# Riemannian Geometry 2017/2018

• The lectures will be given in English.
• Fridays, 8:00 - 10:00. Room: Math II Lonza.
• The poster of the course can be found here.
• Office hours: by appointment.

## 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. ---- ---- ---- 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