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, existence and the geodesic flow. 
 
 
 
14 
23.02.2018 

15 
02.03.2018 

16 
09.03.2018 

17 
16.03.2018 

18 
23.03.2018 

19 
13.04.2018 

20 
20.04.2018 

21 
27.04.2018 

22 
04.05.2018 

23 
18.05.2018 

24 
25.05.2018 

25 
01.06.2018 
