流形导论

Preface to the Second Edition

Preface to the First Edition

A Brief Introduction

Chapter 1 Euclidean Spaces

1 Smooth Functions on a Euclidean Space

1.1 C∞ Versus Analytic Functions

1.2 Taylor's Theorem with Remainder

Problems

2 Tangent Vectors in Rn as Derivations

2.1 The Directional Derivative

2.2 Germs of Functions

2.3 Derivations at a Point

2.4 Vector Fields

2.5 Vector Fields as Derivations

Problems

3 The Exterior Algebra of Multicovectors

3.1 Dual Space

3.2 Permutations

3.3 Multilinear Functions

3.4 The Permutation Action on Multilinear Functions

3.5 The Symmetrizing and Alternating Operators

3.6 The Tensor Product

3.7 The Wedge Product

3.8 Anticommutativity of the Wedge Product

3.9 Associativity of the Wedge Product

3.10 A Basis for k—Covectors

Problems

4 Differential Forms on Rn

4.1 Differential 1—Forms and the Differential of a Function

4.2 Differential k—Forms

4.3 Differential Forms as Multilinear Functions on Vector Fields

4.4 The Exterior Derivative

4.5 Closed Forms and Exact Forms

4.6 Applications to Vector Calculus

4.7 Convention on Subscripts and Superscripts

Problems

Chapter 2 Manifolds

5 Manifolds

5.1 Topological Manifolds

5.2 Compatible Charts

5.3 Smooth Manifolds

5.4 Examples of Smooth Manifolds

Problems

6 Smooth Maps on a Manifold

6.1 Smooth Functions on a Manifold

6.2 Smooth Maps Between Manifolds

6.3 Diffeomorphisms

6.4 Smoothness in Terms of Components

6.5 Examples of Smooth Maps

6.6 Partial Derivatives

6.7 The Inverse Function Theorem

Problems

7 Quotients

7.1 The Quotient Topology

7.2 Continuity of a Map on a Quotient

7.3 Identification of a Subset to a Point

7.4 A Necessary Condition for a Hausdorff Quotient

7.5 Open Equivalence Relations

7.6 Real Projective Space

7.7 The Standard C∞ Atlas on a Real Projective Space

Problems

Chapter 3 The Tangent Space

8 The Tangent Space

8.1 The Tangent Space at a Point

8.2 The Differential of a Map

8.3 The Chain Rule

8.4 Bases for the Tangent Space at a Point

8.5 A Local Expression for the Differential

8.6 Curves in a Manifold

8.7 Computing the Differential Using Curves

8.8 Immersions and Submersions

8.9 Rank， and Critical and Regular Points

Problems

9 Submanifolds

9.1 Submanifolds

9.2 Level Sets of a Function

9.3 The Regular Level Set Theorem

9.4 Examples of Regular Submanifolds

Problems

10 Categories and Functors

10.1 Categories

10.2 Functors

10.3 The Dual Functor and the Multicovector Functor

Problems

11 The Rank of a Smooth Map

11.1 Constant Rank Theorem

11.2 The Immersion and Submersion Theorems

11.3 Images of Smooth Maps

11.4 Smooth Maps into a Submanifold

11.5 The Tangent Plane to a Surface in R3

Problems

12 The Tangent Bundle

12.1 The Topology of the Tangent Bundle

12.2 The Manifold Structure on the Tangent Bundle

12.3 Vector Bundles

12.4 Smooth Sections

12.5 Smooth Frames

Problems

13 Bump Functions and Partitions of Unity

13.1 C∞ Bump Functions

13.2 Partitions of Unity

13.3 Existence of a Partition of Unity

Problems

14 Vector Fields

14.1 Smoothness of a Vector Field

14.2 Integral Curves

14.3 Local Flows

14.4 The Lie Bracket

14.5 The Pushforward of Vector Fields

14.6 Related Vector Fields

Problems

Chapter 4 Lie Groups and Lie Algebras

15 Lie Groups

15.1 Examples of Lie Groups

15.2 Lie Subgroups

15.3 The Matrix Exponential

15.4 The Trace of a Matrix

15.5 The Differential of det at the Identity

Problems

16 Lie Algebras

16.1 Tangent Space at the Identity of a Lie Group

16.2 Left—Invariant Vector Fields on a Lie Group

16.3 The Lie Algebra of a Lie Group

16.4 The Lie Bracket on gl（n，R）

16.5 The Pushforward of Left—Invariant Vector Fields

16.6 The Differential as a Lie Algebra Homomorphism

Problems

Chapter 5 Differential Forms

17 Differential 1—Forms

17.1 The Differential of a Function

17.2 Local Expression for a Differential 1—Form

17.3 The Cotangent Bundle

17.4 Characterization of C∞ l—Forms

17.5 Pullback of l—Forms

17.6 Restriction of l—Forms to an Immersed Submanifold

Problems

18 Differential k—Forms

18.1 Differential Forms

18.2 Local Expression for a k—Form

18.3 The Bundle Point of View

18.4 Smooth k—Forms

18.5 Pullback of k—Forms

18.6 The Wedge Product

18.7 Differential Forms on a Circle

18.8 Invariant Forms on a Lie Group

Problems

19 The Exterior Derivative

19.1 Exterior Derivative on a Coordinate Chart

19.2 Local Operators

19.3 Existence of an Exterior Derivative on a Manifold

19.4 Uniqueness of the Exterior Derivative

19.5 Exterior Differentiation Under a Pullback

19.6 Restriction of k—Forms to a Submanifold

19.7 A Nowhere—Vanishing 1—Form on the Circle

Problems

20 The Lie Derivative and Interior Multiplication

20.1 Families of Vector Fields and Differential Forms

20.2 The Lie Derivative of a Vector Field

20.3 The Lie Derivative of a Differential Form

20.4 Interior Multiplication

20.5 Properties of the Lie Derivative

20.6 Global Formulas for the Lie and Exterior Derivatives

Problems

Chapter 6 Integration

21 Orientations

21.1 Orientations of a Vector Space

21.2 Orientations and n—Covectors

21.3 Orientations on a Manifold

21.4 Orientations and Differential Forms

21.5 Orientations and Atlases

Problems

22 Manifolds with Boundary

22.1 Smooth Invariance of Domain in Rn

22.2 Manifolds with Boundary

22.3 The Boundary of a Manifold with Boundary

22.4 Tangent Vectors， Differential Forms， and Orientations

22.5 Outward—Pointing Vector Fields

22.6 Boundary Orientation

Problems

23 Integration on Manifolds

23.1 The Riemann Integral of a Function on Rn

23.2 Integrability Conditions

23.3 The Integral of an n—Form on Rn

23.4 Integral of a Differential Form over a Manifold

23.5 Stokes's Theorem

23.6 Line Integrals and Green's Theorem

Problems

Chapter 7 De Rham Theory

24 De Rham Cohomology

24.1 De Rharn Cohomology

24.2 Examples of de Rham Cohomology

24.3 Diffeomorphism Invariance

24.4 The Ring Structure on de Rham Cohomology

Problems

25 The Long Exact Sequence in Cohomology

25.1 Exact Sequences

25.2 Cohomology of Cochain Complexes

25.3 The Connecting Homomorphism

25.4 The Zig—Zag Lemma

Problems

26 The Mayer—Vietoris Sequence

26.1 The Mayer—Vietoris Sequence

26.2 The Cohomology of the Circle

26.3 The Euler Characteristic

Problems

27 Homotopy Invariance

27.1 Smooth Homotopy

27.2 Homotopy Type

27.3 Deformation Retractions

27.4 The Homotopy Axiom for de Rham Cohomology

Problems

28 Computation of de Rham Cohomology

28.1 Cohomology Vector Space of a Torus

28.2 The Cohomology Ring of a Torus

28.3 The Cohomology of a Surface of Genus g

Problems

29 Proofof Homotopy Invarianee

29.1 Reduction to Two Sections

29.2 Cochain Homotopies

29.3 Differential Forms on M × R

29.4 A Cochain Homotopy Between i0* and i1*

29.5 Verification of Cochain Homotopy

Problems

Appendices

A Point—Set Topology

A.1 Topological Spaces

A.2 Subspace Topology

A.3 Bases

A.4 First and Second Countability

A.5 Separation Axioms

A.6 Product Topology

A.7 Continuity

A.8 Compactness

A.9 Boundedness in Rn

A.10 Connectedness

A.11 Connected Components

A.12 Closure

A.13 Convergence

Problems

B The Inverse Function Theorem on Rn and Related Results

B.1 The Inverse Function Theorem

B.2 The Implicit Function Theorem

B.3 Constant Rank Theorem

Problems

C Existence of a Partition of Unity in General

D Linear Algebra

D.1 Quotient Vector Spaces

D.2 Linear Transformations

D.3 Direct Product and Direct Sum

Problems

E Quaternions and the Symplectic Group

E.1 Representation of Linear Maps by Matrices

E.2 Quaternionic Conjugation

E.3 Quaternionic Inner Product

This is a completely revised edition, with more than fifty pages of new material scattered throughout. In keeping with the conventional meaning of chapters and sections, I have reorgaruzed the book into twenty-nine sections in seven chapters. The main additions are Section 20 0n the Lie derivative and interior multiplication, two intrinsic operations on a manifold too important to leave out, new criteria in Section 21 for the boundary orientation, and a new appendix on quaternions and the symplectic group.
　　Apart from correcting errors and misprints, I have thought through every proof again, clarified many passages, and added new examples, exercises, hints, and solutions. In the process, every section has been rewritten, sometimes quite drastically. The revisions are so extensive that it is not possible to enumerate them all here. Each chapter now comes with an introductory essay giving an overview of what is to come. To provide a timeline for the development ofideas, I have indicated whenever possi- ble the historical origin of the concepts, and have augmented the bibliography with historical references.