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＜E-Book＞
Topological, Differential and Conformal Geometry of Surfaces / by Norbert A'Campo
(Universitext. ISSN:21916675)

Edition	1st ed. 2021.
Publisher	Cham : Springer International Publishing : Imprint: Springer
Year	2021
Size	X, 284 p. 26 illus., 23 illus. in color : online resource
Authors	*A'Campo, Norbert author SpringerLink (Online service)
Subjects	LCSH:Geometry, Differential LCSH:Algebraic topology LCSH:Algebraic geometry FREE:Differential Geometry FREE:Algebraic Topology FREE:Algebraic Geometry
Notes	-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 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 HTTP:URL=https://doi.org/10.1007/978-3-030-89032-2

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