Contents ix

b. The cross-ratio 193

c. Circles in the hyperbolic plane 196

Lecture 28 198

a. Three approaches to hyperbolic geometry 198

b. Characterisation of isometries 199

Lecture 29 204

a. Classification of isometries 204

b. Geometric interpretation of isometries 213

Lecture 30 217

a. Area of triangles in different geometries 217

b. Area and angular defect in hyperbolic geometry 218

Lecture 31 224

a. Hyperbolic metrics on surfaces of higher genus 224

b. Curvature, area, and Euler characteristic 228

Lecture 32 231

a. Geodesic polar coordinates 231

b. Curvature as an error term in the circle length formula 233

c. The Gauss-Bonnet Theorem 235

d. Comparison with traditional approach 240

Chapter 5. Topology and Smooth Structure Revisited 243

Lecture 33 243

a. Back to degree and index 243

b. The Fundamental Theorem of Algebra 246

Lecture 34 249

a. Jordan Curve Theorem 249

b. Another interpretation of genus 253

Lecture 35 255

a. A remark on tubular neighbourhoods 255

b. Proving the Jordan Curve Theorem 256

c. Poincar´ e-Hopf Index Formula 259

Lecture 36 260

a. Proving the Poincar´ e-Hopf Index Formula 260

b. Gradients and index formula for general functions 265

c. Fixed points and index formula for maps 267

d. The ubiquitous Euler characteristic 269