The theory of Riemann surfaces is experiencing a strong revival. An indication of this is the number of books appearing in the field. None were published during the four decades following Weyl's classic [Die Idee der Riemannschen Fläche, Teubner, Leipzig-Berlin, 1913], but the last five years have given birth to ten books largely devoted to this theory. This awakened interest has created a need for an introductory text in English. The book under consideration fills this gap. 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. Regarding closed Riemann surfaces, the title ``introduction'' is modest: these surfaces have received, in the largest chapter of the book, a quite comprehensive treatment. It is true that this part of the theory reached a rather final shape decades ago and that no new presentation can add much. Also, the modern doctrine of abstract Riemann surfaces could be approached without knowledge of algebraic functions, even without the foggiest idea of the Riemann surface of √z. But the pedagogical value of such ``concrete'' surfaces is undeniable, the beautiful classical theory being particularly suitable for developing the student's ability of geometric thinking. Furthermore, the theory on differentials on closed Riemann surfaces is a good starting point for modern literature on the space of Riemann surfaces and the Riemann-Roch theorem on complex manifolds. As to open surfaces, the author informs us that the book ``is not meant to be a survey of the current work being done in the realm of Riemann surfaces''. Specifically, he restricts his attention to topological aspects and the uniformization theory, which he treats with rigor, using modern tools from combinatorial topology and orthogonal projection. Constructive methods would lead somewhat quicker to a more general function theory and related extremal problems, but a knowledge of Hilbert space methods on Riemann surfaces is a must, and the uniformization is as good a context as any to introduce it. It also was time that someone incorporated in a text the elegant mechanism of Cartan's exterior differential calculus, which should be a tool of every graduate student. Concerning earlier literature the author acknowledges the influence of Weyl's 1913 book and of lectures delivered by Ahlfors at Harvard University in 1948. The former perhaps contributes primarily to the general set-up of the book, the latter to chapters dealing with topology and exterior differential calculus. The book is, roughly speaking, divided into two equal parts, the first (Ch. 1–5) devoted to Riemann surfaces, the second (Ch. 6–10) to functions and differentials. The introductory Ch. 1 is largely heuristic and presents main goals, methods, and concepts of the book. Ch. 2, General Topology, is a brief introduction to Hausdorff spaces, their subsets and mappings, and to the concepts of manifold and abstract Riemann surface. In Ch. 3, Riemann Surface of an Analytic Function, the author gives the definitions of complete analytic function, the corresponding analytic manifold, the analytic configuration, and its geometric counterpart, the Riemann surface of an analytic function. The last is shown to be a special case of an abstract Riemann surface. Ch. 4, Covering Manifolds, is a discussion of smooth and unlimited covering manifolds, the monodromy theorem, the fundamental group, and covering transformations. Ch. 5, Combinatorial Topology, is given over to global properties of triangulable manifolds. Barycentric co-ordinates and subdivision are introduced, and the orientability of manifolds is considered. It is shown that every Riemann surface is orientable and that every orientable triangulable manifold can be made into a Riemann surface by defining an analytic structure on it. Differentiable and analytic curves are introduced and their properties briefly discussed. Every compact orientable surface is shown to be homeomorphic to a sphere with g handles. The genus g is then characterized by homology groups and Betti numbers, and relations are given between the fundamental group and the first homology group. The chapter ends with an explicit computation of the homology groups on compact surfaces. Ch. 6, Differentials and Integrals, is an exposition of the usual material centering around Stokes' theorem. An introduction to exterior differential calculus is given and harmonic and analytic differentials briefly dealt with. Ch. 7, Hilbert Space on Differentials, introduces smoothing operators, Weyl's lemma, and orthogonal projections, to be used in the existence proofs of Ch. 8. In the latter chapter, the existence of exterior harmonic differentials is established, with given singularities and a finite norm over a boundary neighborhood. It is also shown that every Riemann surface has a countable basis. The existence theorems of Ch. 8 are then used in Ch. 9 to establish the parallel slit mapping and the general uniformization theorems. Automorphic functions and non-Euclidian geometry on the uniformizing disk are considered, and every Riemann surface is shown to be triangulable. Ch. 10, Compact Riemann Surfaces, is an account of bilinear relations, the Riemann-Roch theorem, Weier-strass points, Abel's theorem, the Jacobi inversion problem, and the field of algebraic functions. For illustration, the hyperelliptic case is treated in detail.
Reviewer: Sario, L. [form MathSciNet]