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About this book
This book grew out of lectures on Riemann surfaces which the author gave at the universities of Munich, Regensburg and Munster. Its aim is to give an introduction to this rich and beautiful subject, while presenting methods from the theory of complex manifolds which, in the special case of one complex variable, turn out to be particularly elementary and transparent. The book is divided into three chapters. In the first chapter we consider Riemann surfaces as covering spaces and develop a few basics from topology which are needed for this. Then we construct the Riemann surfaces which arise via analytic continuation of function germs. In particular this includes the Riemann surfaces of algebraic functions. As well we look more closely at analytic functions which display a special multi-valued behavior. Examples of this are the primitives of holomorphic i-forms and the solutions of linear differential equations. The second chapter is devoted to compact Riemann surfaces. The main classical results, like the Riemann-Roch Theorem, Abel's Theorem and the Jacobi inversion problem, are presented. Sheaf cohomology is an important technical tool. But only the first cohomology groups are used and these are comparatively easy to handle. The main theorems are all derived, following Serre, from the finite dimensionality of the first cohomology group with coefficients in the sheaf of holomorphic functions. And the proof of this is based on the fact that one can locally solve inhomogeneous Cauchy Riemann equations and on Schwarz' Lemma.
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Table of contents (3 chapters)
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Front Matter
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Back Matter
Reviews
O. Forster and B. Gilligan
Lectures on Riemann Surfaces
"A very attractive addition to the list in the form of a well-conceived and handsomely produced textbook based on several years' lecturing experience . . . This book deserves very serious consideration as a text for anyone contemplating giving a course on Riemann surfaces. The reviewer is inclined to think that it may well become a favorite."—MATHEMATICAL REVIEWS
Authors and Affiliations
Bibliographic Information
Book Title: Lectures on Riemann Surfaces
Authors: Otto Forster
Series Title: Graduate Texts in Mathematics
DOI: https://doi.org/10.1007/978-1-4612-5961-9
Publisher: Springer New York, NY
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eBook Packages: Springer Book Archive
Copyright Information: Springer-Verlag New York Inc. 1981
Hardcover ISBN: 978-0-387-90617-1Published: 02 November 1981
Softcover ISBN: 978-1-4612-5963-3Published: 12 October 2011
eBook ISBN: 978-1-4612-5961-9Published: 06 December 2012
Series ISSN: 0072-5285
Series E-ISSN: 2197-5612
Edition Number: 1
Number of Pages: VIII, 256
Additional Information: Original German edition was published as volume 184 of the series: Heidelberger Taschenbücher, 1977
Topics: Analysis