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Catalogue Information
Field name
Details
Dewey Class
515.223
Title
Lectures on Riemann Surfaces ([EBook]) / by Otto Forster.
Author
Forster, Otto , 1937-
Other name(s)
SpringerLink (Online service)
Publication
New York, NY : Springer , 1981
Physical Details
VIII, 256 pages : online resource.
Series
Graduate texts in mathematics
0072-5285 ; ; 81
ISBN
9781461259619
Summary Note
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.:
Contents note
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.
System details note
Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site
http://dx.doi.org/10.1007/978-1-4612-5961-9
Links to Related Works
Subject References:
Analysis (Mathematics)
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Mathematical analysis
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Riemann surfaces
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Authors:
Forster, Otto 1937-
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Corporate Authors:
SpringerLink (Online service)
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Series:
Graduate texts in mathematics
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GTM
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Classification:
515.223
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