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I have indicated whenever possi- ble the historical origin of the concepts, and have augmented the bibliography with historical references.
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
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