دانلود کتاب Differential Geometry: Connections, Curvature, and Characteristic Classes: 275

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Preface
Contents
Chapter 1 Curvature and Vector Fields
1 Riemannian Manifolds
1.1 Inner Products on a Vector Space
1.2 Representations of Inner Products bySymmetric Matrices
1.3 Riemannian Metrics
1.4 Existence of a Riemannian Metric
Problems
2 Curves
2.1 Regular Curves
2.2 Arc Length Parametrization
2.3 Signed Curvature of a Plane Curve
2.4 Orientation and Curvature
Problems
3 Surfaces in Space
3.1 Principal, Mean, and Gaussian Curvatures
3.2 Gauss\'s Theorema Egregium
3.3 The Gauss–Bonnet Theorem
Problems
4 Directional Derivatives in Euclidean Space
4.1 Directional Derivatives in Euclidean Space
4.2 Other Properties of the Directional Derivative
4.3 Vector Fields Along a Curve
4.4 Vector Fields Along a Submanifold
4.5 Directional Derivatives on a Submanifold of Rn
Problems
5 The Shape Operator
5.1 Normal Vector Fields
5.2 The Shape Operator
5.3 Curvature and the Shape Operator
5.4 The First and Second Fundamental Forms
5.5 The Catenoid and the Helicoid
Problems
6 Affine Connections
6.1 Affine Connections
6.2 Torsion and Curvature
6.3 The Riemannian Connection
6.4 Orthogonal Projection on a Surface in R3
6.5 The Riemannian Connection on a Surface in R3
Problems
7 Vector Bundles
7.1 Definition of a Vector Bundle
7.2 The Vector Space of Sections
7.3 Extending a Local Section to a Global Section
7.4 Local Operators
7.5 Restriction of a Local Operator to an Open Subset
7.6 Frames
7.7 F-Linearity and Bundle Maps
7.8 Multilinear Maps over Smooth Functions
Problems
8 Gauss\'s Theorema Egregium
8.1 The Gauss and Codazzi–Mainardi Equations
8.2 A Proof of the Theorema Egregium
8.3 The Gaussian Curvature in Termsof an Arbitrary Basis
Problems
9 Generalizations to Hypersurfaces in Rn+1
9.1 The Shape Operator of a Hypersurface
9.2 The Riemannian Connection of a Hypersurface
9.3 The Second Fundamental Form
9.4 The Gauss Curvature and Codazzi–Mainardi Equations
Chapter 2 Curvature and Differential Forms
10 Connections on a Vector Bundle
10.1 Connections on a Vector Bundle
10.2 Existence of a Connection on a Vector Bundle
10.3 Curvature of a Connection on a Vector Bundle
10.4 Riemannian Bundles
10.5 Metric Connections
10.6 Restricting a Connection to an Open Subset
10.7 Connections at a Point
Problems
11 Connection, Curvature, and Torsion Forms
11.1 Connection and Curvature Forms
11.2 Connections on a Framed Open Set
11.3 The Gram–Schmidt Process
11.4 Metric Connection Relative to anOrthonormal Frame
11.5 Connections on the Tangent Bundle
Problems
12 The Theorema Egregium Using Forms
12.1 The Gauss Curvature Equation
12.2 The Theorema Egregium
12.3 Skew-Symmetries of the Curvature Tensor
12.4 Sectional Curvature
12.5 Poincaré Half-Plane
Problems
Chapter 3 Geodesics
13 More on Affine Connections
13.1 Covariant Differentiation Along a Curve
13.2 Connection-Preserving Diffeomorphisms
13.3 Christoffel Symbols
Problems
14 Geodesics
14.1 The Definition of a Geodesic
14.2 Reparametrization of a Geodesic
14.3 Existence of Geodesics
14.4 Geodesics in the Poincaré Half-Plane
14.5 Parallel Translation
14.6 Existence of Parallel Translation Along a Curve
14.7 Parallel Translation on a Riemannian Manifold
Problems
15 Exponential Maps
15.1 The Exponential Map of a Connection
15.2 The Differential of the Exponential Map
15.3 Normal Coordinates
15.4 Left-Invariant Vector Fields on a Lie Group
15.5 Exponential Map for a Lie Group
15.6 Naturality of the Exponential Map for a Lie Group
15.7 Adjoint Representation
15.8 Associativity of a Bi-Invariant Metric on a Lie Group
Problems
15.9 Addendum. The Exponential Map as aNatural Transformation
16 Distance and Volume
16.1 Distance in a Riemannian Manifold
16.2 Geodesic Completeness
16.3 Dual 1-Forms Under a Change of Frame
16.4 Volume Form
16.5 The Volume Form in Local Coordinates
Problems
17 The Gauss–Bonnet Theorem
17.1 Geodesic Curvature
17.2 The Angle Function Along a Curve
17.3 Signed Geodesic Curvature on an Oriented Surface
17.4 Gauss–Bonnet Formula for a Polygon
17.5 Triangles on a Riemannian 2-Manifold
17.6 Gauss–Bonnet Theorem for a Surface
17.7 Gauss–Bonnet Theorem for a Hypersurface in R2n+1
Problems
Chapter 4 Tools from Algebra and Topology
18 The Tensor Product and the Dual Module
18.1 Construction of the Tensor Product
18.2 Universal Mapping Property for Bilinear Maps
18.3 Characterization of the Tensor Product
18.4 A Basis for the Tensor Product
18.5 The Dual Module
18.6 Identities for the Tensor Product
18.7 Functoriality of the Tensor Product
18.8 Generalization to Multilinear Maps
18.9 Associativity of the Tensor Product
18.10 The Tensor Algebra
Problems
19 The Exterior Power
19.1 The Exterior Algebra
19.2 Properties of the Wedge Product
19.3 Universal Mapping Property for Alternating k-Linear Maps
19.4 A Basis for kV
19.5 Nondegenerate Pairings
19.6 A Nondegenerate Pairing of k(V) with kV
19.7 A Formula for the Wedge Product
Problems
20 Operations on Vector Bundles
20.1 Vector Subbundles
20.2 Subbundle Criterion
20.3 Quotient Bundles
20.4 The Pullback Bundle
20.5 Examples of the Pullback Bundle
20.6 The Direct Sum of Vector Bundles
20.7 Other Operations on Vector Bundles
Problems
21 Vector-Valued Forms
21.1 Vector-Valued Forms as Sections of a Vector Bundle
21.2 Products of Vector-Valued Forms
21.3 Directional Derivative of a Vector-Valued Function
21.4 Exterior Derivative of a Vector-Valued Form
21.5 Differential Forms with Values in a Lie Algebra
21.6 Pullback of Vector-Valued Forms
21.7 Forms with Values in a Vector Bundle
21.8 Tensor Fields on a Manifold
21.9 The Tensor Criterion
21.10 Remark on Signs Concerning Vector-Valued Forms
Problems
Chapter 5 Vector Bundles and Characteristic Classes
22 Connections and Curvature Again
22.1 Connection and Curvature Matrices Under a Change of Frame
22.2 Bianchi Identities
22.3 The First Bianchi Identity in Vector Form
22.4 Symmetry Properties of the Curvature Tensor
22.5 Covariant Derivative of Tensor Fields
22.6 The Second Bianchi Identity in Vector Form
22.7 Ricci Curvature
22.8 Scalar Curvature
22.9 Defining a Connection Using Connection Matrices
22.10 Induced Connection on a Pullback Bundle
Problems
23 Characteristic Classes
23.1 Invariant Polynomials on gl(r,R)
23.2 The Chern–Weil Homomorphism
23.3 Characteristic Forms Are Closed
23.4 Differential Forms Depending on a Real Parameter
23.5 Independence of Characteristic Classes of a Connection
23.6 Functorial Definition of a Characteristic Class
23.7 Naturality
Problems
24 Pontrjagin Classes
24.1 Vanishing of Characteristic Classes
24.2 Pontrjagin Classes
24.3 The Whitney Product Formula
25 The Euler Class and Chern Classes
25.1 Orientation on a Vector Bundle
25.2 Characteristic Classes of an Oriented Vector Bundle
25.3 The Pfaffian of a Skew-Symmetric Matrix
25.4 The Euler Class
25.5 Generalized Gauss–Bonnet Theorem
25.6 Hermitian Metrics
25.7 Connections and Curvature on a ComplexVector Bundle
25.8 Chern Classes
Problems
26 Some Applications of Characteristic Classes
26.1 The Generalized Gauss–Bonnet Theorem
26.2 Characteristic Numbers
26.3 The Cobordism Problem
26.4 The Embedding Problem
26.5 The Hirzebruch Signature Formula
26.6 The Riemann–Roch Problem
Chapter 6 Principal Bundles and Characteristic Classes
27 Principal Bundles
27.1 Principal Bundles
27.2 The Frame Bundle of a Vector Bundle
27.3 Fundamental Vector Fields of a Right Action
27.4 Integral Curves of a Fundamental Vector Field
27.5 Vertical Subbundle of the Tangent Bundle TP
27.6 Horizontal Distributions on a Principal Bundle
Problems
28 Connections on a Principal Bundle
28.1 Connections on a Principal Bundle
28.2 Vertical and Horizontal Componentsof a Tangent Vector
28.3 The Horizontal Distribution of an Ehresmann Connection
28.4 Horizontal Lift of a Vector Field to a Principal Bundle
28.5 Lie Bracket of a Fundamental Vector Field
Problems
29 Horizontal Distributions on a Frame Bundle
29.1 Parallel Translation in a Vector Bundle
29.2 Horizontal Vectors on a Frame Bundle
29.3 Horizontal Lift of a Vector Field to a Frame Bundle
29.4 Pullback of a Connection on a Frame Bundle Under a Section
30 Curvature on a Principal Bundle
30.1 Curvature Form on a Principal Bundle
30.2 Properties of the Curvature Form
Problems
31 Covariant Derivative on a Principal Bundle
31.1 The Associated Bundle
31.2 The Fiber of the Associated Bundle
31.3 Tensorial Forms on a Principal Bundle
31.4 Covariant Derivative
31.5 A Formula for the Covariant Derivative of a Tensorial Form
Problems
32 Characteristic Classes of Principal Bundles
32.1 Invariant Polynomials on a Lie Algebra
32.2 The Chern–Weil Homomorphism
Problems
Appendix
A Manifolds
A.1 Manifolds and Smooth Maps
A.2 Tangent Vectors
A.3 Vector Fields
A.4 Differential Forms
A.5 Exterior Differentiation on a Manifold
A.6 Exterior Differentiation on R3
A.7 Pullback of Differential Forms
Problems
B Invariant Polynomials
B.1 Polynomials Versus Polynomial Functions
B.2 Polynomial Identities
B.3 Invariant Polynomials on gl(r,F)
B.4 Invariant Complex Polynomials
B.5 L-Polynomials, Todd Polynomials, and the Chern Character
B.6 Invariant Real Polynomials
B.7 Newton\'s Identities
Problems
Hints and Solutions to Selected End-of-Section Problems
List of Notations
References
Index