Introduction To Geometry And Topology

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Introduction to Geometry and Topology

Werner Ballmann
Introduction to Geometry and Topology Book PDF

Author : Werner Ballmann
Publisher : Birkhäuser
Page : 169 pages
File Size : 40,8 Mb
Release : 2018-07-18
Category : Mathematics
ISBN : 9783034809832

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Introduction to Geometry and Topology by Werner Ballmann Pdf

This book provides an introduction to topology, differential topology, and differential geometry. It is based on manuscripts refined through use in a variety of lecture courses. The first chapter covers elementary results and concepts from point-set topology. An exception is the Jordan Curve Theorem, which is proved for polygonal paths and is intended to give students a first glimpse into the nature of deeper topological problems. The second chapter of the book introduces manifolds and Lie groups, and examines a wide assortment of examples. Further discussion explores tangent bundles, vector bundles, differentials, vector fields, and Lie brackets of vector fields. This discussion is deepened and expanded in the third chapter, which introduces the de Rham cohomology and the oriented integral and gives proofs of the Brouwer Fixed-Point Theorem, the Jordan-Brouwer Separation Theorem, and Stokes's integral formula. The fourth and final chapter is devoted to the fundamentals of differential geometry and traces the development of ideas from curves to submanifolds of Euclidean spaces. Along the way, the book discusses connections and curvature--the central concepts of differential geometry. The discussion culminates with the Gauß equations and the version of Gauß's theorema egregium for submanifolds of arbitrary dimension and codimension. This book is primarily aimed at advanced undergraduates in mathematics and physics and is intended as the template for a one- or two-semester bachelor's course.

Introduction to Topology and Geometry

Saul Stahl,Catherine Stenson
Introduction to Topology and Geometry Book PDF

Author : Saul Stahl,Catherine Stenson
Publisher : John Wiley & Sons
Page : 536 pages
File Size : 43,6 Mb
Release : 2014-08-21
Category : Mathematics
ISBN : 9781118546147

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Introduction to Topology and Geometry by Saul Stahl,Catherine Stenson Pdf

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.

Geometry with an Introduction to Cosmic Topology

Michael P. Hitchman
Geometry with an Introduction to Cosmic Topology Book PDF

Author : Michael P. Hitchman
Publisher : Jones & Bartlett Learning
Page : 255 pages
File Size : 41,6 Mb
Release : 2009
Category : Mathematics
ISBN : 9780763754570

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Geometry with an Introduction to Cosmic Topology by Michael P. Hitchman Pdf

The content of Geometry with an Introduction to Cosmic Topology is motivated by questions that have ignited the imagination of stargazers since antiquity. What is the shape of the universe? Does the universe have and edge? Is it infinitely big? Dr. Hitchman aims to clarify this fascinating area of mathematics. This non-Euclidean geometry text is organized intothree natural parts. Chapter 1 provides an overview including a brief history of Geometry, Surfaces, and reasons to study Non-Euclidean Geometry. Chapters 2-7 contain the core mathematical content of the text, following the ErlangenProgram, which develops geometry in terms of a space and a group of transformations on that space. Finally chapters 1 and 8 introduce (chapter 1) and explore (chapter 8) the topic of cosmic topology through the geometry learned in the preceding chapters.

Geometry and Topology

Miles Reid,Balazs Szendroi
Geometry and Topology Book PDF

Author : Miles Reid,Balazs Szendroi
Publisher : Cambridge University Press
Page : 218 pages
File Size : 40,9 Mb
Release : 2005-11-10
Category : Mathematics
ISBN : 052184889X

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Geometry and Topology by Miles Reid,Balazs Szendroi Pdf

Geometry aims to describe the world around us. It is central to many branches of mathematics and physics, and offers a whole range of views on the universe. This is an introduction to the ideas of geometry and includes generous helpings of simple explanations and examples. The book is based on many years teaching experience so is thoroughly class-tested, and as prerequisites are minimal, it is suited to newcomers to the subject. There are plenty of illustrations; chapters end with a collection of exercises, and solutions are available for teachers.

Topology and Geometry

Glen E. Bredon
Topology and Geometry Book PDF

Author : Glen E. Bredon
Publisher : Springer Science & Business Media
Page : 571 pages
File Size : 51,6 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9781475768480

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Topology and Geometry by Glen E. Bredon Pdf

This book offers an introductory course in algebraic topology. Starting with general topology, it discusses differentiable manifolds, cohomology, products and duality, the fundamental group, homology theory, and homotopy theory. From the reviews: "An interesting and original graduate text in topology and geometry...a good lecturer can use this text to create a fine course....A beginning graduate student can use this text to learn a great deal of mathematics."—-MATHEMATICAL REVIEWS

An Introduction to the Geometry and Topology of Fluid Flows

Renzo L. Ricca
An Introduction to the Geometry and Topology of Fluid Flows Book PDF

Author : Renzo L. Ricca
Publisher : Springer Science & Business Media
Page : 346 pages
File Size : 49,5 Mb
Release : 2012-12-06
Category : Science
ISBN : 9789401004466

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An Introduction to the Geometry and Topology of Fluid Flows by Renzo L. Ricca Pdf

Leading experts present a unique, invaluable introduction to the study of the geometry and typology of fluid flows. From basic motions on curves and surfaces to the recent developments in knots and links, the reader is gradually led to explore the fascinating world of geometric and topological fluid mechanics. Geodesics and chaotic orbits, magnetic knots and vortex links, continual flows and singularities become alive with more than 160 figures and examples. In the opening article, H. K. Moffatt sets the pace, proposing eight outstanding problems for the 21st century. The book goes on to provide concepts and techniques for tackling these and many other interesting open problems.

Introduction to Topology

Bert Mendelson
Introduction to Topology Book PDF

Author : Bert Mendelson
Publisher : Courier Corporation
Page : 226 pages
File Size : 53,8 Mb
Release : 2012-04-26
Category : Mathematics
ISBN : 9780486135090

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Introduction to Topology by Bert Mendelson Pdf

Concise undergraduate introduction to fundamentals of topology — clearly and engagingly written, and filled with stimulating, imaginative exercises. Topics include set theory, metric and topological spaces, connectedness, and compactness. 1975 edition.

Introduction to Topological Manifolds

John M. Lee
Introduction to Topological Manifolds Book PDF

Author : John M. Lee
Publisher : Springer Science & Business Media
Page : 395 pages
File Size : 48,8 Mb
Release : 2006-04-06
Category : Mathematics
ISBN : 9780387227276

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Introduction to Topological Manifolds by John M. Lee Pdf

Manifolds play an important role in topology, geometry, complex analysis, algebra, and classical mechanics. Learning manifolds differs from most other introductory mathematics in that the subject matter is often completely unfamiliar. This introduction guides readers by explaining the roles manifolds play in diverse branches of mathematics and physics. The book begins with the basics of general topology and gently moves to manifolds, the fundamental group, and covering spaces.

An Introduction to Contact Topology

Hansjörg Geiges
An Introduction to Contact Topology Book PDF

Author : Hansjörg Geiges
Publisher : Cambridge University Press
Page : 8 pages
File Size : 42,6 Mb
Release : 2008-03-13
Category : Mathematics
ISBN : 9781139467957

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An Introduction to Contact Topology by Hansjörg Geiges Pdf

This text on contact topology is a comprehensive introduction to the subject, including recent striking applications in geometric and differential topology: Eliashberg's proof of Cerf's theorem via the classification of tight contact structures on the 3-sphere, and the Kronheimer-Mrowka proof of property P for knots via symplectic fillings of contact 3-manifolds. Starting with the basic differential topology of contact manifolds, all aspects of 3-dimensional contact manifolds are treated in this book. One notable feature is a detailed exposition of Eliashberg's classification of overtwisted contact structures. Later chapters also deal with higher-dimensional contact topology. Here the focus is on contact surgery, but other constructions of contact manifolds are described, such as open books or fibre connected sums. This book serves both as a self-contained introduction to the subject for advanced graduate students and as a reference for researchers.

Geometry and Topology of Manifolds: Surfaces and Beyond

Vicente Muñoz,Ángel González-Prieto,Juan Ángel Rojo
Geometry and Topology of Manifolds Surfaces and Beyond Book PDF

Author : Vicente Muñoz,Ángel González-Prieto,Juan Ángel Rojo
Publisher : American Mathematical Soc.
Page : 408 pages
File Size : 47,9 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.

Topology and Geometry for Physicists

Charles Nash,Siddhartha Sen
Topology and Geometry for Physicists Book PDF

Author : Charles Nash,Siddhartha Sen
Publisher : Courier Corporation
Page : 302 pages
File Size : 49,7 Mb
Release : 2013-08-16
Category : Mathematics
ISBN : 9780486318363

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Topology and Geometry for Physicists by Charles Nash,Siddhartha Sen Pdf

Written by physicists for physics students, this text assumes no detailed background in topology or geometry. Topics include differential forms, homotopy, homology, cohomology, fiber bundles, connection and covariant derivatives, and Morse theory. 1983 edition.

Knots, Molecules, and the Universe

Erica Flapan
Knots Molecules and the Universe Book PDF

Author : Erica Flapan
Publisher : American Mathematical Soc.
Page : 386 pages
File Size : 50,7 Mb
Release : 2015-12-22
Category : Algebraic topology
ISBN : 9781470425357

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Knots, Molecules, and the Universe by Erica Flapan Pdf

This book is an elementary introduction to geometric topology and its applications to chemistry, molecular biology, and cosmology. It does not assume any mathematical or scientific background, sophistication, or even motivation to study mathematics. It is meant to be fun and engaging while drawing students in to learn about fundamental topological and geometric ideas. Though the book can be read and enjoyed by nonmathematicians, college students, or even eager high school students, it is intended to be used as an undergraduate textbook. The book is divided into three parts corresponding to the three areas referred to in the title. Part 1 develops techniques that enable two- and three-dimensional creatures to visualize possible shapes for their universe and to use topological and geometric properties to distinguish one such space from another. Part 2 is an introduction to knot theory with an emphasis on invariants. Part 3 presents applications of topology and geometry to molecular symmetries, DNA, and proteins. Each chapter ends with exercises that allow for better understanding of the material. The style of the book is informal and lively. Though all of the definitions and theorems are explicitly stated, they are given in an intuitive rather than a rigorous form, with several hundreds of figures illustrating the exposition. This allows students to develop intuition about topology and geometry without getting bogged down in technical details.

Handbook of Geometric Topology

R.B. Sher,R.J. Daverman
Handbook of Geometric Topology Book PDF

Author : R.B. Sher,R.J. Daverman
Publisher : Elsevier
Page : 1145 pages
File Size : 43,9 Mb
Release : 2001-12-20
Category : Mathematics
ISBN : 9780080532851

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Handbook of Geometric Topology by R.B. Sher,R.J. Daverman Pdf

Geometric Topology is a foundational component of modern mathematics, involving the study of spacial properties and invariants of familiar objects such as manifolds and complexes. This volume, which is intended both as an introduction to the subject and as a wide ranging resouce for those already grounded in it, consists of 21 expository surveys written by leading experts and covering active areas of current research. They provide the reader with an up-to-date overview of this flourishing branch of mathematics.

Differential Geometry and Topology

Keith Burns,Marian Gidea
Differential Geometry and Topology Book PDF

Author : Keith Burns,Marian Gidea
Publisher : CRC Press
Page : 400 pages
File Size : 53,7 Mb
Release : 2005-05-27
Category : Mathematics
ISBN : 9781420057539

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Differential Geometry and Topology by Keith Burns,Marian Gidea Pdf

Accessible, concise, and self-contained, this book offers an outstanding introduction to three related subjects: differential geometry, differential topology, and dynamical systems. Topics of special interest addressed in the book include Brouwer's fixed point theorem, Morse Theory, and the geodesic flow. Smooth manifolds, Riemannian metrics, affine connections, the curvature tensor, differential forms, and integration on manifolds provide the foundation for many applications in dynamical systems and mechanics. The authors also discuss the Gauss-Bonnet theorem and its implications in non-Euclidean geometry models. The differential topology aspect of the book centers on classical, transversality theory, Sard's theorem, intersection theory, and fixed-point theorems. The construction of the de Rham cohomology builds further arguments for the strong connection between the differential structure and the topological structure. It also furnishes some of the tools necessary for a complete understanding of the Morse theory. These discussions are followed by an introduction to the theory of hyperbolic systems, with emphasis on the quintessential role of the geodesic flow. The integration of geometric theory, topological theory, and concrete applications to dynamical systems set this book apart. With clean, clear prose and effective examples, the authors' intuitive approach creates a treatment that is comprehensible to relative beginners, yet rigorous enough for those with more background and experience in the field.

The Geometry and Topology of Coxeter Groups

Michael Davis
The Geometry and Topology of Coxeter Groups Book PDF

Author : Michael Davis
Publisher : Princeton University Press
Page : 601 pages
File Size : 50,5 Mb
Release : 2008
Category : Mathematics
ISBN : 9780691131382

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The Geometry and Topology of Coxeter Groups by Michael Davis Pdf

The Geometry and Topology of Coxeter Groups is a comprehensive and authoritative treatment of Coxeter groups from the viewpoint of geometric group theory. Groups generated by reflections are ubiquitous in mathematics, and there are classical examples of reflection groups in spherical, Euclidean, and hyperbolic geometry. Any Coxeter group can be realized as a group generated by reflection on a certain contractible cell complex, and this complex is the principal subject of this book. The book explains a theorem of Moussong that demonstrates that a polyhedral metric on this cell complex is nonpositively curved, meaning that Coxeter groups are "CAT(0) groups." The book describes the reflection group trick, one of the most potent sources of examples of aspherical manifolds. And the book discusses many important topics in geometric group theory and topology, including Hopf's theory of ends; contractible manifolds and homology spheres; the Poincaré Conjecture; and Gromov's theory of CAT(0) spaces and groups. Finally, the book examines connections between Coxeter groups and some of topology's most famous open problems concerning aspherical manifolds, such as the Euler Characteristic Conjecture and the Borel and Singer conjectures.