Riemannian Geometry (Hardcover)

~ Manfredo Perdigao do Carmo (Author), Francis Flaherty (Translator) "The notion of a differentiable manifold is necessary for extending the methods of differential calculus to spaces more general than Rn..." (more)
Key Phrases: Index Lemma, Preissman's Theorem, Theorem of Rauch (more...)

Editorial Reviews

Review

"This is one of the best (if even not just the best) book for those who want to get a good, smooth and quick, but yet thorough introduction to modern Riemannian geometry."

–Publicationes Mathematicae

"This is a very nice introduction to global Riemannian geometry, which leads the reader quickly to the heart of the topic. Nevertheless, classical results are also discussed on many occasions, and almost 60 pages are devoted to exercises."

–Newsletter of the EMS

"In the reviewer's opinion, this is a superb book which makes learning a real pleasure."

—Revue Romaine de Mathematiques Pures et Appliquees

"This mainstream presentation of differential geometry serves well for a course on Riemannian geometry, and it is complemented by many annotated exercises."

—Monatshefte F. Mathematik

Product Description

This text has been adopted at:

University of Pennsylvania, Philadelphia University of Connecticut, Storrs Duke University, Durham, NC California Institute of Technology, Pasadena University of Washington, Seattle Swarthmore College, Swarthmore, PA University of Chicago, IL University of Michigan, Ann Arbor

"In the reviewer's opinion, this is a superb book which makes learning a real pleasure."

- Revue Romaine de Mathematiques Pures et Appliquees

"This main-stream presentation of differential geometry serves well for a course on Riemannian geometry, and it is complemented by many annotated exercises."

- Monatshefte F. Mathematik

"This is one of the best (if even not just the best) book for those who want to get a good, smooth and quick, but yet thorough introduction to modern Riemannian geometry."

- Publicationes Mathematicae

Contents: Differential Manifolds * Riemannian Metrics * Affine Connections; Riemannian Connections * Geodesics; Convex Neighborhoods * Curvature * Jacobi Fields * Isometric Immersions * Complete Manifolds; Hopf-Rinow and Hadamard Theorems * Spaces of Constant Curvature * Variations of Energy * The Rauch Comparison Theorem * The Morse Index Theorem * The Fundamental Group of Manifolds of Negative Curvature * The Sphere Theorem * Index

Series: Mathematics: Theory and Applications

See all Editorial Reviews

Product Details

• Hardcover: 320 pages
• Publisher: Birkhäuser Boston; 1 edition (January 1, 1992)
• Language: English
• ISBN-10: 0817634908
• ISBN-13: 978-0817634902
• Product Dimensions: 9.3 x 6.6 x 0.8 inches
• Shipping Weight: 1.2 pounds (View shipping rates and policies)
• Average Customer Review: 4.9 out of 5 stars  See all reviews (7 customer reviews)
• Amazon.com Sales Rank: #151,478 in Books (See Bestsellers in Books)

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Customer Reviews

Most Helpful Customer Reviews

28 of 30 people found the following review helpful:

5.0 out of 5 stars Excellent start~!, November 27, 2003

By A Customer

I have gone through many books about riemannian geometry, only to find that most of them are playing magic in front of me. When it comes to curvature and variation of energy (arc length), most of the book are just playing around with the notations without drawing any geometric insight. When defining Levi-Civita connections, many books simply list out 4 meaningless formulae. I was so happy to read this book since it explains everything in riemannian geometry in a clear and concise way. Theoretical facts and geometrical interpretations are both having their place in this book.

Only one thing to notice: This book is a basic elementary introductory text in riemannian geometry. Those who want to know more should consult other book. Yet, as a first book in riemannian geometry, this book is undoubtedly the best.

11 of 11 people found the following review helpful:

5.0 out of 5 stars Best 1st semester Riemannian Geometry book after 1 semester DG, October 26, 2006

By	Christina Sormani (NYC) - See all my reviews (REAL NAME)

This is the best Riemannian Geometry book after students have finished a semester of differential geometry. It gives geometric intuition, has plenty of exercises and
is excellent preparation for more advanced books like Cheeger-Ebin.

Students should already know differential geometry (Spivak "Calculus on manifolds" and Spivak "Differential Geometry Volume I" might be used there)

Warning: the curvature tensor is defined backwards as compared to Cheeger-Ebin.

16 of 18 people found the following review helpful:

5.0 out of 5 stars Probably the best introduction to the subject., March 25, 2005

By	a reader - See all my reviews

I had the pleasure of taking a course in Riemannian Geometry from the author himself, using the Portuguese version of this book. Do Carmo managed to cover the whole thing in one semester without breaking a sweat; I don't know how he managed, or how we did. The fact is that the book is extremely well-written. It provides geometric insight but doesn't avoid computations. Also, the choice of topics is great, and they are ordered in a way that enhances the logical unity of the whole. The English translation seems to be every bit as good as the original. For a first course in Riemannian Geometry, this book might make a geometer out of you.