Lee J. - Introduction to Smooth Manifolds.pdf - Topology - Kuya

INTRODUCTION TO

SMOOTH MANIFOLDS

by John M. Lee

University of Washington

Department of Mathematics

John M. Lee

Introduction to

Smooth Manifolds

Version 3.0

December 31, 2000

John M. Lee

University of Washington

Department of Mathematics

Seattle, WA 98195-4350

USA

lee@math.washington.edu

http://www.math.washington.edu/~lee

2000 by John M. Lee

Preface

This book is an introductory graduate-level textbook on the theory of

smooth manifolds, for students who already have a solid acquaintance with

general topology, the fundamental group, and covering spaces, as well as

basic undergraduate linear algebra and real analysis. It is a natural sequel

to my earlier book on topological manifolds [Lee00].

This subject is often called \dierential geometry." I have mostly avoided

this term, however, because it applies more properly to the study of smooth

manifolds endowed with some extra structure, such as a Riemannian met-

ric, a symplectic structure, a Lie group structure, or a foliation, and of the

properties that are invariant under maps that preserve the structure. Al-

though I do treat all of these subjects in this book, they are treated more as

interesting examples to which to apply the general theory than as objects

of study in their own right. A student who nishes this book should be

well prepared to go on to study any of these specialized subjects in much

greater depth.

The book is organized roughly as follows. Chapters 1 through 4 are

mainly denitions. It is the bane of this subject that there are so many

denitions that must be piled on top of one another before anything in-

teresting can be said, much less proved. I have tried, nonetheless, to bring

in signicant applications as early and as often as possible. The rst one

comes at the end of Chapter 4, where I show how to generalize the classical

theory of line integrals to manifolds.

The next three chapters, 5 through 7, present the rst of four major

foundational theorems on which all of smooth manifolds theory rests|the

inverse function theorem|and some applications of it: to submanifold the-

Lee J. - Introduction to Smooth Manifolds.pdf