Overview
- New edition of a successful Serge Lang title
- Written in the authors unique and engaging style, with clear and elegant proofs
- Covers the fundamentals of differential geometry, differential topology, and differential equations
- Includes new chapters on Jacobi lifts, tensorial splitting of the double tangent bundle, curvature and the variation formula, and an example of semi-negative curvature
- New chapters, sections, examples, and exercises have been added
Part of the book series: Graduate Texts in Mathematics (GTM, volume 191)
Access this book
Tax calculation will be finalised at checkout
Other ways to access
About this book
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.
Similar content being viewed by others
Keywords
Table of contents (18 chapters)
-
Front Matter
-
General Differential Theory
-
Front Matter
-
-
Metrics, Covariant Derivatives, and Riemannian Geometry
-
Front Matter
-
-
Volume Forms and Integration
-
Front Matter
-
Reviews
"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
Authors and Affiliations
About the author
Bibliographic Information
Book Title: Fundamentals of Differential Geometry
Authors: Serge Lang
Series Title: Graduate Texts in Mathematics
DOI: https://doi.org/10.1007/978-1-4612-0541-8
Publisher: Springer New York, NY
-
eBook Packages: Springer Book Archive
Copyright Information: Springer Science+Business Media New York 1999
Hardcover ISBN: 978-0-387-98593-0Published: 30 December 1998
Softcover ISBN: 978-1-4612-6810-9Published: 05 October 2012
eBook ISBN: 978-1-4612-0541-8Published: 06 December 2012
Series ISSN: 0072-5285
Series E-ISSN: 2197-5612
Edition Number: 1
Number of Pages: XVII, 540
Topics: Algebraic Topology, Analysis