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Graduate Texts in Mathematics

Fundamentals of Differential Geometry

Authors: Lang, Serge

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About this Textbook

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.

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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 differential forms are expounded. The book ends with the Stokes theorem and some of its applications."-- MATHEMATICAL REVIEWS


Table of contents (18 chapters)

Table of contents (18 chapters)

Buy this book

eBook $59.99
price for USA in USD
  • ISBN 978-1-4612-0541-8
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $109.99
price for USA in USD
Softcover $74.99
price for USA in USD
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Bibliographic Information

Bibliographic Information
Book Title
Fundamentals of Differential Geometry
Authors
Series Title
Graduate Texts in Mathematics
Series Volume
191
Copyright
1999
Publisher
Springer-Verlag New York
Copyright Holder
Springer Science+Business Media New York
eBook ISBN
978-1-4612-0541-8
DOI
10.1007/978-1-4612-0541-8
Hardcover ISBN
978-0-387-98593-0
Softcover ISBN
978-1-4612-6810-9
Series ISSN
0072-5285
Edition Number
1
Number of Pages
XVII, 540
Topics

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