Lecturers: Dr. Mario Micallef
Term(s): Term 1
Commitment: 30 lectures
Assessment: Oral Exam
Prerequisites: A knowledge of manifolds, e.g. MA3H5 Manifolds is required. These topics will be covered rapidly in the first few lectures. A thorough knowledge of linear algebra, including bilinear forms, dual spaces, eigenvalues and eigenvectors is essential, as is a thorough knowledge of differentiation of functions of several variables, including the Chain Rule and Inverse and Implicit Function theorems. Familiarity with basic point set topology, including quotient/identification topology, will be assumed, as well as the statement of the theorem on the existence and uniqueness of solutions to ODEs and their smooth dependence on parameters, in particular on initial conditions.
Outline: The core of this course will be an introduction to Riemannian geometry - the study of Riemannian metrics on abstract manifolds. This is a classical subject, but is required knowledge for research in diverse areas of modern mathematics. We will try to present the material in order to prepare for the study of some of the other geometric structures one can put on manifolds.
- Review of basic notions on smooth manifolds; tensor fields.
- Riemannian metrics.
- Affine connections; Levi-Civita connection; parallel transport.
- Geodesics; exponential map; minimising properties of geodesics.
- The curvature tensor; sectional, Ricci and scalar curvatures.
- Training in making calculations: switching covariant derivatives; Bochner/Weitzenböck formula.
- Jacobi fields; geometric interpretation of curvature; second variation of length.
- Classical theorems in Riemannian Geometry: Bonnet-Myers, Hopf-Rinow and Cartan-Hadamard.
Lee, J. M.: Riemannian Manifolds: An Introduction to Curvature. Graduate Texts in Mathematics, 176. Springer-Verlag, 1997.
Gallot, S., Hulin, D., Lafontaine, J.: Riemannian geometry. Springer. 2nd edition (1993)
Jost, J.: Riemannian Geometry and Geometric Analysis 5th edition. Springer-Verlag, 2008
Petersen, P.: Riemannian Geometry Graduate Texts in Mathematics, 171. Springer-Verlag, 1998
Kobayashi, S., Nomizu, K.: Foundations of differential geometry.
do Carmo, M: Riemannian geometry. Birkhäuser, Boston, MA, 1992.