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Title:
Fundamentals of Differential Geometry [electronic resource] / by Serge Lang.
Author:
Lang, Serge.
Published:
New York, NY : Springer New York : Imprint: Springer, 1999.
Description:
XVII, 540 p. digital.
Series:
Graduate Texts in Mathematics, 191
Summary:
This is the new edition of Serge Lang's "Differential and Riemannian Manifolds." This text provides an introduction to basic concepts in differential topology, differential geometry, and differential equations, and some of the main basic theorems in all three areas: for instance, the existence, uniqueness, and smoothness theorems for differential equations and the flow of a vector field; the basic theory of vector bundles including the existence of tubular neighborhoods for a submanifold; the calculus of differential forms; basic notions of symplectic manifolds, including the canonical 2-form; sprays and covariant derivatives for Riemannian and pseudo-Riemannian manifolds; applications to the exponential map, including the Cartan-Hadamard theorem and the first basic theorem of calculus of variations.
Contents:
I.: General Differential Theory: Differential Calculus -- Manifolds. Vector Bundles. Vector Fields and Differential Equations. Operations on Vector Fields and Differential Forms. The Theorem of Frobenius -- II: Metrics, Covariant Derivatives and Riemannian Geometry: Metrics. Covariant Derivatives and Geodesics. Curvature. Jacobi Lifts and Tensorial Splitting of the Double Tangent Bundle. Curvature and the Variation Formula. An Example of Seminegative Curvature. Automorphisms and Symmetries -- III: Volume Forms and Integration: Volume Forms. Integration of Differential Forms. Stokes' Theorem. Applications of Stokes' Theorem -- Appendix: The Spectral Theorem.
Notes:
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SpringerLink (Online service)
ISBN:
9781461205418
Online Access:
e-book

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