This well-known book is a self-contained treatment of the classical theory of abstract Riemann surfaces. The first five chapters cover the requisite function theory and topology for Riemann surfaces. The second five chapters cover differentials and uniformization. For compact Riemann surfaces, there are clear treatments of divisors, Weierstrass points, the Riemann-Roch theorem and other important topics. Springer's book is an excellent text for an introductory course on Riemann surfaces. It includes exercises after each chapter and is illustrated with a beautiful set of figures.
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.