Topics: Manifolds, vector bundles, embeddings and submersions, integral curves and flows, basic Lie groups, differential forms and integration, Riemannian metrics, geodesics, connections, curvature.
Course goals: Learn the modern language of smooth manifolds; become comfortable performing calculations in Riemannian geometry in many concrete examples; understand how the local invariants of a Riemannian manifold constrain its global topology.
Recommended prerequisites: Analysis I and II (MA1, MA2) and Linear Algebra I and II (MA4, MA5)
The second final exam is a written exam held on Friday, September 29 from 9:15 - 10:45 in room SR4 in the Mathematikon. The first final exam was a written exam held on Friday, July 28.
Exercises are due on Friday by the beginning of class. They can be submitted on MaMpf. They should be sumbitted in groups of 3-5 people. Please include the name of every group member on the uploaded file. Write clearly in large letters in English.
| Exersise sheet | Due date |
|---|---|
| Exercise sheet 1 | Friday, 28.04 |
| Exercise sheet 2 | Friday, 05.05 |
| Week 3: no exercises | |
| Exercise sheet 3 | Friday, 19.05 |
| Exercise sheet 4 | Friday, 26.05 |
| Week 6: no exercises | |
| Exercise sheet 5 | Friday, 09.06 |
| Exercise sheet 6 | Friday, 16.06 |
| Exercise sheet 7 | Monday, 26.06 |
| Exercise sheet 8 | Monday, 03.07 |
| Exercise sheet 9 | Friday, 14.07 |
| Date | Content | Reference | Notes |
|---|---|---|---|
| 17.04 | Course overview, definition of manifold. | Guillemin-Pollack Ch. 1.1, Boothby Ch. 1 | Handwritten notes |
| 21.04 | Inverse function theorem | Guillemin-Pollack Ch. 1.2-3, Boothby Ch. 2 | Handwritten notes |
| 24.04 | Abstract manifolds and quotients | Boothby Ch. 3 | Handwritten notes |
| 28.04 | Vector fields and flows | Boothby Ch. 4 | Handwritten notes |
| 05.05 | Vector fields and flows cont'd | Boothby Ch. 4 | Handwritten notes |
| 08.05 | Lie algebras and Lie groups | Boothby Ch. 4, Lee (Smooth Manifold) Ch. 7 | Handwritten notes |
| 12.05 | Derivations and Frobenius | Boothby Ch. 4 | Handwritten notes |
| 15.05 | Covectors and Riemannian metrics, isometry | Boothby Ch. 5.1-3 | Handwritten notes |
| 19.05 | Tensors and exterior algebra | Boothby Ch 5.5-8 | Handwritten notes |
| 22.05 | Tensor fields on manifolds | Boothby Ch 5.5-8 | Handwritten notes |
| 26.05 | Tensors and exterior algebra | Boothby Ch 5.5-8 | Handwritten notes |
| 02.06 | Orientation and integration | Boothby Ch. 6.1-4, Lee Ch. 16.1-2, 16.5 | Handwritten notes |
| 05.06 | Stokes theorem | Boothby Ch. 6.1-5, Lee 16.3 | Handwritten notes |
| 09.06 | Plane curves | Do Carmo (Curves and Surfaces) Ch. 1 | Handwritten notes |
| 12.06 | Space curves | Do Carmo Ch. 1, Spivak (Vol 2) Ch. 1 | Handwritten notes |
| 14.06 | The Gauss map, curvature of surfaces | Do Carmo (Curves and Surfaces) Ch. 3 | Handwritten notes |
| 19.06 | Connections and Theorem Egregium | Do Carmo 4.3, Lee (Riemannian manifolds) ch. 4 | Handwritten notes |
| 23.06 | The Levi-Civita connection | Lee (R.M.) ch. 5.1-2 | Handwritten notes |
| 26.06 | Geodesics, exponential map | Lee (R.M.) Ch. 5-6, Do Carmo (C.S) Ch. 4.4-6 | Handwritten notes |
| 30.06 | Metric spaces and the Hopf-Rinow theorem | Lee (R.M) Ch. 6, Do Carmo (C.S) Ch. 3.3 | Handwritten notes |
| 03.07 | The curvature tensor | Lee (R.M) Ch. 7.1-2 | Handwritten notes |
| 07.07 | The Gauss-Bonnet theorem | Lee (R.M) Ch. 9, Do Carmo (C.S.) Ch. 4.5 | Handwritten notes |
| 10.07 | Curvature and distance functions | Lee (R.M) Ch. 9, Do Carmo (C.S.) Ch. 4.5 | Handwritten notes |
| 14.07 | Symmetries of the curvature tensor | Lee (R.M) Ch. 7.1-4 | Handwritten notes |
| 17.07 | Jacobi fields and curvature of spaceforms | Lee (R.M) Ch. 3.2-4, Ch. 10.1-2 | Handwritten notes |
| 19.07 | Comparison theory | Lee (R.M) Ch. 10.3-4, Ch. 11 |