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  • Produktbild: Differential and Riemannian Manifolds
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Band 160

Differential and Riemannian Manifolds

Aus der Reihe Graduate Texts in Mathematics

Fr. 86.90

inkl. gesetzl. MwSt., Versandkostenfrei

Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

04.02.2012

Abbildungen

XIV, 364 p.

Verlag

Springer Us

Seitenzahl

364

Maße (L/B/H)

23.5/15.5/2.1 cm

Gewicht

581 g

Auflage

Third Edition 1995

Sprache

Englisch

ISBN

978-1-4612-8688-2

Beschreibung

Rezension

S. Lang


Differential and Riemannian Manifolds


"An introduction to differential geometry, starting from recalling differential calculus and going through all the basic topics such as manifolds, vector bundles, vector fields, the theorem of Frobenius, Riemannian metrics and curvature. Useful to the researcher wishing to learn about infinite-dimensional geometry."


—MATHEMATICAL REVIEWS

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

04.02.2012

Abbildungen

XIV, 364 p.

Verlag

Springer Us

Seitenzahl

364

Maße (L/B/H)

23.5/15.5/2.1 cm

Gewicht

581 g

Auflage

Third Edition 1995

Sprache

Englisch

ISBN

978-1-4612-8688-2

Herstelleradresse

Springer-Verlag KG
Sachsenplatz 4-6
1201 Wien
AT

Email: ProductSafety@springernature.com

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  • Produktbild: Differential and Riemannian Manifolds
  • Produktbild: Differential and Riemannian Manifolds
  • I Differential Calculus.- §1. Categories.- §2. Topological Vector Spaces.- §3. Derivatives and Composition of Maps.- §4. Integration and Taylor’s Formula.- §5. The Inverse Mapping Theorem.- II Manifolds.- §1. Atlases, Charts, Morphisms.- §2. Submanifolds, Immersions, Submersions.- §3. Partitions of Unity.- §4. Manifolds with Boundary.- III Vector Bundles.- §1. Definition, Pull Backs.- §2. The Tangent Bundle.- §3. Exact Sequences of Bundles.- §4. Operations on Vector Bundles.- §5. Splitting of Vector Bundles.- IV Vector Fields and Differential Equations.- §1. Existence Theorem for Differential Equations.- §2. Vector Fields, Curves, and Flows.- §3. Sprays.- §4. The Flow of a Spray and the Exponential Map.- §5. Existence of Tubular Neighborhoods.- §6. Uniqueness of Tubular Neighborhoods.- V Operations on Vector Fields and Differential Forms.- §1. Vector Fields, Differential Operators, Brackets.- §2. Lie Derivative.- $3. Exterior Derivative.- §4. The Poincaré Lemma.- §5. Contractions and Lie Derivative.- §6. Vector Fields and 1-Forms Under Self Duality.- §7. The Canonical 2-Form.- §8. Darboux’s Theorem.- VI The Theorem of Frobenius.- §1. Statement of the Theorem.- §2. Differential Equations Depending on a Parameter.- §3. Proof of the Theorem.- §4. The Global Formulation.- §5. Lie Groups and Subgroups.- VII Metrics.- §1. Definition and Functoriality.- §2. The Hilbert Group.- §3. Reduction to the Hilbert Group.- §4. Hilbertian Tubular Neighborhoods.- §5. The Morse—Palais Lemma.- §6. The Riemannian Distance.- §7. The Canonical Spray.- VIII Covariant Derivatives and Geodesics.- §1. Basic Properties.- §2. Sprays and Covariant Derivatives.- §3. Derivative Along a Curve and Parallelism.- §4. The Metric Derivative.- §5. More LocalResults on the Exponential Map.- §6. Riemannian Geodesic Length and Completeness.- IX Curvature.- §1. The Riemann Tensor.- §2. Jacobi Lifts.- §3. Application of Jacobi Lifts to dexpx.- §4. The Index Form, Variations, and the Second Variation Formula.- §5. Taylor Expansions.- X Volume Forms.- §1. The Riemannian Volume Form.- §2. Covariant Derivatives.- §3. The Jacobian Determinant of the Exponential Map.- §4. The Hodge Star on Forms.- §5. Hodge Decomposition of Differential Forms.- XI Integration of Differential Forms.- §1. Sets of Measure 0.- §2. Change of Variables Formula.- §3. Orientation.- §4. The Measure Associated with a Differential Form.- XII Stokes’ Theorem.- §1. Stokes’ Theorem for a Rectangular Simplex.- §2. Stokes’ Theorem on a Manifold.- §3. Stokes’ Theorem with Singularities.- XIII Applications of Stokes’ Theorem.- §1. The Maximal de Rham Cohomology.- §2. Moser’s Theorem.- §3. The Divergence Theorem.- §4. The Adjoint of d for Higher Degree Forms.- §5. Cauchy’s Theorem.- §6. The Residue Theorem.- Appendix The Spectral Theorem.- §1. Hilbert Space.- §2. Functionals and Operators.- §3. Hermitian Operators.