Introduction to Riemann Surfaces.

Abstract

Introduction: 1-1 Algebraic functions and Riemann surfaces 1-2 Plane fluid flows 1-3 Fluid flows on surfaces 1-4 Regular potentials 1-5 Meromorphic functions 1-6 Function theory on a torus General Topology: 2-1 Topological spaces 2-2 Functions and mappings 2-3 Manifolds Riemann Surface of an Analytic Function: 3-1 The complete analytic function 3-2 The analytic configuration Covering Manifolds: 4-1 Covering manifolds 4-2 Monodromy theorem 4-3 Fundamental group 4-4 Covering transformations Combinatorial Topology: 5-1 Triangulation 5-2 Barycentric coordinates and subdivision 5-3 Orientability 5-4 Differentiable and analytic curves 5-5 Normal forms of compact orientable surfaces 5-6 Homology groups and Betti numbers 5-7 Invariance of the homology groups 5-8 Fundamental group and first homology group 5-9 Homology on compact surfaces Differentials and Integrals: 6-1 Second-order differentials and surface integrals 6-2 First-order differentials and line integrals 6-3 Stokes' theorem 6-4 The exterior differential calculus 6-5 Harmonic and analytic differentials The Hilbert Space of Differentials: 7-1 Definition and properties of Hilbert space 7-2 Smoothing operators 7-3 Weyl's lemma and orthogonal projections Existence of Harmonic and Analytic Differentials: 8-1 Existence theorems 8-2 Countability of a Riemann surface Uniformization: 9-1 Schlichtartig surfaces 9-2 Universal covering surfaces 9-3 Triangulation of a Riemann surface 9-4 Mappings of a Riemann surface onto itself Compact Riemann Surfaces: 10-1 Regular harmonic differentials 10-2 The bilinear relations of Riemann 10-3 Bilinear relations for differentials with singularities 10-4 Divisors 10-5 The Riemann-Roch theorem 10-6 Weierstrass points 10-7 Abel's theorem 10-8 Jacobi inversion problem 10-9 The field of algebraic functions 10-10 The hyperelliptic case References Index.