Differential Geometry I Autumn 2025

Week	Date	Topics
1	16.09	Arc length and reparametrization of curves, Frenet curves and curvatures, Frenet equations.
19.09	The fundamental theorem of local curve theory, rotation index, Hopf Umlaufsatz and total curvature of plane curves.
2	23.9	Fenchel-Borsuk theorem. Submanifolds and immersions. Regular value theorem and statement of immersion theorem.
26.9	Proof of immersion theorem, local parametrizations and parameter transformation. Tangent space, normal space, differentiability and differential of maps on submanifolds.
3	30.9	Orientability and a proposition on orientable hypersurfaces, Jordan-Brouwer Separation theorem. Discussed the statement of Schonflies theorem.
2.10	Section 3.1, Urs Lang'a notes.
4	7.10	First fundamental form of submanifolds and immersions, examples. Area of graphs, isometries, Christoffel symbols.
9.10	Covariant derivative and parallel vector field, 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. Umbilical points, Gauss curvature and mean curvature.
16.10	Integrability conditions and Gauss's theorema egregium.
6	21.10	Geodesic Parallel & Fermi coordinates (features, existence). Example with a surface of rotation. Constant Gauss curvature in Fermi coordinates. Proof that constant, equal Gauss curvature implies everywhere local isometry. Surfaces of rotation with constant Gauss curvature.
23.10	Ruled surfaces: definition and curvature. Examples. Every 0-curvature with no planar points surface is locally a ruled surface. Minimal surfaces. Variation of the area functional under local normal perturbations. Isothermal coordinates. Isothermal coordinates and their Laplacian. Examples of isothermal parametrisation: catenoid and helicoid.
7	28.10	Gauss-Bonet Theorem
30.10	Poincaré-Hopf Theorem
8	4.11	Minkowski space, the hyperboloid model of hyperbolic m-space, isometries and geodesics.
6.11	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.
9	11.11	Differential topology: differentiable manifolds, smooth structures, differentiable maps and tangent spaces.
13.11	Tangent bundle, differential and partition of unity. Equivalent definitions of submanifolds, regular value theorem.
10	18.11	Embeddability of compact manilfolds. Tangent vectors as derivations. Sets of measure zero, statement of Sard's Theorem.
20.11	Halfspace, manifolds with boundary, regular value theorem for manifolds with boundary, no retraction theorem and Brouwer fixed point theorem.
11	25.11	Smooth homotopies and isotopies, mapping degree mod 2, homotopy invariance, mapping degree, hairy ball theorem and statement of Hopf's theorem.
27.11	Transverse maps and a generalization of regular value theorem. Parametric transversality theorem, existence of homotopy to a transverse map, intersection number modulo 2
12	2.12	Vector bundles, bundle maps and isomorphisms, sections, criterion for triviality,
4.12	transition maps, structure group, Cotangent bundle, pull-back bundle
13	9.12	Whitney sum, tensor bundles, tensor fields. Flow of vector fields, Lie brackets.
11.12	Lie derivative of vector fields, differential forms, exterior product, exterior derivative.
14	16.12	A coordinate-free expression for the exterior derivative and pull-back forms. Measurable decomposition, integrability and integration of forms.
18.12	Stokes' theorem, volume form and integration without orientation.

19.09

The fundamental theorem of local curve theory, rotation index, Hopf Umlaufsatz and total curvature of plane curves.

2

23.9

Fenchel-Borsuk theorem. Submanifolds and immersions. Regular value theorem and statement of immersion theorem.

26.9

Proof of immersion theorem, local parametrizations and parameter transformation. Tangent space, normal space, differentiability and differential of maps on submanifolds.

3

30.9

Orientability and a proposition on orientable hypersurfaces, Jordan-Brouwer Separation theorem. Discussed the statement of Schonflies theorem.

2.10

Section 3.1, Urs Lang'a notes.

4

7.10

First fundamental form of submanifolds and immersions, examples. Area of graphs, isometries, Christoffel symbols.

9.10

Covariant derivative and parallel vector field, 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. Umbilical points, Gauss curvature and mean curvature.

16.10

Integrability conditions and Gauss's theorema egregium.

6

21.10

Geodesic Parallel & Fermi coordinates (features, existence). Example with a surface of rotation. Constant Gauss curvature in Fermi coordinates. Proof that constant, equal Gauss curvature implies everywhere local isometry. Surfaces of rotation with constant Gauss curvature.

23.10

Ruled surfaces: definition and curvature. Examples. Every 0-curvature with no planar points surface is locally a ruled surface. Minimal surfaces. Variation of the area functional under local normal perturbations. Isothermal coordinates. Isothermal coordinates and their Laplacian. Examples of isothermal parametrisation: catenoid and helicoid.

7

28.10

Gauss-Bonet Theorem

30.10

Poincaré-Hopf Theorem

8

4.11

Minkowski space, the hyperboloid model of hyperbolic m-space, isometries and geodesics.

6.11

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.

9

11.11

Differential topology: differentiable manifolds, smooth structures, differentiable maps and tangent spaces.

13.11

Tangent bundle, differential and partition of unity. Equivalent definitions of submanifolds, regular value theorem.

10

18.11

Embeddability of compact manilfolds. Tangent vectors as derivations. Sets of measure zero, statement of Sard's Theorem.

20.11

Halfspace, manifolds with boundary, regular value theorem for manifolds with boundary, no retraction theorem and Brouwer fixed point theorem.

11

25.11

Smooth homotopies and isotopies, mapping degree mod 2, homotopy invariance, mapping degree, hairy ball theorem and statement of Hopf's theorem.

27.11

Transverse maps and a generalization of regular value theorem. Parametric transversality theorem, existence of homotopy to a transverse map, intersection number modulo 2

12

2.12

Vector bundles, bundle maps and isomorphisms, sections, criterion for triviality,

4.12

transition maps, structure group, Cotangent bundle, pull-back bundle

13

9.12

Whitney sum, tensor bundles, tensor fields. Flow of vector fields, Lie brackets.

11.12

Lie derivative of vector fields, differential forms, exterior product, exterior derivative.

14

16.12

A coordinate-free expression for the exterior derivative and pull-back forms. Measurable decomposition, integrability and integration of forms.

18.12

Stokes' theorem, volume form and integration without orientation.

Exercise Sheet	Due By	Upload Link	Solutions
Exercise sheet 1	September 26, 23:59	Upload Sheet 1	Solution 1
Exercise sheet 2	October 4, 23:59	Upload Sheet 2	Solution 2
Exercise sheet 3	October 11, 23:59	Upload Sheet 3	Solution 3
Exercise sheet 4	October 17, 23:59	Upload Sheet 4	Solution 4
Exercise sheet 5	October 24, 23:59	Upload Sheet 5	Solution 5
Exercise sheet 6	October 31, 23:59	Upload Sheet 6	Solution 6
Exercise sheet 7	November 6, 23:59	Upload Sheet 7	Solution 7
Exercise sheet 8	November 13, 23:59	Upload Sheet 8	Solution 8
Exercise sheet 9	November 21, 23:59	Upload Sheet 9	Solution 9
Exercise sheet 10	November 28, 23:59	Upload Sheet 10	Solution 10
Exercise sheet 11	December 5, 23:59	Upload Sheet 11	Solution 11
Exercise sheet 12	December 12, 23:59	Upload Sheet 12	Solution 12
Exercise sheet 13	December 19, 23:59	Upload Sheet 13	Solution 13

Time	Room	Assistant	Language
Th 13:15-14:00	HG E 22	Michler Finn	en
Th 16:15-17:00	IFW C 33	Michler Finn	en
Fr 12:15-13:00	HG E 21	Steffens Derk	en
Fr 13:15-14:00	HG E 21	Steffens Derk	en

Differential Geometry I Autumn 2025

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