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Introduction to Riemann Surfaces
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AMS Chelsea Publishing
1981; 309 pp; hardcover
Volume: 313
Reprint/Revision History:
first AMS printing 2002
ISBN-10: 0-8218-3156-9
ISBN-13: 978-0-8218-3156-4
List Price: US$41 Member Price: US$36.90
Order Code: CHEL/313.H

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.

Reviews

"Written with unusual clearness. As in the Introduction, which outlines the whole book, similar [outlines] appear in each chapter ... a modern treatment in a self-contained manner with a minimum assumption of knowledge. He is most successful in this magnificent project ... It is highly recommended."

-- American Mathematical Monthly

"The book is written specifically with graduate (and advanced undergraduate) students in mind. There are no prerequisites beyond standard first courses in complex variables, real variables, and algebra. What is needed of topology and Hilbert space theory is derived from the beginning. Concepts and theorems are illuminated by examples and excellent figures, proofs are clarified by heuristic remarks, and the inventiveness of even the good student is challenged by a well chosen problem collection. The style, while very readable, never becomes "insultingly simple" and even the specialist can derive pleasure from reviewing basic material in a well-organized form."

-- Mathematical Reviews

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