| 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. |
| 08.06.2018 | Exam. | |
| 13.09.2018 | Exam. |