AMS Chelsea Publishing 1981; 309 pp; hardcover Volume: 313 Reprint/Revision History: first AMS printing 2002 ISBN10: 0821831569 ISBN13: 9780821831564 List Price: US$41 Member Price: US$36.90 Order Code: CHEL/313.H
 This wellknown book is a selfcontained 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 RiemannRoch 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 selfcontained 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 wellorganized form."  Mathematical Reviews Table of Contents Introduction  11 Algebraic functions and Riemann surfaces
 12 Plane fluid flows
 13 Fluid flows on surfaces
 14 Regular potentials
 15 Meromorphic functions
 16 Function theory on a torus
General Topology  21 Topological spaces
 22 Functions and mappings
 23 Manifolds
Riemann Surface of an Analytic Function  31 The complete analytic function
 32 The analytic configuration
Covering Manifolds  41 Covering manifolds
 42 Monodromy theorem
 43 Fundamental group
 44 Covering transformations
Combinatorial Topology  51 Triangulation
 52 Barycentric coordinates and subdivision
 53 Orientability
 54 Differentiable and analytic curves
 55 Normal forms of compact orientable surfaces
 56 Homology groups and Betti numbers
 57 Invariance of the homology groups
 58 Fundamental group and first homology group
 59 Homology on compact surfaces
Differentials and Integrals  61 Secondorder differentials and surface integrals
 62 Firstorder differentials and line integrals
 63 Stokes' theorem
 64 The exterior differential calculus
 65 Harmonic and analytic differentials
The Hilbert Space of Differentials  71 Definition and properties of Hilbert space
 72 Smoothing operators
 73 Weyl's lemma and orthogonal projections
Existence of Harmonic and Analytic Differentials  81 Existence theorems
 82 Countability of a Riemann surface
Uniformization  91 Schlichtartig surfaces
 92 Universal covering surfaces
 93 Triangulation of a Riemann surface
 94 Mappings of a Riemann surface onto itself
Compact Riemann Surfaces  101 Regular harmonic differentials
 102 The bilinear relations of Riemann
 103 Bilinear relations for differentials with singularities
 104 Divisors
 105 The RiemannRoch theorem
 106 Weierstrass points
 107 Abel's theorem
 108 Jacobi inversion problem
 109 The field of algebraic functions
 1010 The hyperelliptic case
 References
 Index
