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Metric Structures in Differential Geometry - Gerard Walschap

Metric Structures in Differential Geometry

(Autor)

Buch | Softcover
229 Seiten
2010 | Softcover reprint of the original 1st ed. 2004
Springer-Verlag New York Inc.
978-1-4419-1913-7 (ISBN)
CHF 97,35 inkl. MwSt
They are first told that these are derivations on certain equiv­ alence classes of functions, but later that the tangent space of ffi.n is "the same" n as ffi. We have tried to keep these spaces separate and to carefully explain how a vector space E is canonically isomorphic to its tangent space at a point.
This text is an elementary introduction to differential geometry. Although it was written for a graduate-level audience, the only requisite is a solid back­ ground in calculus, linear algebra, and basic point-set topology. The first chapter covers the fundamentals of differentiable manifolds that are the bread and butter of differential geometry. All the usual topics are cov­ ered, culminating in Stokes' theorem together with some applications. The stu­ dents' first contact with the subject can be overwhelming because of the wealth of abstract definitions involved, so examples have been stressed throughout. One concept, for instance, that students often find confusing is the definition of tangent vectors. They are first told that these are derivations on certain equiv­ alence classes of functions, but later that the tangent space of ffi.n is "the same" n as ffi. . We have tried to keep these spaces separate and to carefully explain how a vector space E is canonically isomorphic to its tangent space at a point. This subtle distinction becomes essential when later discussing the vertical bundle of a given vector bundle.

1. Differentiable Manifolds.- 1. Basic Definitions.- 2. Differentiable Maps.- 3. Tangent Vectors.- 4. The Derivative.- 5. The Inverse and Implicit Function Theorems.- 6. Submanifolds.- 7. Vector Fields.- 8. The Lie Bracket.- 9. Distributions and Frobenius Theorem.- 10. Multilinear Algebra and Tensors.- 11. Tensor Fields and Differential Forms.- 12. Integration on Chains.- 13. The Local Version of Stokes’ Theorem.- 14. Orientation and the Global Version of Stokes’ Theorem.- 15. Some Applications of Stokes’ Theorem.- 2. Fiber Bundles.- 1. Basic Definitions and Examples.- 2. Principal and Associated Bundles.- 3. The Tangent Bundle of Sn.- 4. Cross-Sections of Bundles.- 5. Pullback and Normal Bundles.- 6. Fibrations and the Homotopy Lifting/Covering Properties.- 7. Grassmannians and Universal Bundles.- 3. Homotopy Groups and Bundles Over Spheres.- 1. Differentiable Approximations.- 2. Homotopy Groups.- 3. The Homotopy Sequence of a Fibration.- 4. Bundles Over Spheres.- 5. The Vector Bundles Over Low-Dimensional Spheres.- 1. Connections on Vector Bundles.- 4. Connections and Curvature.- 2. Covariant Derivatives.- 3. The Curvature Tensor of a Connection.- 4. Connections on Manifolds.- 5. Connections on Principal Bundles.- 5. Metric Structures.- 1. Euclidean Bundles and Riemannian Manifolds.- 2. Riemannian Connections.- 3. Curvature Quantifiers.- 4. Isometric Immersions.- 5. Riemannian Submersions.- 6. The Gauss Lemma.- 7. Length-Minimizing Properties of Geodesics.- 8. First and Second Variation of Arc-Length.- 9. Curvature and Topology.- 10. Actions of Compact Lie Groups.- 6. Characteristic Classes.- 1. The Weil Homomorphism.- 2. Pontrjagin Classes.- 3. The Euler Class.- 4. The Whitney Sum Formula for Pontrjagin and Euler Classes.- 5. Some Examples.- 6. The Unit SphereBundle and the Euler Class.- 7. The Generalized Gauss-Bonnet Theorem.- 8. Complex and Symplectic Vector Spaces.- 9. Chern Classes.

Erscheint lt. Verlag 29.11.2010
Reihe/Serie Graduate Texts in Mathematics ; 224
Zusatzinfo 7 Illustrations, black and white; VIII, 229 p. 7 illus.
Verlagsort New York, NY
Sprache englisch
Maße 155 x 235 mm
Themenwelt Mathematik / Informatik Mathematik Analysis
Mathematik / Informatik Mathematik Geometrie / Topologie
ISBN-10 1-4419-1913-9 / 1441919139
ISBN-13 978-1-4419-1913-7 / 9781441919137
Zustand Neuware
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