AMS Chelsea Publishing 1983; 442 pp; hardcover Reprint/Revision History: first AMS printing 1999 ISBN10: 0821813854 ISBN13: 9780821813850 List Price: US$57 Member Price: US$51.30 Order Code: CHEL/316.H
 This book is based on lectures given at Harvard University during the academic year 19601961. The presentation assumes knowledge of the elements of modern algebra (groups, vector spaces, etc.) and pointset 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
