MA6C0 Differential Geometry
Lecturers: Max Stolarski
Term(s): Term 1
Commitment: 30 lectures
Assessment: Oral Exam
Formal registration prerequisites: None
Assumed knowledge:
MA3H5 Manifolds: Key topics from MA3H5 will be covered rapidly in the first few lectures but you should do this module in Term 2 for a thorough discussion of them
- Bilinear forms
- Eigenvalues and Eigenvectors
- Differentiation of functions of several variables, including the Chain Rule
- Inverse and Implicit Function theorems
MA260 Norms, Metrics and Topologies OR MA222 Metric Spaces:
- Basic point set topology
- Existence and uniqueness of solutions to ODEs and their smooth dependence on parameters and initial conditions
Useful background:
Synergies:
Summary:
- 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
Books:
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