vi Contents

Lecture 4 28

a. Remarks on metric spaces and topology 28

b. Homeomorphisms and isometries 31

c. Other notions of dimension 32

d. Geodesics 33

Lecture 5 34

a. Isometries of the Euclidean plane 34

b. Isometries of the sphere and the elliptic plane 38

Lecture 6 39

a. Classification of isometries of the sphere and the

elliptic plane 39

b. Area of a spherical triangle 41

Lecture 7 43

a. Spaces with lots of isometries 43

b. Symmetric spaces 45

c. Remarks concerning direct products 47

Chapter 2. Combinatorial Structure and Topological

Classification of Surfaces 49

Lecture 8 49

a. Topology and combinatorial structure on surfaces 49

b. Triangulation 52

c. Euler characteristic 56

Lecture 9 58

a. Continuation of the proof of Theorem 2.4 58

b. Calculation of Euler characteristic 65

Lecture 10 67

a. From triangulations to maps 67

b. Examples 70

Lecture 11 73

a. Euler characteristic of planar models 73

b. Attaching handles 74

c. Orientability 77

d. Inverted handles and M¨ obius caps 79