Topological, Differential and Conformal Ge...

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Sprache: Englisch

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Beschreibung

This book provides an introduction to the main geometric structures that are carried by compact surfaces, with an emphasis on the classical theory of Riemann surfaces. It first covers the prerequisites, including the basics of differential forms, the Poincaré Lemma, the Morse Lemma, the classification of compact connected oriented surfaces, Stokes¿ Theorem, fixed point theorems and rigidity theorems. There is also a novel presentation of planar hyperbolic geometry. Moving on to more advanced concepts, it covers topics such as Riemannian metrics, the isometric torsion-free connection on vector fields, the Ansatz of Koszul, the Gauss¿Bonnet Theorem, and integrability. These concepts are then used for the study of Riemann surfaces. One of the focal points is the Uniformization Theorem for compact surfaces, an elementary proof of which is given via a property of the energy functional. Among numerous other results, there is also a proof of Chow¿s Theorem on compact holomorphic submanifolds in complex projective spaces.

Based on lecture courses given by the author, the book will be accessible to undergraduates and graduates interested in the analytic theory of Riemann surfaces.

Über den Autor

Norbert A'Campo earned his doctorate in 1972 from the University of Paris-Sud. He was a Professor of Mathematics in Paris (1974-1982) and in Basel (1982-2011) and was an invited speaker at the International Congress of Mathematicians in 1974. In 1988 he was elected president of the Swiss Mathematical Society and in 2012 he became a fellow of the American Mathematical Society. His research focuses on singularity theory.

Zusammenfassung

Provides an introduction to the classical theory of Riemannian surfaces

Includes an elementary proof of the Uniformization Theorem for compact surfaces

Features numerous interesting topological, analytical and algebraic detours

Inhaltsverzeichnis

-1. Basic Differential Geometry.- 2. The Geometry of Manifolds.- 3. Hyperbolic Geometry.- 4. Some Examples and Sources of Geometry.- 5. Differential Topology of Surfaces.- 6. Riemann Surfaces.- 7. Surfaces of Genus g = 0.- 8. Surfaces with Riemannian Metric.- 9. Outline: Uniformization by Spectral Determinant.- 10. Uniformization by Energy.- 11. Families of Spaces.- 12. Functions on Riemann Surfaces.- 13. Line Bundles and Cohomology.- 14. Moduli Spaces and Teichmüller Spaces.- 15. Dimensions of Spaces of Holomorphic Sections.- 16. The Teichmüller Curve and its Universal Property.- 17. Riemann Surfaces and Algebraic Curves.- 18. The Jacobian of a Riemann Surface.- 19. Special Metrics on J-Surfaces.- 20. The Fundamental Group and Coverings.- A. Reminder: Topology.- References.- Index.

Erscheinungsjahr:	2021
Fachbereich:	Geometrie
Genre:	Mathematik
Rubrik:	Naturwissenschaften & Technik
Medium:	Taschenbuch
Seiten:	296
Reihe:	Universitext
Inhalt:	x 284 S. 3 s/w Illustr. 23 farbige Illustr. 284 p. 26 illus. 23 illus. in color.
ISBN-13:	9783030890315
ISBN-10:	3030890317
Sprache:	Englisch
Ausstattung / Beilage:	Paperback
Einband:	Kartoniert / Broschiert
Autor:	A'Campo, Norbert
Auflage:	1st ed. 2021
Hersteller:	Springer International Publishing Universitext
Maße:	235 x 155 x 17 mm
Von/Mit:	Norbert A'Campo
Erscheinungsdatum:	28.10.2021
Gewicht:	0,452 kg

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