John M. Lee Introduction to Smooth Manifolds.pdf

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Introduction to Smooth Manifolds

Preface

Prerequisites

Exercises and Problems

About the Second Edition

Acknowledgments

Contents

Chapter 1: Smooth Manifolds

Topological Manifolds

Coordinate Charts

Examples of Topological Manifolds

Topological Properties of Manifolds

Connectivity

Local Compactness and Paracompactness

Fundamental Groups of Manifolds

Smooth Structures

Local Coordinate Representations

Examples of Smooth Manifolds

The Einstein Summation Convention

More Examples

Manifolds with Boundary

Smooth Structures on Manifolds with Boundary

Problems

Chapter 2: Smooth Maps

Smooth Functions and Smooth Maps

Smooth Functions on Manifolds

Smooth Maps Between Manifolds

Diffeomorphisms

Partitions of Unity

Applications of Partitions of Unity

Problems

Chapter 3: Tangent Vectors

Tangent Vectors

Geometric Tangent Vectors

Tangent Vectors on Manifolds

The Differential of a Smooth Map

Computations in Coordinates

The Differential in Coordinates

Change of Coordinates

The Tangent Bundle

Velocity Vectors of Curves

Alternative Deﬁnitions of the Tangent Space

Tangent Vectors as Derivations of the Space of Germs

Tangent Vectors as Equivalence Classes of Curves

Tangent Vectors as Equivalence Classes of n-Tuples

Categories and Functors

Problems

Chapter 4: Submersions, Immersions, and Embeddings

Maps of Constant Rank

Local Diffeomorphisms

The Rank Theorem

The Rank Theorem for Manifolds with Boundary

Embeddings

Submersions

Smooth Covering Maps

Problems

Chapter 5: Submanifolds

Embedded Submanifolds

Slice Charts for Embedded Submanifolds

Level Sets

Immersed Submanifolds

Restricting Maps to Submanifolds

Uniqueness of Smooth Structures on Submanifolds

Extending Functions from Submanifolds

The Tangent Space to a Submanifold

Submanifolds with Boundary

Problems

Chapter 6: Sard's Theorem

Sets of Measure Zero

Sard's Theorem

The Whitney Embedding Theorem

The Whitney Approximation Theorems

Tubular Neighborhoods

Smooth Approximation of Maps Between Manifolds

Transversality

Problems

Chapter 7: Lie Groups

Basic Deﬁnitions

Lie Group Homomorphisms

The Universal Covering Group

Lie Subgroups

Group Actions and Equivariant Maps

Equivariant Maps

Semidirect Products

Representations

Problems

Chapter 8: Vector Fields

Vector Fields on Manifolds

Local and Global Frames

Vector Fields as Derivations of Cinfty(M)

Vector Fields and Smooth Maps

Vector Fields and Submanifolds

Lie Brackets

The Lie Algebra of a Lie Group

Induced Lie Algebra Homomorphisms

The Lie Algebra of a Lie Subgroup

Problems

Chapter 9: Integral Curves and Flows

Integral Curves

Flows

The Fundamental Theorem on Flows

Complete Vector Fields

Flowouts

Regular Points and Singular Points

Flows and Flowouts on Manifolds with Boundary

Lie Derivatives

Commuting Vector Fields

Commuting Frames

Time-Dependent Vector Fields

First-Order Partial Differential Equations

Linear Equations

Quasilinear Equations

Problems

Chapter 10: Vector Bundles

Vector Bundles

Local and Global Sections of Vector Bundles

Local and Global Frames

Bundle Homomorphisms

Subbundles

Fiber Bundles

Problems

Chapter 11: The Cotangent Bundle

Covectors

Tangent Covectors on Manifolds

Covector Fields

Coframes

The Differential of a Function

Pullbacks of Covector Fields

Restricting Covector Fields to Submanifolds

Line Integrals

Conservative Covector Fields

Problems

Chapter 12: Tensors

Multilinear Algebra

Abstract Tensor Products of Vector Spaces

Covariant and Contravariant Tensors on a Vector Space

Symmetric and Alternating Tensors

Symmetric Tensors

Alternating Tensors

Tensors and Tensor Fields on Manifolds

Pullbacks of Tensor Fields

Lie Derivatives of Tensor Fields

Problems

Chapter 13: Riemannian Metrics

Riemannian Manifolds

Pullback Metrics

Riemannian Submanifolds

The Normal Bundle

The Riemannian Distance Function

The Tangent-Cotangent Isomorphism

Pseudo-Riemannian Metrics

Problems

Chapter 14: Differential Forms

The Algebra of Alternating Tensors

Elementary Alternating Tensors

The Wedge Product

Interior Multiplication

Differential Forms on Manifolds

Exterior Derivatives

Exterior Derivatives and Vector Calculus in R3

An Invariant Formula for the Exterior Derivative

Lie Derivatives of Differential Forms

Problems

Chapter 15: Orientations

Orientations of Vector Spaces

Orientations of Manifolds

Orientations of Hypersurfaces

Boundary Orientations

The Riemannian Volume Form

Hypersurfaces in Riemannian Manifolds

Orientations and Covering Maps

The Orientation Covering

Problems

Chapter 16: Integration on Manifolds

The Geometry of Volume Measurement

Integration of Differential Forms

Integration on Manifolds

Integration on Lie Groups

Stokes's Theorem

Manifolds with Corners

Integration on Riemannian Manifolds

Integration of Functions on Riemannian Manifolds

The Divergence Theorem

Surface Integrals

Densities

The Riemannian Density

Problems

Chapter 17: De Rham Cohomology

The de Rham Cohomology Groups

Elementary Computations

Homotopy Invariance

Computations Using Homotopy Invariance

The Mayer-Vietoris Theorem

Computations Using the Mayer-Vietoris Theorem

Degree Theory

Proof of the Mayer-Vietoris Theorem

Problems

Chapter 18: The de Rham Theorem

Singular Homology

Singular Cohomology

Smooth Singular Homology

The de Rham Theorem

Problems

Chapter 19: Distributions and Foliations

Distributions and Involutivity

Integral Manifolds and Involutivity

Involutivity and Differential Forms

The Frobenius Theorem

Foliations

Lie Subalgebras and Lie Subgroups

Overdetermined Systems of Partial Differential Equations

Problems

Chapter 20: The Exponential Map

One-Parameter Subgroups and the Exponential Map

One-Parameter Subgroups

The Exponential Map

The Closed Subgroup Theorem

Inﬁnitesimal Generators of Group Actions

Left Actions

The Lie Correspondence

Lie's Fundamental Theorems

Normal Subgroups

The Adjoint Representation

Ideals and Normal Subgroups

Problems

Chapter 21: Quotient Manifolds

Quotients of Manifolds by Group Actions

The Quotient Manifold Theorem

Covering Manifolds

Homogeneous Spaces

Sets with Transitive Group Actions

Applications to Lie Theory

Quotient Groups

Connectivity of Lie Groups

Problems

Chapter 22: Symplectic Manifolds

Symplectic Tensors

Symplectic Structures on Manifolds

The Canonical Symplectic Form on the Cotangent Bundle

The Darboux Theorem

Hamiltonian Vector Fields

Poisson Brackets

Hamiltonian Flowouts

Contact Structures

Nonlinear First-Order PDEs

General Nonlinear Equations

Problems

Appendix A: Review of Topology

Topological Spaces

Hausdorff Spaces

Bases and Countability

Subspaces, Products, Disjoint Unions, and Quotients

Subspaces

Product Spaces

Disjoint Union Spaces

Quotient Spaces and Quotient Maps

Open and Closed Maps

Connectedness and Compactness

Locally Compact Hausdorff Spaces

Homotopy and the Fundamental Group

Covering Maps

Appendix B: Review of Linear Algebra

Vector Spaces

Bases and Dimension

Linear Maps

Change of Basis

The Determinant

Inner Products and Norms

Norms

Direct Products and Direct Sums

Appendix C: Review of Calculus

Total and Partial Derivatives

Partial Derivatives

Multiple Integrals

Integrals of Vector-Valued Functions

Sequences and Series of Functions

The Inverse and Implicit Function Theorems

Appendix D: Review of Differential Equations

Existence, Uniqueness, and Smoothness

Nonautonomous Systems

Simple Solution Techniques

Separable Equations

2x2 Constant-Coefﬁcient Linear Systems

Partially Uncoupled Systems

References

Notation Index

Subject Index

Graduate Texts in Mathematics 218

Graduate Texts in Mathematics Series Editors: Sheldon Axler San Francisco State University, San Francisco, CA, USA Kenneth Ribet University of California, Berkeley, CA, USA Advisory Board: Colin Adams, Williams College, Williamstown, MA, USA Alejandro Adem, University of British Columbia, Vancouver, BC, Canada Ruth Charney, Brandeis University, Waltham, MA, USA Irene M. Gamba, The University of Texas at Austin, Austin, TX, USA Roger E. Howe, Yale University, New Haven, CT, USA David Jerison, Massachusetts Institute of Technology, Cambridge, MA, USA Jeffrey C. Lagarias, University of Michigan, Ann Arbor, MI, USA Jill Pipher, Brown University, Providence, RI, USA Fadil Santosa, University of Minnesota, Minneapolis, MN, USA Amie Wilkinson, University of Chicago, Chicago, IL, USA Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathe- matics. The volumes are carefully written as teaching aids and highlight character- istic features of the theory. Although these books are frequently used as textbooks in graduate courses, they are also suitable for individual study. For further volumes: www.springer.com/series/136

John M. Lee Introduction to Smooth Manifolds Second Edition

John M. Lee Department of Mathematics University of Washington Seattle, WA, USA ISSN 0072-5285 ISBN 978-1-4419-9981-8 DOI 10.1007/978-1-4419-9982-5 Springer New York Heidelberg Dordrecht London ISBN 978-1-4419-9982-5 (eBook) Library of Congress Control Number: 2012945172 Mathematics Subject Classiﬁcation: 53-01, 58-01, 57-01 © Springer Science+Business Media New York 2003, 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied speciﬁcally for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of pub- lication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface Manifolds crop up everywhere in mathematics. These generalizations of curves and surfaces to arbitrarily many dimensions provide the mathematical context for un- derstanding “space” in all of its manifestations. Today, the tools of manifold theory are indispensable in most major subﬁelds of pure mathematics, and are becoming increasingly important in such diverse ﬁelds as genetics, robotics, econometrics, statistics, computer graphics, biomedical imaging, and, of course, the undisputed leader among consumers (and inspirers) of mathematics—theoretical physics. No longer the province of differential geometers alone, smooth manifold technology is now a basic skill that all mathematics students should acquire as early as possible. Over the past century or two, mathematicians have developed a wondrous collec- tion of conceptual machines that enable us to peer ever more deeply into the invisi- ble world of geometry in higher dimensions. Once their operation is mastered, these powerful machines enable us to think geometrically about the 6-dimensional solu- tion set of a polynomial equation in four complex variables, or the 10-dimensional manifold of 5 5 orthogonal matrices, as easily as we think about the familiar 2-dimensional sphere in R3. The price we pay for this power, however, is that the machines are assembled from layer upon layer of abstract structure. Starting with the familiar raw materials of Euclidean spaces, linear algebra, multivariable calculus, and differential equations, one must progress through topological spaces, smooth at- lases, tangent bundles, immersed and embedded submanifolds, vector ﬁelds, ﬂows, cotangent bundles, tensors, Riemannian metrics, differential forms, foliations, Lie derivatives, Lie groups, Lie algebras, and more—just to get to the point where one can even think about studying specialized applications of manifold theory such as comparison theory, gauge theory, symplectic topology, or Ricci ﬂow. This book is designed as a ﬁrst-year graduate text on manifold theory, for stu- dents who already have a solid acquaintance with undergraduate linear algebra, real analysis, and topology. I have tried to focus on the portions of manifold theory that will be needed by most people who go on to use manifolds in mathematical or sci- entiﬁc research. I introduce and use all of the standard tools of the subject, and prove most of its fundamental theorems, while avoiding unnecessary generalization v

vi Preface or specialization. I try to keep the approach as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, but without shying away from the powerful tools that modern mathemat- ics has to offer. To ﬁt in all of the basics and still maintain a reasonably sane pace, I have had to omit or barely touch on a number of important topics, such as complex manifolds, inﬁnite-dimensional manifolds, connections, geodesics, curvature, ﬁber bundles, sheaves, characteristic classes, and Hodge theory. Think of them as dessert, to be savored after completing this book as the main course. To convey the book’s compass, it is easiest to describe where it starts and where it ends. The starting line is drawn just after topology: I assume that the reader has had a rigorous introduction to general topology, including the fundamental group and covering spaces. One convenient source for this material is my Introduction to Topological Manifolds [LeeTM], which I wrote partly with the aim of providing the topological background needed for this book. There are other books that cover sim- ilar material well; I am especially fond of the second edition of Munkres’s Topology [Mun00]. The ﬁnish line is drawn just after a broad and solid background has been established, but before getting into the more specialized aspects of any particular subject. In particular, I introduce Riemannian metrics, but I do not go into connec- tions, geodesics, or curvature. There are many Riemannian geometry books for the interested student to take up next, including one that I wrote [LeeRM] with the goal of moving expediently in a one-quarter course from basic smooth manifold theory to nontrivial geometric theorems about curvature and topology. Similar material is covered in the last two chapters of the recent book by Jeffrey Lee (no relation) [LeeJeff09], and do Carmo [dC92] covers a bit more. For more ambitious readers, I recommend the beautiful books by Petersen [Pet06], Sharpe [Sha97], and Chavel [Cha06]. This subject is often called “differential geometry.” I have deliberately avoided using that term to describe what this book is about, however, because the term ap- plies more properly to the study of smooth manifolds endowed with some extra structure—such as Lie groups, Riemannian manifolds, symplectic manifolds, vec- tor bundles, foliations—and of their properties that are invariant under structure- preserving maps. Although I do give all of these geometric structures their due (after all, smooth manifold theory is pretty sterile without some geometric applications), I felt that it was more honest not to suggest that the book is primarily about one or all of these geometries. Instead, it is about developing the general tools for working with smooth manifolds, so that the reader can go on to work in whatever ﬁeld of differential geometry or its cousins he or she feels drawn to. There is no canonical linear path through this material. I have chosen an order- ing of topics designed to establish a good technical foundation in the ﬁrst half of the book, so that I can discuss interesting applications in the second half. Once the ﬁrst twelve chapters have been completed, there is some ﬂexibility in ordering the remaining chapters. For example, Chapter 13 (Riemannian Metrics) can be post- poned if desired, although some sections of Chapters 15 and 16 would have to be postponed as well. On the other hand, Chapters 19–21 (Distributions and Foliations, The Exponential Map, and Quotient Manifolds, respectively) could in principle be

Preface vii inserted any time after Chapter 14, and much of the material can be covered even earlier if you are willing to skip over the references to differential forms. And the ﬁnal chapter (Symplectic Manifolds) would make sense any time after Chapter 17, or even after Chapter 14 if you skip the references to de Rham cohomology. As you might have guessed from the size of the book, and will quickly conﬁrm when you start reading it, my style tends toward more detailed explanations and proofs than one typically ﬁnds in graduate textbooks. I realize this is not to every instructor’s taste, but in my experience most students appreciate having the details spelled out when they are ﬁrst learning the subject. The detailed proofs in the book provide students with useful models of rigor, and can free up class time for dis- cussion of the meanings and motivations behind the deﬁnitions as well as the “big ideas” underlying some of the more difﬁcult proofs. There are plenty of opportuni- ties in the exercises and problems for students to provide arguments of their own. I should say something about my choices of conventions and notations. The old joke that “differential geometry is the study of properties that are invariant under change of notation” is funny primarily because it is alarmingly close to the truth. Every geometer has his or her favorite system of notation, and while the systems are all in some sense formally isomorphic, the transformations required to get from one to another are often not at all obvious to students. Because one of my central goals is to prepare students to read advanced texts and research articles in differential geometry, I have tried to choose notations and conventions that are as close to the mainstream as I can make them without sacriﬁcing too much internal consistency. (One difference between this edition and the previous one is that I have changed a number of my notational conventions to make them more consistent with main- stream mathematical usage.) When there are multiple conventions in common use (such as for the wedge product or the Laplace operator), I explain what the alterna- tives are and alert the student to be aware of which convention is in use by any given writer. Striving for too much consistency in this subject can be a mistake, however, and I have eschewed absolute consistency whenever I felt it would get in the way of ease of understanding. I have also introduced some common shortcuts at an early stage, such as the Einstein summation convention and the systematic confounding of maps with their coordinate representations, both of which tend to drive students crazy at ﬁrst, but pay off enormously in efﬁciency later. Prerequisites This subject draws on most of the topics that are covered in a typical undergraduate mathematics education. The appendices (which most readers should read, or at least skim, ﬁrst) contain a cursory summary of prerequisite material on topology, linear algebra, calculus, and differential equations. Although students who have not seen this material before will not learn it from reading the appendices, I hope readers will appreciate having all of the background material collected in one place. Besides giving me a convenient way to refer to results that I want to assume as known, it also gives the reader a splendid opportunity to brush up on topics that were once (hopefully) understood but may have faded.

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