Overview
- 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
Part of the book series: Universitext (UTX)
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Table of contents (20 chapters)
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Front Matter
Keywords
About this book
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.
Authors and Affiliations
About the author
Bibliographic Information
Book Title: Topological, Differential and Conformal Geometry of Surfaces
Authors: Norbert A'Campo
Series Title: Universitext
DOI: https://doi.org/10.1007/978-3-030-89032-2
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
Softcover ISBN: 978-3-030-89031-5Published: 28 October 2021
eBook ISBN: 978-3-030-89032-2Published: 27 October 2021
Series ISSN: 0172-5939
Series E-ISSN: 2191-6675
Edition Number: 1
Number of Pages: X, 284
Number of Illustrations: 3 b/w illustrations, 23 illustrations in colour
Topics: Differential Geometry, Algebraic Topology, Algebraic Geometry