Contents Preface
Chapter 1. Differentiable Manifolds
1. Basic Definitions
2. Differentiable Maps
3. Tangent Vectors
4. The Derivative
5. The Inverse and Implicit Function Theorems
6. Submanifolds
7. Vector Fields
8. The Lie Bracket
9. Distributions and Frobenius Theorem
10. Multilinear Algebra and Tensors
11. Tensor Fields and Differential Forms
12. Integration on Chains
13. The Local Version of Stokes' Theorem
14. Orientation and the Global Version of Stokes' Theorem
15. Some Applications of Stokes' Theorem Chapter 2. Fiber Bundles
1. Basic Definitions and Examples
2. Principal and Associated Bundles
3. The Tangent Bundle of Sn
4. Cross-Sections of Bundles
5. Pullback and Normal Bundles
6. Fibrations and the Homotopy Lifting/Covering Properties
7. Grassmannians and Universal Bundles Chapter 3. Homotopy Groups and Bundles Over Spheres
1. Differentiable Approximations
2. Homotopy Groups
3. The Homotopy Sequence of a Fibration
4. Bundles Over Spheres
5. The Vector Bundles Over Low-Dimensional Spheres Chapter 4. Connections and Curvature
1. Connections on Vector Bundles
2. Covariant Derivatives
3. The Curvature Tensor of a Connection
4. Connections on Manifolds
5. Connections on Principal Bundles Chapter 5. Metric Structures
1. Euclidean Bundles and Riemannian Manifolds
2. Riemannian Connections
3. Curvature Quantifiers
4. Isometric Immersions
5. Riemannian Submersions
6. The Gauss Lemma
7. Length-Minimizing Properties of Geodesics
8. First and Second Variation of Arc-Length
9. Curvature and Topology
10. Actions of Compact Lie Groups Chapter 6. Characteristic Classes
1. The Weil Homomorphism
2. Pontrjagin Classes
3. The Euler Class
4. The Whitney Sum Formula for Pontrjagin and Euler Classes
5. Some Examples
6. The Unit Sphere Bundle and the Euler Class
7. The Generalized Gauss-Bonnet Theorem
8. Complex and Symplectic Vector Spaces
9. Chern Classes
Bibliography
Index

This book offers an introduction to the theory of differentiable manifolds and fiber bundles. It examines bundles from the point of view of metric differential geometry: Euclidean bundles, Riemannian connections, curvature, and Chern-Weil theory are discussed, including the Pontrjagin, Euler, and Chern characteristic classes of a vector bundle. These concepts are illustrated in detail for bundles over spheres.