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Cover: 9781461259633 | Lectures on Riemann Surfaces | Otto Forster | Taschenbuch | Paperback
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Lectures on Riemann Surfaces
Taschenbuch von Otto Forster
Sprache: Englisch

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Beschreibung
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.
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.
Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 2011
Fachbereich: Analysis
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 268
Reihe: Graduate Texts in Mathematics
Inhalt: viii
256 S.
ISBN-13: 9781461259633
ISBN-10: 1461259630
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Forster, Otto
Übersetzung: Gilligan, Bruce
Auflage: Softcover reprint of the original 1st ed. 1981
Hersteller: Springer US
Springer New York
Graduate Texts in Mathematics
Maße: 235 x 155 x 15 mm
Von/Mit: Otto Forster
Erscheinungsdatum: 12.10.2011
Gewicht: 0,411 kg
preigu-id: 105630134
Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 2011
Fachbereich: Analysis
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 268
Reihe: Graduate Texts in Mathematics
Inhalt: viii
256 S.
ISBN-13: 9781461259633
ISBN-10: 1461259630
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Forster, Otto
Übersetzung: Gilligan, Bruce
Auflage: Softcover reprint of the original 1st ed. 1981
Hersteller: Springer US
Springer New York
Graduate Texts in Mathematics
Maße: 235 x 155 x 15 mm
Von/Mit: Otto Forster
Erscheinungsdatum: 12.10.2011
Gewicht: 0,411 kg
preigu-id: 105630134
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