Introduction to Smooth Manifolds: 218 (Graduate Texts in Mathematics) - Hardcover

Lee, John

Hardcover

ISBN 10: 1441999817 ISBN 13: 9781441999818

Verlag: Springer, 2012

Inhaltsangabe

<P>THIS BOOK IS AN INTRODUCTORY GRADUATE-LEVEL TEXTBOOK ON THE THEORY OF SMOOTH MANIFOLDS. ITS GOAL IS TO FAMILIARIZE STUDENTS WITH THE TOOLS THEY WILL NEED IN ORDER TO USE MANIFOLDS IN MATHEMATICAL OR SCIENTIFIC RESEARCH--- SMOOTH STRUCTURES, TANGENT VECTORS AND COVECTORS, VECTOR BUNDLES, IMMERSED AND EMBEDDED SUBMANIFOLDS, TENSORS, DIFFERENTIAL FORMS, DE RHAM COHOMOLOGY, VECTOR FIELDS, FLOWS, FOLIATIONS, LIE DERIVATIVES, LIE GROUPS, LIE ALGEBRAS, AND MORE. THE APPROACH IS AS CONCRETE AS POSSIBLE, WITH PICTURES AND INTUITIVE DISCUSSIONS OF HOW ONE SHOULD THINK GEOMETRICALLY ABOUT THE ABSTRACT CONCEPTS, WHILE MAKING FULL USE OF THE POWERFUL TOOLS THAT MODERN MATHEMATICS HAS TO OFFER.</P><P><P>THIS SECOND EDITION HAS BEEN EXTENSIVELY REVISED AND CLARIFIED, AND THE TOPICS HAVE BEEN SUBSTANTIALLY REARRANGED. THE BOOK NOW INTRODUCES THE TWO MOST IMPORTANT ANALYTIC TOOLS, THE RANK THEOREM AND THE FUNDAMENTAL THEOREM ON FLOWS, MUCH EARLIER SO THAT THEY CAN BE USED THROUGHOUT THE BOOK. A FEW NEW TOPICS HAVE BEEN ADDED, NOTABLY SARD’S THEOREM AND TRANSVERSALITY, A PROOF THAT INFINITESIMAL LIE GROUP ACTIONS GENERATE GLOBAL GROUP ACTIONS, A MORE THOROUGH STUDY OF FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS, A BRIEF TREATMENT OF DEGREE THEORY FOR SMOOTH MAPS BETWEEN COMPACT MANIFOLDS, AND AN INTRODUCTION TO CONTACT STRUCTURES.</P><P><P>PREREQUISITES INCLUDE A SOLID ACQUAINTANCE WITH GENERAL TOPOLOGY, THE FUNDAMENTAL GROUP, AND COVERING SPACES, AS WELL AS BASIC UNDERGRADUATE LINEAR ALGEBRA AND REAL ANALYSIS.</P>

Über diesen Titel

Críticas

From the reviews of the second edition:

“It starts off with five chapters covering basics on smooth manifolds up to submersions, immersions, embeddings, and of course submanifolds. ... the book under review is laden with excellent exercises that significantly further the reader’s understanding of the material, and Lee takes great pains to motivate everything well all the way through ... . a fine graduate-level text for differential geometers as well as people like me, fellow travelers who always wish they knew more about such a beautiful subject.” (Michael Berg, MAA Reviews, October, 2012)

Reseña del editor

This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research--- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer.

This second edition has been extensively revised and clarified, and the topics have been substantially rearranged. The book now introduces the two most important analytic tools, the rank theorem and the fundamental theorem on flows, much earlier so that they can be used throughout the book. A few new topics have been added, notably Sard’s theorem and transversality, a proof that infinitesimal Lie group actions generate global group actions, a more thorough study of first-order partial differential equations, a brief treatment of degree theory for smooth maps between compact manifolds, and an introduction to contact structures.

Prerequisites include a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis.

Details zur Ausgabe

• Verlag: Springer
• Erscheinungsdatum: 2012
• ISBN 10: 1441999817
• ISBN 13: 9781441999818
• Einband: Tapa dura
• Sprache: Englisch
• Anzahl der Seiten: 726