Geometry And Topology Of Surfaces

Author : Vicente Muñoz,Ángel González-Prieto,Juan Ángel Rojo
Publisher : American Mathematical Soc.
Page : 408 pages
File Size : 42,9 Mb
Release : 2020-10-21
Category : Education
ISBN : 9781470461324

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Geometry and Topology of Manifolds: Surfaces and Beyond by Vicente Muñoz,Ángel González-Prieto,Juan Ángel Rojo Pdf

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.

Author : Sebastian Baader
Publisher : Unknown
Page : 128 pages
File Size : 43,5 Mb
Release : 2021
Category : Electronic
ISBN : 3985470006

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Geometry and Topology of Surfaces by Sebastian Baader Pdf

Author : L.Christine Kinsey
Publisher : Springer Science & Business Media
Page : 290 pages
File Size : 43,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461208990

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Topology of Surfaces by L.Christine Kinsey Pdf

" . . . 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.

Author : Norbert A'Campo
Publisher : Springer Nature
Page : 282 pages
File Size : 41,8 Mb
Release : 2021-10-27
Category : Mathematics
ISBN : 9783030890322

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Topological, Differential and Conformal Geometry of Surfaces by Norbert A'Campo Pdf

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.

Author : John Stillwell
Publisher : Springer Science & Business Media
Page : 244 pages
File Size : 42,5 Mb
Release : 1995-02-03
Category : Mathematics
ISBN : 0387977430

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Geometry of Surfaces by John Stillwell Pdf

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.

Author : Richard Evan Schwartz
Publisher : American Mathematical Soc.
Page : 330 pages
File Size : 47,5 Mb
Release : 2011
Category : Mathematics
ISBN : 9780821853689

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Mostly Surfaces by Richard Evan Schwartz Pdf

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.

Author : Ethan D. Bloch
Publisher : Springer Science & Business Media
Page : 433 pages
File Size : 47,6 Mb
Release : 2011-06-27
Category : Mathematics
ISBN : 9780817681227

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A First Course in Geometric Topology and Differential Geometry by Ethan D. Bloch Pdf

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.

Author : P. A. Firby,Cyril F. Gardiner
Publisher : Halsted Press
Page : 224 pages
File Size : 48,6 Mb
Release : 1982
Category : Mathematics
ISBN : UOM:39015015610564

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Surface Topology by P. A. Firby,Cyril F. Gardiner Pdf

Author : Stephan C. Carlson
Publisher : John Wiley & Sons
Page : 178 pages
File Size : 51,5 Mb
Release : 2001-01-10
Category : Mathematics
ISBN : UOM:39015049686283

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Topology of Surfaces, Knots, and Manifolds by Stephan C. Carlson Pdf

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.

Author : A. T. Fomenko,A. A. Tuzhilin
Publisher : American Mathematical Soc.
Page : 156 pages
File Size : 47,9 Mb
Release : 2005
Category : Minimal surfaces
ISBN : 9780821837917

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Elements of the geometry and topology of minimal surfaces in three-dimensional space by A. T. Fomenko,A. A. Tuzhilin Pdf

This book grew out of lectures presented to students of mathematics, physics, and mechanics by A. T. Fomenko at Moscow University, under the auspices of the Moscow Mathematical Society. The book describes modern and visual aspects of the theory of minimal, two-dimensional surfaces in three-dimensional space. The main topics covered are: topological properties of minimal surfaces, stable and unstable minimal films, classical examples, the Morse-Smale index of minimal two-surfaces in Euclidean space, and minimal films in Lobachevskian space. Requiring only a standard first-year calculus and elementary notions of geometry, this book brings the reader rapidly into this fascinating branch of modern geometry.

Author : P A Firby,C F Gardiner
Publisher : Elsevier
Page : 261 pages
File Size : 52,8 Mb
Release : 2001-06-01
Category : Mathematics
ISBN : 9780857099679

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Surface Topology by P A Firby,C F Gardiner Pdf

This updated and revised edition of a widely acclaimed and successful text for undergraduates examines topology of recent compact surfaces through the development of simple ideas in plane geometry. Containing over 171 diagrams, the approach allows for a straightforward treatment of its subject area. It is particularly attractive for its wealth of applications and variety of interactions with branches of mathematics, linked with surface topology, graph theory, group theory, vector field theory, and plane Euclidean and non-Euclidean geometry. Examines topology of recent compact surfaces through the development of simple ideas in plane geometry Contains a wealth of applications and a variety of interactions with branches of mathematics, linked with surface topology, graph theory, group theory, vector field theory, and plane Euclidean and non-Euclidean geometry

Author : P. A. Firby,Cyril F. Gardiner
Publisher : Unknown
Page : 220 pages
File Size : 49,7 Mb
Release : 1982
Category : Mathematics
ISBN : MINN:31951000556976L

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Surface Topology by P. A. Firby,Cyril F. Gardiner Pdf

Author : Francis Bonahon
Publisher : American Mathematical Soc.
Page : 403 pages
File Size : 50,9 Mb
Release : 2009-07-14
Category : Mathematics
ISBN : 9780821848166

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Low-Dimensional Geometry by Francis Bonahon Pdf

The study of 3-dimensional spaces brings together elements from several areas of mathematics. The most notable are topology and geometry, but elements of number theory and analysis also make appearances. In the past 30 years, there have been striking developments in the mathematics of 3-dimensional manifolds. This book aims to introduce undergraduate students to some of these important developments. Low-Dimensional Geometry starts at a relatively elementary level, and its early chapters can be used as a brief introduction to hyperbolic geometry. However, the ultimate goal is to describe the very recently completed geometrization program for 3-dimensional manifolds. The journey to reach this goal emphasizes examples and concrete constructions as an introduction to more general statements. This includes the tessellations associated to the process of gluing together the sides of a polygon. Bending some of these tessellations provides a natural introduction to 3-dimensional hyperbolic geometry and to the theory of kleinian groups, and it eventually leads to a discussion of the geometrization theorems for knot complements and 3-dimensional manifolds. This book is illustrated with many pictures, as the author intended to share his own enthusiasm for the beauty of some of the mathematical objects involved. However, it also emphasizes mathematical rigor and, with the exception of the most recent research breakthroughs, its constructions and statements are carefully justified.

Author : Mikhail Gersh Katz
Publisher : American Mathematical Soc.
Page : 238 pages
File Size : 52,8 Mb
Release : 2007
Category : Geometry, Algebraic
ISBN : 9780821841778

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Systolic Geometry and Topology by Mikhail Gersh Katz Pdf

The systole of a compact metric space $X$ is a metric invariant of $X$, defined as the least length of a noncontractible loop in $X$. When $X$ is a graph, the invariant is usually referred to as the girth, ever since the 1947 article by W. Tutte. The first nontrivial results for systoles of surfaces are the two classical inequalities of C. Loewner and P. Pu, relying on integral-geometric identities, in the case of the two-dimensional torus and real projective plane, respectively. Currently, systolic geometry is a rapidly developing field, which studies systolic invariants in their relation to other geometric invariants of a manifold. This book presents the systolic geometry of manifolds and polyhedra, starting with the two classical inequalities, and then proceeding to recent results, including a proof of M. Gromov's filling area conjecture in a hyperelliptic setting. It then presents Gromov's inequalities and their generalisations, as well as asymptotic phenomena for systoles of surfaces of large genus, revealing a link both to ergodic theory and to properties of congruence subgroups of arithmetic groups. The author includes results on the systolic manifestations of Massey products, as well as of the classical Lusternik-Schnirelmann category.

Author : J Scott Carter
Publisher : World Scientific
Page : 338 pages
File Size : 51,6 Mb
Release : 1995-05-11
Category : Mathematics
ISBN : 9789814501231

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How Surfaces Intersect in Space by J Scott Carter Pdf

This marvelous book of pictures illustrates the fundamental concepts of geometric topology in a way that is very friendly to the reader. The first chapter discusses the meaning of surface and space and gives the classification of orientable surfaces. In the second chapter we are introduced to the Möbius band and surfaces that can be constructed from this non-orientable piece of fabric. In chapter 3, we see how curves can fit in surfaces and how surfaces can fit into spaces with these curves on their boundary. Basic applications to knot theory are discussed and four-dimensional space is introduced. In Chapter 4 we learn about some 3-dimensional spaces and surfaces that sit inside them. These surfaces help us imagine the structures of the larger space. Chapter 5 is completely new! It contains recent results of Cromwell, Izumiya and Marar. One of these results is a formula relating the rank of a surface to the number of triple points. The other major result is a collection of examples of surfaces in 3-space that have one triple point and 6 branch points. These are beautiful generalizations of the Steiner Roman surface. Chapter 6 reviews the movie technique for examining surfaces in 4-dimensional space. Various movies of the Klein bottle are presented, and the Carter-Saito movie move theorem is explained. The author shows us how to turn the 2-sphere inside out by means of these movie moves and this illustration alone is well worth the price of the book! In the last chapter higher dimensional spaces are examined from an elementary point of view. This is a guide book to a wide variety of topics. It will be of value to anyone who wants to understand the subject by way of examples. Undergraduates, beginning graduate students, and non-professionals will profit from reading the book and from just looking at the pictures. Contents:Front MatterSurface and SpaceNon-orientable SurfacesCurves and KnotsOther Three Dimensional SpacesRelationshipsSurfaces in 4-DimensionsHigher Dimensional SpacesBack Matter Readership: Undergraduates, graduates and mathematicians. keywords:Moving Surfaces;Surfaces;Triple Point;Branch Points “In this excellent book the author teaches us to see a bit more than it meets our eyes. Without hurry he introduces us to the world of topological images. Step by step the reader learns the beauty of topological vision. Surfaces and their intersections, curves and knots, three-dimensional manifolds, surfaces in dimension 4 etc., all these material are presented in an informal easy way, making the exposition available to undergraduate students. As to the pictures, they are really delightful. I especially enjoyed the movies of surfaces and movie moves. On the whole the book is a successful attempt of an introduction to topology focusing on its spirit and skipping its technical side.” Vladimir Turaev Directeur de Recherche au CNRS “This book is a definite enrichment to the literature in low-dimensional topology.” Mathematics Abstracts