Clifford Henry Taubes is the William Petschek Professor of Mathematics at Harvard University. He is a member of the National Academy of Sciences and also the American Academy of Sciences. He was awarded the American Mathematical Society's Oswald Veblen Prize in 1991 for his work in differential geometry and topology. He was also the recipient of the French Academy of Sciences Elie Cartan Prize in 1993, the Clay Research Award in 2008, the National Academy of Sciences' Mathematics Award in 2008, and the Shaw Prize in Mathematics in 2009.

• 1. Smooth manifolds ; 2. Matrices and Lie groups ; 3. Introduction to vector bundles ; 4. Algebra of vector bundles ; 5. Maps and vector bundles ; 6. Vector bundles with fiber C]n ; 7. Metrics on vector bundles ; 8. Geodesics ; 9. Properties of geodesics ; 10. Principal bundles ; 11. Covariant derivatives and connections ; 12. Covariant derivatives, connections and curvature ; 13. Flat connections and holonomy ; 14. Curvature polynomials and characteristic classes ; 15. Covariant derivatives and metrics ; 16. The Riemann curvature tensor ; 17. Complex manifolds ; 18. Holomorphic submanifolds, holomorphic sections and curvature ; 19. The Hodge star ; Indexed list of propositions by subject ; Index