The geometry of surfaces is an ideal starting point for learning geometry, for, among other reasons, the theory of surfaces of constant curvature has maximal connectivity with the rest of mathematics. This text provides the student with the knowledge of a geometry of greater scope than the classical geometry taught today, which is no longer an adequate basis for mathematics or physics, both of which are becoming increasingly geometric. It includes exercises and informal discussions.

Inhalt

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 of Surfaces.- 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.