This text is an introduction to the theory of differentiable manifolds and fiber bundles. The only requisites are a solid background in calculus and linear algebra, together with some basic point-set topology. The first chapter provides a comprehensive overview of differentiable manifolds. The following two chapters are devoted to fiber bundles and homotopy theory of fibrations. Vector bundles have been emphasized, although principal bundles are also discussed in detail. The last three chapters study 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. Chapter 5, with its focus on the tangent bundle, also serves as a basic introduction to Riemannian geometry in the large. This book can be used for a one-semester course on manifolds or bundles, or a two-semester course in differential geometry.

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.- 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.- 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.- 1. Connections on Vector Bundles.- 4. Connections and Curvature.- 2. Covariant Derivatives.- 3. The Curvature Tensor of a Connection.- 4. Connections on Manifolds.- 5. Connections on Principal Bundles.- 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.- 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 SphereBundle and the Euler Class.- 7. The Generalized Gauss-Bonnet Theorem.- 8. Complex and Symplectic Vector Spaces.- 9. Chern Classes.