Differential Geometric Structures

Content: Cover
 Title Page
 Copyright Page
 Dedication
 Contents
 Preface
 Chapter 1: An Introduction to Fiber Bundles
 The Definition of a Fiber Bundle
 Vector Bundles
 The Vertical Bundle
 Operations on Vector Bundles
 Principal and Associated Bundles
 Sections of Fiber Bundles
 Chapter 2: Connection Theory for Vector Bundles
 Parallelism Structures in Vector Bundles
 Holonomy in Vector Bundles
 Connections on Vector Bundles
 The Curvature Tensor
 Covariant Derivative Operators
 The Structure Equation
 The Space of Connections on a Vector Bundle
 Characteristic Classes. The Tangent Bundle: Linear ConnectionsThe Tangent Bundle: Affine Connections
 Affine Transformations
 Chapter 3: Riemannian Vector Bundles
 Riemannian Metrics
 Riemannian Connections
 The Levi-Civita Connection
 The Metric Structure of a Riemannian Manifold
 The Gauss-Bonnet Theorem
 Chapter 4: Harmonic Theory
 The Basic Differential Operators
 Green\'s Theorem and Some Applications
 Weitzenböck\'s Formula for the Laplacian
 Chern\'s Formula for the Laplacian
 Chapter 5: Geometric Vector Fields on Riemannian Manifolds
 Harmonic Fields
 Killing Fields
 Conformal Fields
 Affine Fields. Projective FieldsChapter 6: Lie Groups
 A Negative Curvature Example
 Bi-invariant Metrics
 Some Simple Examples
 Homogeneous Spaces
 Chapter 7: Symmetric Spaces
 Affine Symmetric Spaces
 Locally Affine Symmetric Spaces
 Symmetric Lie Algebras
 Riemannian Symmetric Spaces
 Chapter 8: Symplectic and Hermitian Vector Bundles
 Symplectic Vector Bundles
 Hermitian Vector Bundles
 Complex Manifolds
 The Curvature of Kähler Manifolds
 Chapter 9: Other Differential Geometric Structures
 Parallelism in Principal Fiber Bundles
 Holonomy and Curvature in Principal Fiber Bundles. Characteristic Classes of Principal BundlesParallel Transport in Fiber Bundles
 Cartan Connections
 Spin Structures
 Bibliography
 Index of Notation
 Index.