The present book aims to give a fairly comprehensive account of the fundamentals of differential manifolds and differential geometry. The size of the book influenced where to stop, and there would be enough material for a second volume (this is not a threat). At the most basic level, the book gives an introduction to the basic concepts which are used in differential topology, differential geometry, and differential equations. In differential topology, one studies for instance homotopy classes of maps and the possibility of finding suitable differen tiable maps in them (immersions, embeddings, isomorphisms, etc. ). One may also use differentiable structures on topological manifolds to deter mine the topological structure of the manifold (for example, it la Smale [Sm 67]). In differential geometry, one puts an additional structure on the differentiable manifold (a vector field, a spray, a 2-form, a Riemannian metric, ad lib. ) and studies properties connected especially with these objects. Formally, one may say that one studies properties invariant under the group of differentiable automorphisms which preserve the additional structure. In differential equations, one studies vector fields and their in tegral curves, singular points, stable and unstable manifolds, etc. A certain number of concepts are essential for all three, and are so basic and elementary that it is worthwhile to collect them together so that more advanced expositions can be given without having to start from the very beginnings.
By:
Serge Lang
Imprint: Springer-Verlag New York Inc.
Country of Publication: United States
Edition: Softcover reprint of the original 1st ed. 1999
Volume: 191
Dimensions:
Height: 235mm,
Width: 155mm,
Spine: 29mm
Weight: 854g
ISBN: 9781461268109
ISBN 10: 1461268109
Series: Graduate Texts in Mathematics
Pages: 540
Publication Date: 05 October 2012
Audience:
Professional and scholarly
,
Undergraduate
Format: Paperback
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"
