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Lectures on differential geometry

Author: Shlomo Sternberg; American Mathematical Society.
Publisher: Providence, RI : American Mathematical Society, ©1983
Edition/Format:   Print book : English : 2nd edView all editions and formats
Database:WorldCat
Summary:

Treats various selected topics in differential geometry. This book includes a paper written jointly with V Guillemin at the beginning of a period of intense interest in the equivalence problem.

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Document Type: Book
All Authors / Contributors: Shlomo Sternberg; American Mathematical Society.
ISBN: 0821813854 9780821813850
OCLC Number: 43032711
Notes: Previous edition: 1964.
Description: xi, 442 pages ; 24 cm
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.
Responsibility: Shlomo Sternberg.

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