Topological, differential and conformal geometry of surfaces / Norbert A'Campo.
A'Campo, N. (Norbert), author.
2021
QA649 .A33 2021
Available Online

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Title Topological, differential and conformal geometry of surfaces / Norbert A'Campo.
Author A'Campo, N. (Norbert), author.
ISBN 9783030890322 (electronic bk.)
3030890325 (electronic bk.)
9783030890315
3030890317
DOI https://doi.org/10.1007/978-3-030-89032-2
Published Cham : Springer, [2021]
Copyright ©2021
Language English
Description 1 online resource : illustrations (chiefly color)
Item Number 10.1007/978-3-030-89032-2 doi
Call Number QA649 .A33 2021
Dewey Decimal Classification 516.3/6
Summary 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 Poincare 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 GaussBonnet 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 Chows 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.
Bibliography, etc. Note Includes bibliographical references and index.
Access Note Access limited to authorized users.
Digital File Characteristics text file
PDF
Source of Description Online resource; title from PDF title page (SpringerLink, viewed November 4, 2021).
Series Universitext. 2191-6675
Available in Other Form Topological, differential and conformal geometry of surfaces.
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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 Teichmuller Spaces
15. Dimensions of Spaces of Holomorphic Sections
16. The Teichmuller 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.

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