Topology and Geometry

Topology and Geometry
Title Topology and Geometry PDF eBook
Author Glen E. Bredon
Publisher Springer Science & Business Media
Total Pages 580
Release 1993-06-24
Genre Mathematics
ISBN 0387979263

Download Topology and Geometry Book in PDF, Epub and Kindle

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

Geometry and Topology

Geometry and Topology
Title Geometry and Topology PDF eBook
Author Miles Reid
Publisher Cambridge University Press
Total Pages 218
Release 2005-11-10
Genre Mathematics
ISBN 9780521848893

Download Geometry and Topology Book in PDF, Epub and Kindle

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.

Geometry and Topology of Manifolds: Surfaces and Beyond

Geometry and Topology of Manifolds: Surfaces and Beyond
Title Geometry and Topology of Manifolds: Surfaces and Beyond PDF eBook
Author Vicente Muñoz
Publisher American Mathematical Soc.
Total Pages 408
Release 2020-10-21
Genre Education
ISBN 1470461323

Download Geometry and Topology of Manifolds: Surfaces and Beyond Book in PDF, Epub and Kindle

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.

Systolic Geometry and Topology

Systolic Geometry and Topology
Title Systolic Geometry and Topology PDF eBook
Author Mikhail Gersh Katz
Publisher American Mathematical Soc.
Total Pages 238
Release 2007
Genre Mathematics
ISBN 0821841777

Download Systolic Geometry and Topology Book in PDF, Epub and Kindle

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.

Differential Geometry and Topology

Differential Geometry and Topology
Title Differential Geometry and Topology PDF eBook
Author Keith Burns
Publisher CRC Press
Total Pages 408
Release 2005-05-27
Genre Mathematics
ISBN 9781584882534

Download Differential Geometry and Topology Book in PDF, Epub and Kindle

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.

Geometry and Topology for Mesh Generation

Geometry and Topology for Mesh Generation
Title Geometry and Topology for Mesh Generation PDF eBook
Author Herbert Edelsbrunner
Publisher Cambridge University Press
Total Pages 206
Release 2001-05-28
Genre Computers
ISBN 9780521793094

Download Geometry and Topology for Mesh Generation Book in PDF, Epub and Kindle

The book combines topics in mathematics (geometry and topology), computer science (algorithms), and engineering (mesh generation). The original motivation for these topics was the difficulty faced (both conceptually and in the technical execution) in any attempt to combine elements of combinatorial and of numerical algorithms. Mesh generation is a topic where a meaningful combination of these different approaches to problem solving is inevitable. The book develops methods from both areas that are amenable to combination, and explains recent breakthrough solutions to meshing that fit into this category.The book should be an ideal graduate text for courses on mesh generation. The specific material is selected giving preference to topics that are elementary, attractive, lend themselves to teaching, useful, and interesting.

Topology and Geometry for Physicists

Topology and Geometry for Physicists
Title Topology and Geometry for Physicists PDF eBook
Author Charles Nash
Publisher Courier Corporation
Total Pages 302
Release 2013-08-16
Genre Mathematics
ISBN 0486318362

Download Topology and Geometry for Physicists Book in PDF, Epub and Kindle

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.