonal Jacobi fields
Week 18: Conjugate points and minimizing geodesics. Problems class
Week 19: Cartan-Hadamard theorem. Spaces of constant curvature, comparison triangles, theorem of Alexandrov-Toponogov
Homeworks: There will be weekly homework assignments starting from week 12.