This book was first published in 1913. It contained the essentials of some lectures which I gave at the University of Göttingen in the winter semester of 1911–12. The purpose of the book was to develop the basic ideas of Riemann’s 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 Göttingen colleague P. Koebe of the fundamental uniformization theorems; and Hilbert’s 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 Riemann’s 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 für angewandte Mathematik und Physik (4, 1953, 471–492): Über die kombinatorische und kontinuumsmässige Definition der Überschneidungszahl zweier geschlossener Kurven auf einer Fläche. 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 Riemann’s 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. 50 (1949) 587–665, 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 Chevalley’s 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 André 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 Weil’s review of Chevalley’s book, Bull. Am. Math. Soc. 57 (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, Dirichlet’s 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, The theory and applications of harmonic integrals, Cambridge 1941, and in which the paper of Kodaira cited above is such a landmark. I have restricted myself to the case of one complex dimension, and throughout I use either the real or complex numbers as the coefficient domain, in the spirit of the “transcendental” Riemannian point of view which I adopt.
Influenced by Kodaira’s work, I have hesitated a moment as to whether I should not replace the Dirichlet principle by the essentially equivalent “method of orthogonal projection” which is treated in a paper of mine in the Duke Math. Jour. 7 (1940), 411–444. But for reasons the explication of which would lead too far afield here, I have stuck to the old approach.
I fear that in the preparation of the new edition I have followed my own ideas too much and have paid too little attention to other ideas, especially those in the potent literature of topology. May I not be judged too harshly! For technical assistance in preparation of the manuscript I wish to thank most sincerely Frau Natascha Artin, in whom the mathematicians at Princeton and at New York University always have a friend ready to help. My thanks are also due to the publishing house of B. G. Teubner in Stuttgart who, though still in the middle of recreating their business, expended the same care on this new edition as the old firm granted to the first edition before the first world war.
HERMANN WEYL
Princeton, New Jersey
January 1955
Translator’s note: Dr. Weyl’s (1885–1955) academic affiliations included: Ph. D., Göttingen, 1908; hon. Ph. D., Oslo, 1929; hon. Dr. Ing., Stuttgart, 1929; hon. Dr. Sc., Penn., 1940; hon. Dr. Math., Eid. Tech. Hochs. Zürich, 1945; Privat-Dozent, Göttingen, 1910–1913; Prof. of Math., Eid. Tech. Hochs. Zürich, 1913–1930; Prof. of Math., Göttingen, 1930–1933; Professor, Institute for Advanced Study, 1933–1951; Professor Emeritus, 1951–1955. Professor, Princeton, 1928–1929. Lobatschefski Prize, Kazan, 1925.