Introduction to differential geometry and differential topology. Contents: Curves, (hyper-)surfaces in \(\mathbb R^n\), geodesics, curvature, Theorema Egregium, Theorem of Gauss-Bonnet. Hyperbolic space. Differentiable manifolds, immersions and embeddings, Sard's Theorem, mapping degree and intersection number, vector bundles, vector fields and flows, differential forms, Stokes' Theorem.
| Week | Date | Topics |
|---|---|---|
| 1 | 19.09 | Arc length and reparametrization of curves, Frenet curves and curvatures, Frenet equations. |
| 2 | 23.09 | The fundamental theorem of local curve theory, rotation index, Hopf Umlaufsatz and total curvature of plane curves. |
| 26.09 | Fenchel-Borsuk theorem and statement of Fáry-Milnor theorem. Submanifolds and immersions. Regular value theorem and statement of immersion theorem. | |
| 3 | 30.09 | Proof of immersion theorem, local parametrizations and parameter transformation. Tangent space, normal space, differentiability and differential of maps on submanifolds. Orientability and a proposition on orientable hypersurfaces. |
| 03.10 | Differentiable Jordan-Brouwer separation theorem, statement of Schönflies theorem. First fundamental form of submanifolds and immersions, examples. | |
| 4 | 07.10 | Area of graphs, isometries, Christoffel symbols, covariant derivative and parallel vector field. |
| 10.10 | Existence of parallel vector fields, parallel transport on \(S^2\). Geodesics, Clairaut's relation, first variation of arc length. | |
| 5 | 14.10 | Shape operator and second fundamental form. Normal curvature, principal curvatures and examples. |
| 17.10 | Umbilical points, Gauss curvature and mean curvature. Integrability conditions and Gauss's theorema egregium. | |
| 6 | 21.10 | \(g\) and \(h\) determine \(f\), statement of Bonnet's fundamental theorem of local surface theory. Geodesic parallel coordinates, Fermi coordinates and an existence result. |
| 24.10 | Surfaces with constant Gauss curvature, ruled surfaces, examples and rulings in flat surfaces. | |
| 7 | 28.10 | Minimal surfaces, first variation of area, isothermal minimal surfaces and Alexandrov-Hopf theorem. |
| 31.10 | Geodesic curvature of curves in surfaces, local version of the Gauss-Bonnet theorem, Gauss' theorema elegantissimum. | |
| 8 | 04.11 | Global version of the Gauss-Bonnet theorem, the Poincaré index theorem. |
| 07.11 | Minkowski space, the hyperboloid model of hyperbolic \(m\)-space, isometries and geodesics. | |
| 9 | 11.11 | Beltrami-Klein model, Poincaré disk model and halfspace model of \(H^m\). Hilbert's theorem with a sketch of proof, and the statement of Nash-Kuiper theorem. |
| 14.11 | Differential topology: differentiable manifolds, smooth structures, differentiable maps and tangent spaces. |
The new exercise sheet will be uploaded on this page on Monday after the lecture. You are supposed to have a look at it before the exercise class, so that you can ask questions if you need to. You have time until the following Monday at 12:15 to upload your solutions.
Please, upload your solution via the SAM upload tool.
In order to access the website you will need a NETHZ-account and you will have to be connected to the ETH-network. From outside the ETH network you can connect to the ETH network via VPN. Here are instructions on how to do that.
Make sure that your solution is one PDF file and that its file name is formatted in the following way:
solution_<number of exercise sheet>_<your last name>_<your first name>.pdf
Example:
solution_2_Surname_Name.pdf.
| Exercise Sheet | Due By | Upload Link | Solutions |
|---|---|---|---|
| Exercise sheet 1 | September 30, 12:15 | Upload Sheet 1 | Solution 1 |
| Exercise sheet 2 | October 7, 12:15 | Upload Sheet 2 | Solution 2 |
| Exercise sheet 3 | October 14, 12:15 | Upload Sheet 3 | Solution 3 |
| Exercise sheet 4 | October 21, 12:15 | Upload Sheet 4 | Solution 4 |
| Exercise sheet 5 | October 28, 12:15 | Upload Sheet 5 | Solution 5 |
| Exercise sheet 6 | November 04, 12:15 | Upload Sheet 6 | Solution 6 |
| Exercise sheet 7 | November 11, 12:15 | Upload Sheet 7 | Solution 7 |
| Exercise sheet 8 | November 18, 12:15 | Upload Sheet 8 | Solution 8 |
| Exercise sheet 9 | November 25, 12:15 | Upload Sheet 9 |
| Time | Room | Assistant | Language |
|---|---|---|---|
| Th 13:15-14:00 | HG E 22 | Asaf Amitai | en |
| Th 16:15-17:00 | IFW C 33 | Asaf Amitai | en |
| Fr 12:15-13:00 | HG E 21 | Dorian Martino | en |
| Fr 13:15-14:00 | HG E 21 | Dorian Martino | en |