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Metric Structures in Differential Geometry

This text is an introduction to the theory of differentiable manifolds and fiber bundles. The only requisites are a solid background in calculus and linear algebra, together with some basic point-set topology. The first chapter provides a comprehensive overview of differentiable manifolds. The follo...

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Bibliographic Details
Main Author: Walschap, Gerard
Format: eBook
Language:English
Published: New York, NY Springer New York 2004, 2004
Edition:1st ed. 2004
Series:Graduate Texts in Mathematics
Subjects:
Geometry, Differential
Manifolds And Cell Complexes
Manifolds (mathematics)
Differential Geometry
Global Analysis (mathematics)
Global Analysis And Analysis On Manifolds
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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

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