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. Leftinvariant 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 LeviCivita connection. 
12 
01.12.2017 
The Christoffel symbols of the LeviCivita connection. 
13 
05.12.2017 
Geodesics: definition, existenceuniqueness 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 
HopfRinow theorem. Nonpositive curvature: CartanHadamard 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: BonnetMyers theorem. 

01.06.2018 
NO CLASS 

08.06.2018 
EXAM 