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Fundamentals of Differential Geometry (Graduate Texts in Mathematics)
 
 
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Fundamentals of Differential Geometry (Graduate Texts in Mathematics) [Hardcover]

Serge Lang (Author)
4.0 out of 5 stars  See all reviews (1 customer review)

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Book Description

038798593X 978-0387985930 December 30, 1998 Corrected
This book provides an introduction to the basic concepts in differential topology, differential geometry, and differential equations, and some of the main basic theorems in all three areas. This new edition includes new chapters, sections, examples, and exercises. From the 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." --EMS NEWSLETTER

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Editorial Reviews

Review

"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

Product Details

  • Hardcover: 581 pages
  • Publisher: Springer; Corrected edition (December 30, 1998)
  • Language: English
  • ISBN-10: 038798593X
  • ISBN-13: 978-0387985930
  • Product Dimensions: 9.3 x 6.4 x 1.4 inches
  • Shipping Weight: 2.1 pounds (View shipping rates and policies)
  • Average Customer Review: 4.0 out of 5 stars  See all reviews (1 customer review)
  • Amazon Best Sellers Rank: #1,646,209 in Books (See Top 100 in Books)

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17 of 22 people found the following review helpful:
4.0 out of 5 stars Complete but lacking unity, March 28, 2000
This review is from: Fundamentals of Differential Geometry (Graduate Texts in Mathematics) (Hardcover)
This book looks more like a big collection of essays than really a treatise on one subject. If you have read other books by Lang you will notice that some material included in previous works appears reproduced here word by word. Also, there are some inconsistencies in the notation of different sections, making it obvious that different parts of the book were written at different times, and perhaps by different persons.

All these shortcomings don't mean that the book is bad. Quite the opposite: It is a very complete survey on modern differential geometry, including from the fundamentals up to recent results. The graduate student and the working mathematician will find it very useful.

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Inside This Book (learn more)
First Sentence:
We shall recall briefly the notion of derivative and some of its useful properties. Read the first page
Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
toplinear isomorphism, seminegative curvature, semi parallelogram law, tensorial derivative, toplinear automorphism, metric increasing property, canonical spray, differential isomorphism, double tangent bundle, tangent subbundle, symmetric submanifold, vector bundle chart, local uniqueness theorem, vector fields and differential equations, normal submanifold, map expx, implicit mapping theorem, block spanned, inverse mapping theorem, metric isomorphism, geodesic submanifold, locally finite open covering, positive definite scalar product, oriented chart, exponential map
Key Phrases - Capitalized Phrases (CAPs): (learn more)
Proof Let, Proof Immediate, Using Lemma
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