Riemannian Geometry 2017/2018

Announcements

The (oral) exam will be on Friday 08.06.2018.


Lecture number Date Content
1 22.09.2017 Smooth manifolds and smooth maps.
2 29.09.2017 Tangent space, the differential of a map, immersions and submersions.
3 06.10.2017 The tangent bundle, vector fields and the Lie bracket (I).
4 13.10.2017 Lie bracket (II) and partitions of unity. Riemannian metrics.
5 20.10.2017 Existence of metrics, (local) isometries, lengths of curves.
6 31.10.2017 Examples of lengths of curves and partitions of unity. The isometry group.
7 03.11.2017 Smooth group actions and quotients. Isometric actions.
8 07.11.2017 Riemannian quotients. Left-invariant metrics on Lie groups.
9 14.11.2017 Affine connections and the covariant derivative (I).
10 17.11.2017 The covariant derivative (II) and parallel transport.
11 28.11.2017 Symmetric and compatible connections. The Levi-Civita connection.
12 01.12.2017 The Christoffel symbols of the Levi-Civita connection.
13 05.12.2017 Geodesics: definition, existence-uniqueness and the geodesic flow.
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14 23.02.2018 The exponential map, examples. Distance functions.
15 02.03.2018 Minimizing properties of geodesics.
16 09.03.2018 The curvature tensor: definition and properties.
17 16.03.2018 Sectional curvature (I).
18 23.03.2018 Sectional curvature (II) and Jacobi fields (I).
19 13.04.2018 Jacobi fields (II).
20 20.04.2018 Conjugate points. Completeness.
21 27.04.2018 Hopf-Rinow theorem. Non-positive curvature: Cartan-Hadamard theorem.
22 04.05.2018 Isometric immersions. The curvature of the sphere.
23 18.05.2018 The curvature of the hyperbolic space. Manifolds with constant curvature.
24 25.05.2018 Variations of energy. Positive curvature: Bonnet-Myers theorem.
01.06.2018 NO CLASS
08.06.2018 EXAM