Period of duration of course
Basic knowledge of differentiable manifolds, vector andtensorial fields,differential forms, integration on manifolds.Suggested for Master students, or for Ph.D. students if they have not attended related courses.
- Riemannian Metrics
- Affine connections
- Riemannian and Ricci curvatures
- Parallel transport
- Geodesics and exponential map
- First and second variation of lenght
- Jacobi fields and conjugate points
- Isometric immersions
- Hopf-Rinow and Hadamard theorems
- Constant curvature spaces
- Bonnet-Myers theorem
- Negative curvature spaces
Time permitting: Sphere Theorems
The purpose of the course is to exmploy analytical tools and differential calculus to study the geometry and topology of manifolds.
- M. Do Carmo: Riemannian Geometry.
- P. Petersen: Riemannian Geometry.
- M. Spivak: A comprehensive introduction to differentialgeometry