Riemannian Geometry is an expanded edition of a highly acclaimed and successful textbook (originally published in Portuguese) for first-year graduate students in mathematics and physics. The author's treatment goes very directly to the basic language of Riemannian geometry and immediately presents some of its most fundamental theorems. It is elementary, assuming only a modest background from readers, making it suitable for a wide variety of students and course structures. Its selection of topics has been deemed "superb" by teachers who have used the text. A significant feature of the book is its powerful and revealing structure, beginning simply with the definition of a differentiable manifold and ending with one of the most important results in Riemannian geometry, a proof of the Sphere Theorem. The text abounds with basic definitions and theorems, examples, applications, and numerous exercises to test the student's understanding and extend knowledge and insight into the subject. Instructors and students alike will find the work to be a significant contribution to this highly applicable and stimulating subject.
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"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
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First Sentence
The notion of a differentiable manifold is necessary for extending the methods of differential calculus to spaces more general than Rn. Read the first page
Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
The notion of a differentiable manifold is necessary for extending the methods of differential calculus to spaces more general than Rn. Read the first page
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Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
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Customer Reviews
Most helpful customer reviews
By A Customer
Format:Hardcover
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.
Most Helpful Customer Reviews on Amazon.com (beta)
Amazon.com:
4.9 out of 5 stars
9 reviews
43 of 45 people found the following review helpful
5.0 out of 5 stars
Excellent start~!
Nov 27 2003
By A Customer
- Published on Amazon.com
Format:Hardcover
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.
22 of 22 people found the following review helpful
5.0 out of 5 stars
Best 1st semester Riemannian Geometry book after 1 semester DG
Oct 26 2006
By Christina Sormani
- Published on Amazon.com
Format:Hardcover
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.
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.
22 of 24 people found the following review helpful
5.0 out of 5 stars
Probably the best introduction to the subject.
Mar 25 2005
By a reader
- Published on Amazon.com
Format:Hardcover
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.
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4.9 out of 5 stars
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