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Riemannian Geometry [Hardcover]

Manfredo P. do Carmo , Francis Flaherty
4.9 out of 5 stars  See all reviews (11 customer reviews)

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Book Description

September 23, 2011 0817634908 978-0817634902 1st ed. 1992. Corr. 14th printing 2013

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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Riemannian Geometry + Differential Geometry of Curves and Surfaces + A Comprehensive Introduction to Differential Geometry, Vol. 1, 3rd Edition
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Editorial Reviews


"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

Language Notes

Text: English (translation)
Original Language: Portugese

Product Details

  • Series: Mathematics: Theory & Applications
  • Hardcover: 320 pages
  • Publisher: Birkhäuser; 1st ed. 1992. Corr. 14th printing 2013 edition (September 23, 2011)
  • Language: English
  • ISBN-10: 0817634908
  • ISBN-13: 978-0817634902
  • Product Dimensions: 9.3 x 6.3 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 (11 customer reviews)
  • Amazon Best Sellers Rank: #82,760 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews
46 of 48 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.
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24 of 24 people found the following review helpful
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.
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24 of 26 people found the following review helpful
5.0 out of 5 stars Probably the best introduction to the subject. March 25, 2005
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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21 of 23 people found the following review helpful
5.0 out of 5 stars Definitely a good start November 30, 2005
This book is definitely a solid way to start in Riemannian geometry. The topics chosen give a glimpse of more advanced topics that the reader can venture to next, and the order covered leaves little confusion. The book is to the point, with little conversation about the concepts except at the very beginning of each chapter.

I only have two complaints, but neither would cause me to lower the rating to 4 stars.

1. There could be more "deep" exercises that allow the reader to explore more of the subtleties of the subject. And for what exercises there are, the author sometimes gives far too much away in "hints."

2. The book does not take a unified approach to the subject that fits nicely with the full generality of the theory. This is probably what makes the book good to start with, but there is still going to be a somewhat difficult transition from this book to a typical differential/riemannian geometry book. Namely, the basic language of vector bundles, pull backs/push forwards, tensors and tensor fields are either covered in a very specific framework or not at all.
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9 of 12 people found the following review helpful
5.0 out of 5 stars Concise and clear November 13, 2006
By Arzi
This is really a very good book to start Riemannian Geometry (RG). Exposition of key concepts of RG (affine connection, riemannian connection,geodesics, parallelism and sectional curvature, ...) are well motivated and concisely explained with numerous motivating and not so difficult execises. The book is self contained convenient for self study. It contains an introductory chapter on mathematical background explaining basic concepts as differentiable manifolds, immersion, embedding and so on, which are necessary to deal with RG. I have essentially one basic remark about this book. Formulation of RG as presented in it, is a little bit dated. Now, with the development of geometric algebra and Geometric calculus most, if not all, mathematical concepts needed to study RG like covariant derivative, curvature, and general tensors can be formulated without ressort to coordinates and in a manner to highlight their essential geometric features. Moreover derivation of certain formulae can be much easier and natural. For example the author defines the formula for |x^y| as sqrt(sqr(|x|).sqr(|y|)-sqr(inner product(x,y))). Then explains that it is the area of two dimensional parallelogram determined by the pair of vectors x and y. The reader might be puzzled as to how this formula is obtained. In the context of geometric algebra this is derived very naturally from basic concepts. Anyway, this remark does not diminish the value of this book.
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2 of 2 people found the following review helpful
I bought this book in order to understand, at my amateur/hobbyist level, the fundamental ideas behind geodesics and curvature of manifolds. I have finished 'Analysis on Manifolds' by Munkres fairly rigorously, solving most of the exercises. The transition between Munkres treatment of manifolds and do Carmo's was a little hard for me, possibly because I am studying on my own and don't have a teacher to help. In any case, once you get past that, do carmo's development using vector fields has been excellent - I am now studying the chapter on curvature. The concept of covariant derivative was a little difficult; I had some trouble mapping it to my understanding of derivatives from say, vector calculus. This was better explained in other material I found on the net.
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