Fundamentals of Differential Geometry

Serge Lang

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Hardback

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English

Springer Verlag
01 September 2021

Mathematics & Sciences; Calculus & mathematical analysis; Differential & Riemannian geometry; Algebraic topology

Series: Graduate Texts in Mathematics

"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."

By: Serge Lang
Imprint: Springer Verlag
Country of Publication: United States
Edition: 4th Revised edition
Volume: v. 191
Dimensions: Height: 235mm, Width: 155mm, Spine: 31mm
Weight: 2.130kg
ISBN: 9780387985930
ISBN 10: 038798593X
Series: Graduate Texts in Mathematics
Pages: 581
Publication Date: 01 September 2021
Audience: College/higher education , Professional and scholarly , Further / Higher Education , Undergraduate
Format: Hardback
Publisher's Status: Active

I General Differential Theory.- I Differential Calculus.- II Manifolds.- III Vector Bundles.- IV Vector Fields and Differential Equations.- V Operations on Vector Fields and Differential Forms.- VI The Theorem of Frobenius.- II Metrics, Covariant Derivatives, and Riemannian Geometry.- VII Metrics.- VIII Covariant Derivatives and Geodesics.- IX Curvature.- X Jacobi Lifts and Tensorial Splitting of the Double Tangent Bundle.- XI Curvature and the Variation Formula.- XII An Example of Seminegative Curvature.- XIII Automorphisms and Symmetries.- XIV Immersions and Submersions.- III Volume Forms and Integration.- XV Volume Forms.- XVI Integration of Differential Forms.- XVII Stokes’ Theorem.- XVIII Applications of Stokes’ Theorem.

Reviews for Fundamentals of Differential Geometry

"""There are many books on the fundamentals of differential geometry, but this one is quite exceptional; this is not surprising for those who know Serge Lang's books. ... It can be warmly recommended to a wide audience."" EMS Newsletter, Issue 41, September 2001 ""The text provides a valuable introduction to basic concepts and fundamental results in differential geometry. A special feature of the book is that it deals with infinite-dimensional manifolds, modeled on a Banach space in general, and a Hilbert space for Riemannian geometry. The set-up works well on basic theorems such as the existence, uniqueness and smoothness theorem for differential equations and the flow of a vector field, existence of tubular neighborhoods for a submanifold, and the Cartan-Hadamard theorem. A major exception is the Hopf-Rinow theorem. Curvature and basic comparison theorems are discussed. In the finite-dimensional case, volume forms, the Hodge star operator, and integration of differentialforms are expounded. The book ends with the Stokes theorem and some of its applications.""-- MATHEMATICAL REVIEWS"