Topological, Differential and Conformal Geometry of Surfaces

This book provides an introduction to the main geometric structures carried by compact surfaces,with an emphasis on the classical theory of Riemann surfaces. It covers prerequisites such as differential forms,Poincaré Lemma,Morse Lemma,and more advanced concepts such as Riemannian metrics,isometric torsion-free connection,Ansatz of Koszul,Gauss–Bonnet Theorem,and integrability. It also includes a proof of Chows Theorem on compact holomorphic submanifolds in complex projective spaces.

Format: Paperback / softback
Length: 284 pages
Publication date: 28 October 2021
Publisher: Springer Nature Switzerland AG

This comprehensive book offers an in-depth exploration of the fundamental geometric structures associated with compact surfaces. With a strong emphasis on the classical theory of Riemann surfaces, it provides a solid foundation for understanding these intricate mathematical entities.

The book begins by covering the necessary prerequisites, including differential forms, the Poincaré Lemma, the Morse Lemma, the classification of compact connected oriented surfaces, Stokes Theorem, fixed point theorems, and rigidity theorems. These foundational concepts form the basis for the subsequent discussions.

In addition to the core theoretical aspects, the book presents a novel and engaging presentation of planar hyperbolic geometry. This innovative approach sheds new light on the study of hyperbolic surfaces and their applications.

As the book progresses, it delves into more advanced 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 serve as the building blocks for studying Riemann surfaces and their various properties.

One of the central themes of the book is the Uniformization Theorem for compact surfaces. This theorem, which is an elementary proof, is presented through a property of the energy functional. Numerous other remarkable results are also discussed, including a proof of Chows Theorem on compact holomorphic submanifolds in complex projective spaces.

Written in a clear and concise manner, the book is accessible to undergraduates and graduates interested in the analytic theory of Riemann surfaces. It serves as a valuable resource for both students and researchers in the field.

In conclusion, this book is a must-read for anyone seeking to gain a deeper understanding of the geometric structures carried by compact surfaces. With its comprehensive coverage, innovative presentation, and rigorous analysis, it provides a solid foundation for further research and exploration in this exciting area of mathematics.

Weight: 456g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030890315
Edition number: 1st ed. 2021