Lectures on Differential Geometry: (Chelsea Publications)

This book is based on lectures given at Harvard University during the academic year 1960-1961. The presentation assumes knowledge of the elements of modern algebra (groups, vector spaces, etc.) and point-set topology and some elementary analysis. Rather than giving all the basic information or touching upon every topic in the field, this work treats various selected topics in differential geometry. The author concisely addresses standard material and spreads exercises throughout the text. His reprint has two additions to the original volume: a paper written jointly with V. Guillemin at the beginning of a period of intense interest in the equivalence problem and a short description from the author on results in the field that occurred between the first and the second printings.

Table of Contents:
Algebraic Preliminaries: 1. Tensor products of vector spaces; 2. The tensor algebra of a vector space; 3. The contravariant and symmetric algebras; 4. Exterior algebra; 5. Exterior equations Differentiable Manifolds: 1. Definitions; 2. Differential maps; 3. Sard's theorem; 4. Partitions of unity, approximation theorems; 5. The tangent space; 6. The principal bundle; 7. The tensor bundles; 8. Vector fields and Lie derivatives Integral Calculus on Manifolds: 1. The operator $d$; 2. Chains and integration; 3. Integration of densities; 4. $0$ and $n$-dimensional cohomology, degree; 5. Frobenius' theorem; 6. Darboux's theorem; 7. Hamiltonian structures The Calculus of Variations: 1. Legendre transformations; 2. Necessary conditions; 3. Conservation laws; 4. Sufficient conditions; 5. Conjugate and focal points, Jacobi's condition; 6. The Riemannian case; 7. Completeness; 8. Isometries Lie Groups: 1. Definitions; 2. The invariant forms and the Lie algebra; 3. Normal coordinates, exponential map; 4. Closed subgroups; 5. Invariant metrics; 6. Forms with values in a vector space Differential Geometry of Euclidean Space: 1. The equations of structure of Euclidean space; 2. The equations of structure of a submanifold; 3. The equations of structure of a Riemann manifold; 4. Curves in Euclidean space; 5. The second fundamental form; 6. Surfaces The Geometry of $G$-Structures: 1. Principal and associated bundles, connections; 2. $G$-structures; 3. Prolongations; 4. Structures of finite type; 5. Connections on $G$-structures; 6. The spray of a linear connection Appendix I: Two existence theorems Appendix II: Outline of theory of integration on $E^n$ Appendix III: An algebraic model of transitive differential geometry Appendix IV: The integrability problem for geometrical structures References Index.