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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"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
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17 of 22 people found the following review helpful:
4.0 out of 5 stars
Complete but lacking unity,
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
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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 New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
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 New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 52
books:
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94
books
cite this book:
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- Real and Functional Analysis (Graduate Texts in Mathematics) (v. 142) by Serge Lang in Front Matter, and Back Matter
- Undergraduate Analysis (Undergraduate Texts in Mathematics) by Serge Lang in Front Matter, and Back Matter
- Introduction to Complex Hyperbolic Spaces by Serge Lang in Front Matter, and Back Matter
- Advanced Calculus, Revised Edition by Loomis in Back Matter
- Algebraic Number Theory (Cambridge Studies in Advanced Mathematics) by A. Fröhlich in Front Matter
See all 52 books this book cites
- Algebra by Serge Lang on 4 pages
- Complex Analysis (Graduate Texts in Mathematics) by Serge Lang in Front Matter, and Back Matter
- Analyse mathématique III: Fonctions analytiques, différentielles et variétés, surfaces de Riemann (French Edition) by Roger Godement on page 146, and page 230
- Positive Definite Matrices (Princeton in Applied Mathematics) by Rajendra Bhatia on page 232, and Back Matter
- Introduction to Differentiable Manifolds by Serge Lang in Front Matter, and Back Matter
See all 94 books citing this book
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