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Lectures on Riemann Surfaces

  • Textbook
  • © 1981


Part of the book series: Graduate Texts in Mathematics (GTM, volume 81)

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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)


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

  • Mathematishes Institut, Univerität München, Federal Republic Of Germany

    Otto Forster

Bibliographic Information

  • Book Title: Lectures on Riemann Surfaces

  • Authors: Otto Forster

  • Series Title: Graduate Texts in Mathematics

  • DOI:

  • Publisher: Springer New York, NY

  • 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

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