Overview

This book grew out of lectures on Riemann surfaces given by Otto Forster at the universities of Munich, Regensburg, and Munster. It provides a concise modern introduction to this rewarding subject, as well as presenting methods used in the study of complex manifolds in the special case of complex dimension one. The material corresponds roughly to three semesters of lectures, arranged in a flexible sequence involving a minimum of prerequisites. In the first chapter, the author considers Riemann surfaces as covering spaces, develops the pertinent basics of topology, and focuses on algebraic functions. The next chapter is devoted to the theory of compact Riemann surfaces and cohomology groups, with the main classical results (including the Riemann-Roch theorem, Abel's theorem, and Jacobi's inversion problem). The final section covers the Riemann mapping theorem for simply connected Riemann surfaces, and the main theorems of Behnke-Stein for non-compact Riemann surfaces (the Runge approximation theorem and the theorems of Mittag-Leffler and Weierstrass). The value of this translation is enhanced by newly prepared exercises.

Full Product Details

Author:   Otto Forster ,  Bruce Gilligan
Publisher:   Springer-Verlag New York Inc.
Imprint:   Springer-Verlag New York Inc.
Edition:   1st ed. 1981. Corr. 4th printing 1999
Volume:   81
Dimensions:   Width: 15.60cm , Height: 1.50cm , Length: 23.40cm
Weight:   1.220kg
ISBN:  

9780387906171

ISBN 10:   0387906177
Pages:   256
Publication Date:   02 November 1981
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Out of print, replaced by POD  
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Table of Contents

1 Covering Spaces.- §1. The Definition of Riemann Surfaces.- §2. Elementary Properties of Holomorphic Mappings.- §3. Homotopy of Curves. The Fundamental Group.- §4. Branched and Unbranched Coverings.- §5. The Universal Covering and Covering Transformations.- §6. Sheaves.- §7. Analytic Continuation.- §8. Algebraic Functions.- §9. Differential Forms.- §10. The Integration of Differential Forms.- §11. Linear Differential Equations.- 2 Compact Riemann Surfaces.- §12. Cohomology Groups.- §13. Dolbeault’s Lemma.- §14. A Finiteness Theorem.- §15. The Exact Cohomology Sequence.- §16. The Riemann-Roch Theorem.- §17. The Serre Duality Theorem.- §18. Functions and Differential Forms with Prescribed Principal Parts.- §19. Harmonic Differential Forms.- §20. Abel’s Theorem.- §21. The Jacobi Inversion Problem.- 3 Non-compact Riemann Surfaces.- §22. The Dirichlet Boundary Value Problem.- §23. Countable Topology.- §24. Weyl’s Lemma.- §25. The Runge Approximation Theorem.- §26. The Theorems of Mittag-Leffler and Weierstrass.- §27. The Riemann Mapping Theorem.- §28. Functions with Prescribed Summands of Automorphy.- §29. Line and Vector Bundles.- §30. The Triviality of Vector Bundles.- §31. The Riemann-Hilbert Problem.- A. Partitions of Unity.- B. Topological Vector Spaces.- References.- Symbol Index.- Author and Subject Index.

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

O. Forster and B. Gilligan <p>Lectures on Riemann Surfaces <p> 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. a MATHEMATICAL 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

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