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by Serge Lang
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Mathematics
  • Author:
    Serge Lang
  • ISBN:
    3540943382
  • ISBN13:
    978-3540943389
  • Genre:
  • Publisher:
    Springer-Verlag Berlin and Heidelberg GmbH & Co. K; 3rd edition (April 1, 1995)
  • Pages:
    377 pages
  • Subcategory:
    Mathematics
  • Language:
  • FB2 format
    1209 kb
  • ePUB format
    1425 kb
  • DJVU format
    1877 kb
  • Rating:
    4.6
  • Votes:
    417
  • Formats:
    lrf rtf lit azw


An introduction to differential geometry, starting from recalling differential calculus and going through all the basic topics such as manifolds, vector bundles, vector fields, the theorem of Frobenius, Riemannian metrics and curvature.

An introduction to differential geometry, starting from recalling differential calculus and going through all the basic topics such as manifolds, vector bundles, vector fields, the theorem of Frobenius, Riemannian metrics and curvature.

ourse in Differential Geometry. Graduate texts in mathematics ;176). A Course in Mathematical Logic. ISBN 0-387-98271-X (hardcover : alk. paper) 1. Reimannian manifolds. The purpose of this book is to introduce the theory of Riemannian manifolds: these are smooth manifolds equipped with Riemannian met-rics (smoothly varying choices of inner products on tangent spaces), which allow one to measure geometric quantities such as distances and angles.

Part of the Graduate Texts in Mathematics book series (GTM, volume 160). This is the third version of a book on differential manifolds. The first version appeared in 1962, and was written at the very beginning of a period of great expansion of the subject. At the time, I found no satisfactory book for the foundations of the subject, for multiple reasons. I expanded the book in 1971, and I expand it still further today. De Rham cohomology Hodge decomposition Riemannian geometry cohomology curvature differential geometry exterior derivative homology manifold vector bundle.

This is the third version of a book on differential manifolds Differential and Riemannian Manifolds Graduate Texts in Mathematics (Том 160). Издание: 3, иллюстрированное.

This is the third version of a book on differential manifolds. Differential and Riemannian Manifolds Graduate Texts in Mathematics (Том 160).

Differential and Riemannian Manifolds book Differential and Riemannian Manifolds (Graduate Texts in Mathematics).

Differential and Riemannian Manifolds book. These are all covered for manifolds, modeled on Banach and Hilbert spaces, at no cost in complications, and some gain in the elegance of the proofs. In the finite-dimensional case, differential forms of top degree are discussed, leading to Stokes' theorem (even for manifolds with singular boundary), and several of its applications to the differential or Riemannian case. Differential and Riemannian Manifolds (Graduate Texts in Mathematics).

Graduate Texts in Mathematics 10.

Parts of the theory are contained in various books of Lang, especially and, and there are books of Koblitz and Robert (the latter now out of print) that concentrate on the analytic and modular theory.

This book is an introductory graduate-level textbook on the theory of. .Graduate Texts in Mathematics.

This book is an introductory graduate-level textbook on the theory of smooth manifolds.

This is the third version of a book on Differential Manifolds; in this latest expansion three chapters have been added on Riemannian and pseudo-Riemannian .

This is the third version of a book on Differential Manifolds; in this latest expansion three chapters have been added on Riemannian and pseudo-Riemannian geometry, and the section on sprays and Stokes' theorem have been rewritten. This text provides an introduction to basic concepts in differential topology, differential geometry and differential equations. In differential topology one studies classes of maps and the possibility of finding differentiable maps in them, and one uses differentiable structures on manifolds to determine their topological structure.

This is the third version of a book on Differential Manifolds; in this latest expansion three chapters have been added on Riemannian and pseudo-Riemannian geometry, and the section on sprays and Stokes' theorem have been rewritten. This text provides an introduction to basic concepts in differential topology, differential geometry and differential equations. In differential topology one studies classes of maps and the possibility of finding differentiable maps in them, and one uses differentiable structures on manifolds to determine their topological structure. In differential geometry one adds structures to the manifold (vector fields, sprays, a metric, and so forth) and studies their properties. In differential equations one studies vector fields and their integral curves, singular points, stable and unstable manifolds, and the like.

Tisicai
As other reviewers have indicated this text is a rarity in that it presents and analyzes infinite dimensional differential manifolds, hence the need for the Banach space setting at the outset. Adequate preparation is given in the author's text on real and functional analysis. He uses the language of categories to tidy up the various theories in the reviewed text but this is used more in a descriptive way. Category theory merely formalizes common notions of mathematical objects and their correspondences or mappings. Examples are sets and functions or mappings between them, groups and homomorphisms, etc. Properties of the mappings involve composition as well as inverse and more extended constructs. Categories just axiomatize this common behavior. His brief review is really enough as the main results are proven independent of categories-but category theory will allow or show the results to hold in other settings or theories. The author's famous text on algebra gives a rigorous and self contained presentation on categories if you're not satisfied. Lang believed in giving you the most bang for your buck-being a Bourbakist. He even proves the full version of Sard's theorem which most texts on differential topology relegate to references. If you've been through a more elementary text like Boothby and would like to see a more chiseled, formal, and modern approach get this book. Not for newbies.
Vikus
Well, we have here another book on differential manifolds, and another book by Serge Lang. Lang is well-known by writing (lots of) books on different topics in analysis and algebra, all of them in a quite "Bourbaki-like" style: attaining maximum generality, with less motivation than most students would like. This is no surprise, because Lang himself is a Bourbakist.
So, what's interesting about D&RM? It's a book very much like Lang's other books, only that here the Bourbakist's approach is quite happy: it's one of the very few books on his subject to present most of his results in infinite-dimensional (Banach) version, a must if you are interested in nonlinear functional analysis or dynamical systems. The exposition is very clean and clear: Lang uses categories all the way to estabilish the main relations between the different differential-topological structures and tools, and he does not hesitate in stating and using tools from analysis, such as Lebesgue measure and functional analysis' main theorems. The proofs are very polished and, in a certain sense, beautiful, a philosophy that permeates most of the book. As if it weren't enough, the book still contains an appendix with a Von Neumann's seminar about the spectral theorem.
All things considered, it's a quite "state-of-the-art" book about the basics of differential manifolds, from an analyst's perspective. This perspective provides differential topology with a lot of additional clarity and power. I don't know if most physicists would like this book, because its motivations, if any, are sparse and sometimes quite obscure, as long as physical applications are concerned. For a mathematician, however, this book is a gem: it's Lang at its best, and the perfect opening door to global analysis (the nonlinear analysis on infinite dimensional manifolds, a vast field of mathematics that encompasses dynamical systems and nonlinear functional analysis). Despite all that, I would also recommend to physicists to at least tackle this book, as an antidote to all the crap that the so-called "differential topology for physicists" books put on their heads, because I don't know a cleaner and more precise presentation of differential manifolds so far.
Welen
Lang's book is definitely not useful as textbook for classes or for self-guided study (learnt this the hard way). He is rather abstract and provides zero motivation for the theory. The book is obviously made for people who learnt diff. geometry elsewhere but want to read a cleaner and more modern treatment. To this end, Lang's book is useful. The best part is that manifolds are infinite-dimensional right away. This is probably the only reason for buying Lang instead of/in addition to Dieudonne as a reference. Otherwise, the book is a little too terse; fiber bundles are merely hinted at. Moreover, I think some of the proofs are unnecessarily complicated, such as the one for Frobenius theorem.
Yanki
This book is a proper subset of Lang's later book "Fundamentals of Differential Geometry (Graduate Texts in Mathematics, 191)".