Geometry of Surfaces

1. The Euclidean Plane.- 1.1 Approaches to Euclidean Geometry.- 1.2 Isometries.- 1.3 Rotations and Reflections.- 1.4 The Three Reflections Theorem.- 1.5 Orientation-Reversing Isometries.- 1.6 Distinctive Features of Euclidean Geometry.- 1.7 Discussion.- 2. Euclidean Surfaces.- 2.1 Euclid on Manifolds.- 2.2 The Cylinder.- 2.3 The Twisted Cylinder.- 2.4 The Torus and the Klein Bottle.- 2.5 Quotient Surfaces.- 2.6 A Nondiscontinuous Group.- 2.7 Euclidean Surfaces.- 2.8 Covering a Surface by the Plane.- 2.9 The Covering Isometry Group.- 2.10 Discussion.- 3. The Sphere.- 3.1 The Sphere S2 in ?3.- 3.2 Rotations.- 3.3 Stereographic Projection.- 3.4 Inversion and the Complex Coordinate on the Sphere.- 3.5 Reflections and Rotations as Complex Functions.- 3.6 The Antipodal Map and the Elliptic Plane.- 3.7 Remarks on Groups, Spheres and Projective Spaces.- 3.8 The Area of a Triangle.- 3.9 The Regular Polyhedra.- 3.10 Discussion.- 4. The Hyperbolic Plane.- 4.1 Negative Curvature and the Half-Plane.- 4.2 The Half-Plane Model and the Conformai Disc Model.- 4.3 The Three Reflections Theorem.- 4.4 Isometries as Complex Functions.- 4.5 Geometric Description of Isometries.- 4.6 Classification of Isometries.- 4.7 The Area of a Triangle.- 4.8 The Projective Disc Model.- 4.9 Hyperbolic Space.- 4.10 Discussion.- 5. Hyperbolic Surfaces.- 5.1 Hyperbolic Surfaces and the Killing-Hopf Theorem.- 5.2 The Pseudosphere.- 5.3 The Punctured Sphere.- 5.4 Dense Lines on the Punctured Sphere.- 5.5 General Construction of Hyperbolic Surfaces from Polygons.- 5.6 Geometric Realization of Compact Surfaces.- 5.7 Completeness of Compact Geometric Surfaces.- 5.8 Compact Hyperbolic Surfaces.- 5.9 Discussion.- 6. Paths and Geodesies.- 6.1 Topological Classification of Surfaces.- 6.2 Geometric Classification ofSurfaces.- 6.3 Paths and Homotopy.- 6.4 Lifting Paths and Lifting Homotopies.- 6.5 The Fundamental Group.- 6.6 Generators and Relations for the Fundamental Group.- 6.7 Fundamental Group and Genus.- 6.8 Closed Geodesic Paths.- 6.9 Classification of Closed Geodesic Paths.- 6.10 Discussion.- 7. Planar and Spherical Tessellations.- 7.1 Symmetric Tessellations.- 7.2 Conditions for a Polygon to Be a Fundamental Region.- 7.3 The Triangle Tessellations.- 7.4 Poincaré’s Theorem for Compact Polygons.- 7.5 Discussion.- 8. Tessellations of Compact Surfaces.- 8.1 Orbifolds and Desingularizations.- 8.2 Prom Desingularization to Symmetric Tessellation.- 8.3 Desingularizations as (Branched) Coverings.- 8.4 Some Methods of Desingularization.- 8.5 Reduction to a Permutation Problem.- 8.6 Solution of the Permutation Problem.- 8.7 Discussion.- References.