Differential Geometry I Autumn 2024

Lecturer
Urs Lang
Coordinator
Tian Lan
Time and Location:
Monday, 14:15 - 16:00 in CAB G 11
Thursday, 10:15 - 12:00 in ML H 44
Forum
Link

Content

Introduction to differential geometry and differential topology. Contents: Curves, (hyper-)surfaces in Rn\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.

Lectures

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.
26.09Fenchel-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.
10.10Existence of parallel vector fields, parallel transport on S2S^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.
17.10Umbilical points, Gauss curvature and mean curvature. Integrability conditions and Gauss's theorema egregium.
6 21.10 gg and hh determine ff, statement of Bonnet's fundamental theorem of local surface theory. Geodesic parallel coordinates, Fermi coordinates and an existence result.
24.10Surfaces 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 mm-space, isometries and geodesics.
9 11.11 Beltrami-Klein model, Poincaré disk model and halfspace model of HmH^m. Hilbert's theorem with a sketch of proof, and the statement of Nash-Kuiper theorem.
14.11Differential topology: differentiable manifolds, smooth structures, differentiable maps and tangent spaces.
10 18.11 Tangent bundle, differential and partition of unity. Equivalent definitions of submanifolds, regular value theorem.
21.11Embeddability of compact manilfolds into Rn\mathbb R^n. Tangent vectors as derivations. Sets of measure zero, statement of Sard's Theorem.
11 25.11Halfspace, manifolds with boundary, regular value theorem for manifolds with boundary, no retraction theorem and Brouwer fixed point theorem.
28.11Smooth homotopies and isotopies, mapping degree mod 22, homotopy invariance, mapping degree, hairy ball theorem and statement of Hopf's theorem.
12 02.12Transverse maps and a generalization of regular value theorem. Parametric transversality theorem, existence of homotopy to a transverse map, intersection number modulo 2.
05.12Vector bundles, bundle maps and isomorphisms, sections, criterion for triviality, transition maps, structure group.
13 09.12Cotangent bundle, pull-back bundle, Whitney sum, tensor bundles, tensor fields. Flow of vector fields, Lie brackets.
12.12Lie derivative of vector fields, differential forms, exterior product, exterior derivative.
14 16.12A coordinate-free expression for the exterior derivative and pull-back forms. Measurable decomposition, integrability and integration of forms.
19.12Stokes' theorem, volume form and integration without orientation.

Exercises

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 classes

TimeRoomAssistantLanguage
Th 13:15-14:00HG E 22Asaf Amitaien
Th 16:15-17:00IFW C 33Asaf Amitaien
Fr 12:15-13:00HG E 21Dorian Martinoen
Fr 13:15-14:00HG E 21Dorian Martinoen

Literature

Differential Geometry in Rn\mathbb R^n: Differential Topology: Partial lecture notes from the course taught in Fall 2019 are available from Prof. Lang's website.