Weyl Hermann - Concept of a Riemann Surface

The Concept of a
Riemann Surface

THIRD EDITION

Hermann Weyl

Translated from the German by

Gerald R. MacLane

Dover Publications, Inc.
Mineola, New York

Bibliographical Note

This Dover edition, first published in 2009, is an unabridged republication of the work originally published by Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, in 1955.

Library of Congress Cataloging-in-Publication Data

Weyl, Hermann, 18851955.

[Ider der Riemannschen Flche German]

The concept of a Riemann surface / Hermann Weyl.Dover ed.

p. cm.

Originally published: 3rd ed. Reading, Mass. : Addison-Wesley, 1955.

Includes index.

eISBN-13: 978-0-486-13167-2

1. Riemann surfaces. I. Title.

QA333.W413 2009

515'.93dc22

2008051084

Manufactured in the United States by Courier Corporation
47004002
www.doverpublications.com

Dedicated to my sons

Joachim and Michael

PREFACE

This book was first published in 1913. It contained the essentials of some lectures which I gave at the University of Gttingen in the winter semester of 191112. The purpose of the book was to develop the basic ideas of Riemanns theory of algebraic functions and their integrals and also to treat the requisite ideas and theorems of analysis situs in a fashion satisfying modern demands of rigor. This had not been done before. For example, the concept of a curve is never clarified in the classic book of Hensel and Landsberg, Theorie der algebraischen Funktionen einer Variablen und ihre Anwendung auf algebraische Kurven und Abelsche Integrale (Leipzig 1902, Teubner). Three events had a decisive influence on the form of my book: the fundamental papers of Brouwer on topology, commencing in 1909; the recent proofs by my Gttingen colleague P. Koebe of the fundamental uniformization theorems; and Hilberts establishment of the foundation on which Riemann had built his structure and which was now available for uniformization theory, the Dirichlet principle. The book was dedicated to Felix Klein in Dankbarkeit und Verehrung. Klein had been the first to develop the freer conception of a Riemann surface, in which the surface is no longer a covering of the complex plane; thereby he endowed Riemanns basic ideas with their full power. It was my fortune to discuss this thoroughly with Klein in divers conversations. I shared his conviction that Riemann surfaces are not merely a device for visualizing the many-valuedness of analytic functions, but rather an indispensable essential component of the theory; not a supplement, more or less artificially distilled from the functions, but their native land, the only soil in which the functions grow and thrive. Even more than the text, the enthusiastic preface betrayed the youth of the author.

In 1923 Teubner published an anastatic reprint to which were added a page of corrections and additions and an appendix, A rigorous foundation of the theory of characteristics on two-sided surfaces. This second edition has been distributed since 1947 in an American reprint, authorized by the Attorney General, by the Chelsea Publishing Co.

In the more than forty years which have passed since the appearance of this book the face of mathematics has changed noticeably. Above all, the young shoot analysis situs has become the tree topology, affording shade to large parts of our science. When German mathematicians and the publishing house of Teubner approached me with the invitation to prepare a new edition, since requests for the book continued, it at first seemed appropriate to treat the book more or less as an historical document and send it into the world again unchanged except for a few minor improvements. But as I attempted to merge the appendix with the main text, I became ever more conscious of the deficiencies of both the appendix and the text. The way in which I then undertook to rework the first topological half of the book more thoroughly, emphasizing the combinatorial aspect even more than formerly, may be seen from my paper in Die Zeitschrift fr angewandte Mathematik und Physik (, 1953, 471492): ber die kombinatorische und kontinuumsmssige Definition der berschneidungszahl zweier geschlossener Kurven auf einer Flche. However, there occurred to me during the course of the work the idea, worked out in the end of that paper, of defining the intersection number by topologizing the construction I used in 1913 to define the Abelian integrals of the first kind. This resulted in a clearer structure for the second function-theoretic part of the book; in this development the following important fact no longer had to be suppressed, namely that the imaginary parts of the integrals of the first kind associated with the paths of a basis are the coefficients of a positive quadratic form. Also I found myself in agreement with the recent trend of topology, to replace the dissection of a manifold by a covering by overlapping neighborhoods. To be sure, it then turned out to be natural to subsume Riemann surfaces under differentiable surfaces, rather than the most general topological surfaces. The separation into real and imaginary parts, used systematically in the first edition, which obviates reference to a canonical dissection in Riemanns sense, remained fundamental. The effectiveness of this step is shown most clearly by the generalization to higher dimensions which was completed in the interim: the real harmonic, not the complex holomorphic, linear differential forms have established themselves as the starting point of the theory, and the great paper of Kunihiko Kodaira, Harmonic fields in Riemannian manifolds (generalized potential theory), Ann. of Math. (1949) 587665, specifically invokes the prototype in my old book.

I remark further that in using a covering of the manifold by neighborhoods I find it necessary neither to normalize the neighborhoods (e.g., by the condition of convexity) nor to replace the covering by always finer ones. Also, the cycles, that is, the continuous closed curves, and the integrals along them, take their place in the most natural fashion alongside cocycles, which are so much more accommodating in general topology.

Only subsequently did I observe how much closer my new presentation has come to Claude Chevalleys treatment of Riemann surfaces in chapter VII of his Introduction to the theory of algebraic functions of one variable (Math. Surveys VI, Am. Math. Soc., New York 1951). With Chevalley I must accept the reproach of Andr Weil that the route we chose (without triangulation) has barred, or at least made more difficult, the way to certain classical results; for example, the structure of the fundamental group. (See Weils review of Chevalleys book, Bull. Am. Math. Soc. (1951) 384398; in particular p. 390.) Since these results are beyond the scope of my book I felt that I should accept these limitations for the sake of the advantages of the method. Who says to us that we have already reached the end of the methodical development of topology?

As for references, I have preserved the old-fashioned aspect of the book. I have retained the citations to the literature of the 19th century; the younger generation is only too inclined to forget the connection of the new with the old! The references have been completed by citations of the newer literature, but I have not troubled to prepare a bibliography which is complete in any sense. The reader will find a welcome supplement to the algebraic aspects in the book of Chevalley noted above. Related subjects are treated in: R. Courant, Dirichlets principle, conformal mapping, and minimal surfaces, New York 1950, and Rolf Nevanlinna, Uniformisierung, Berlin 1953; these works contain extensive references to the literature. There is as yet no comprehensive source for the splendid developments in several variables which were introduced in the book of W. V. D. Hodge,