Riemannian Geometry
Prerequisites
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.
Programme
- 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
Educational aims
The purpose of the course is to exmploy analytical tools and differential calculus to study the geometry and topology of manifolds.
Bibliographical references
- M. Do Carmo: Riemannian Geometry.
- P. Petersen: Riemannian Geometry.
- M. Spivak: A comprehensive introduction to differentialgeometry