Geometry Of Surfaces B3 Course 2004 (Lecture notes)

GEOMETRY OF SURFACES b3 course 2004......Page 1
1 Introduction......Page 2
2.1 The de nition of a surface......Page 7
2.2 Planar models and connected sums......Page 12
2.3 The classi cation of surfaces......Page 19
2.4 Orientability......Page 21
2.5 The Euler characteristic......Page 25
3.1 Definitions and examples......Page 29
3.2 Meromorphic functions......Page 33
3.3 Multi-valued functions......Page 37
4.1 Definitions......Page 42
4.2 The first fundamental for......Page 47
4.3 Isometric surfaces......Page 53
4.4 The second fundamental form......Page 56
4.5 The Gaussian curvature......Page 61
4.6 The Gauss-Bonnet theorem......Page 63
4.7 Geodesics......Page 69
4.8 Gaussian curvature revisited......Page 74
5.1 Isometries......Page 77
5.2 Geodesics......Page 78
5.3 Angles and distances......Page 80
5.4 Hyperbolic triangles......Page 81
5.5 Non-Euclidean geometry......Page 84
EUCLID’S AXIOMS......Page 86
5.6 Complex analysis and the hyperbolic plane......Page 88
6.1 A: The inverse function theorem......Page 92
6.2 B: Existence of solutions of ordinary di erential equations......Page 94
6.3 B’: Smooth dependence......Page 95
References......Page 98