Geometry and Topology of Surfaces
| Author | : Sebastian Baader |
| Publisher | : |
| Total Pages | : |
| Release | : 2021 |
| Genre | : |
| ISBN | : 9783985470006 |
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Topology of Surfaces
| Author | : L.Christine Kinsey |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 290 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 1461208998 |
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" . . . that famous pedagogical method whereby one begins with the general and proceeds to the particular only after the student is too confused to understand even that anymore. " Michael Spivak This text was written as an antidote to topology courses such as Spivak It is meant to provide the student with an experience in geomet describes. ric topology. Traditionally, the only topology an undergraduate might see is point-set topology at a fairly abstract level. The next course the average stu dent would take would be a graduate course in algebraic topology, and such courses are commonly very homological in nature, providing quick access to current research, but not developing any intuition or geometric sense. I have tried in this text to provide the undergraduate with a pragmatic introduction to the field, including a sampling from point-set, geometric, and algebraic topology, and trying not to include anything that the student cannot immediately experience. The exercises are to be considered as an in tegral part of the text and, ideally, should be addressed when they are met, rather than at the end of a block of material. Many of them are quite easy and are intended to give the student practice working with the definitions and digesting the current topic before proceeding. The appendix provides a brief survey of the group theory needed.
Geometry and Topology of Manifolds: Surfaces and Beyond
| Author | : Vicente Muñoz |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 408 |
| Release | : 2020-10-21 |
| Genre | : Education |
| ISBN | : 1470461323 |
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This book represents a novel approach to differential topology. Its main focus is to give a comprehensive introduction to the classification of manifolds, with special attention paid to the case of surfaces, for which the book provides a complete classification from many points of view: topological, smooth, constant curvature, complex, and conformal. Each chapter briefly revisits basic results usually known to graduate students from an alternative perspective, focusing on surfaces. We provide full proofs of some remarkable results that sometimes are missed in basic courses (e.g., the construction of triangulations on surfaces, the classification of surfaces, the Gauss-Bonnet theorem, the degree-genus formula for complex plane curves, the existence of constant curvature metrics on conformal surfaces), and we give hints to questions about higher dimensional manifolds. Many examples and remarks are scattered through the book. Each chapter ends with an exhaustive collection of problems and a list of topics for further study. The book is primarily addressed to graduate students who did take standard introductory courses on algebraic topology, differential and Riemannian geometry, or algebraic geometry, but have not seen their deep interconnections, which permeate a modern approach to geometry and topology of manifolds.
Topological, Differential and Conformal Geometry of Surfaces
| Author | : Norbert A'Campo |
| Publisher | : Springer Nature |
| Total Pages | : 282 |
| Release | : 2021-10-27 |
| Genre | : Mathematics |
| ISBN | : 3030890325 |
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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.
Topology of Surfaces, Knots, and Manifolds
| Author | : Stephan C. Carlson |
| Publisher | : John Wiley & Sons |
| Total Pages | : 178 |
| Release | : 2001-01-10 |
| Genre | : Mathematics |
| ISBN | : |
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This textbook contains ideas and problems involving curves, surfaces, and knots, which make up the core of topology. Carlson (mathematics, Rose-Hulman Institute of Technology) introduces some basic ideas and problems concerning manifolds, especially one- and two- dimensional manifolds. A sampling of topics includes classification of compact surfaces, putting more structure on the surfaces, graphs and topology, and knot theory. It is assumed that the reader has a background in calculus. Annotation copyrighted by Book News Inc., Portland, OR.
Geometry of Surfaces
| Author | : John Stillwell |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 225 |
| Release | : 2012-12-06 |
| Genre | : Mathematics |
| ISBN | : 1461209293 |
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The geometry of surfaces is an ideal starting point for learning geometry, for, among other reasons, the theory of surfaces of constant curvature has maximal connectivity with the rest of mathematics. This text provides the student with the knowledge of a geometry of greater scope than the classical geometry taught today, which is no longer an adequate basis for mathematics or physics, both of which are becoming increasingly geometric. It includes exercises and informal discussions.
Surface Topology
| Author | : P. A. Firby |
| Publisher | : Halsted Press |
| Total Pages | : 224 |
| Release | : 1982 |
| Genre | : Mathematics |
| ISBN | : |
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Mostly Surfaces
| Author | : Richard Evan Schwartz |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 330 |
| Release | : 2011 |
| Genre | : Mathematics |
| ISBN | : 0821853686 |
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The goal of the book is to present a tapestry of ideas from various areas of mathematics in a clear and rigorous yet informal and friendly way. Prerequisites include undergraduate courses in real analysis and in linear algebra, and some knowledge of complex analysis. --from publisher description.
A First Course in Geometric Topology and Differential Geometry
| Author | : Ethan D. Bloch |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 433 |
| Release | : 2011-06-27 |
| Genre | : Mathematics |
| ISBN | : 0817681221 |
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The uniqueness of this text in combining geometric topology and differential geometry lies in its unifying thread: the notion of a surface. With numerous illustrations, exercises and examples, the student comes to understand the relationship of the modern abstract approach to geometric intuition. The text is kept at a concrete level, avoiding unnecessary abstractions, yet never sacrificing mathematical rigor. The book includes topics not usually found in a single book at this level.
Introduction to Topology and Geometry
| Author | : Saul Stahl |
| Publisher | : John Wiley & Sons |
| Total Pages | : 430 |
| Release | : 2014-08-21 |
| Genre | : Mathematics |
| ISBN | : 1118546148 |
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An easily accessible introduction to over three centuries of innovations in geometry Praise for the First Edition “. . . a welcome alternative to compartmentalized treatments bound to the old thinking. This clearly written, well-illustrated book supplies sufficient background to be self-contained.” —CHOICE This fully revised new edition offers the most comprehensive coverage of modern geometry currently available at an introductory level. The book strikes a welcome balance between academic rigor and accessibility, providing a complete and cohesive picture of the science with an unparalleled range of topics. Illustrating modern mathematical topics, Introduction to Topology and Geometry, Second Edition discusses introductory topology, algebraic topology, knot theory, the geometry of surfaces, Riemann geometries, fundamental groups, and differential geometry, which opens the doors to a wealth of applications. With its logical, yet flexible, organization, the Second Edition: • Explores historical notes interspersed throughout the exposition to provide readers with a feel for how the mathematical disciplines and theorems came into being • Provides exercises ranging from routine to challenging, allowing readers at varying levels of study to master the concepts and methods • Bridges seemingly disparate topics by creating thoughtful and logical connections • Contains coverage on the elements of polytope theory, which acquaints readers with an exposition of modern theory Introduction to Topology and Geometry, Second Edition is an excellent introductory text for topology and geometry courses at the upper-undergraduate level. In addition, the book serves as an ideal reference for professionals interested in gaining a deeper understanding of the topic.
