viii Contents

c. Conformal property of holomorphic functions and

invariance of angles on Riemann surfaces 127

d. Complex tori and the modular surface 129

Lecture 20 130

a. Differentiable functions on real surfaces 130

b. Morse functions 135

c. The third incarnation of Euler characteristic 138

Lecture 21 141

a. Functions with degenerate critical points 141

b. Degree of a circle map 145

c. Brouwer’s fixed point theorem 149

Lecture 22 150

a. Zeroes of a vector field and their indices 150

b. Calculation of index 153

c. Tangent vectors, tangent spaces, and the tangent

bundle 155

Chapter 4. Riemannian Metrics and Geometry of Surfaces 159

Lecture 23 159

a. Definition of a Riemannian metric 159

b. Partitions of unity 163

Lecture 24 165

a. Existence of partitions of unity 165

b. Global properties from local and infinitesimal 169

c. Lengths, angles, and areas 170

Lecture 25 172

a. Geometry via a Riemannian metric 172

b. Differential equations 174

c. Geodesics 175

Lecture 26 178

a. First glance at curvature 178

b. The hyperbolic plane: two conformal models 181

c. Geodesics and distances on H2 186

Lecture 27 189

a. Detailed discussion of geodesics and isometries in the

upper half-plane model 189