Customer Reviews

Riemannian Geometry

5 star:		(7)
4 star:		(1)
3 star:		(0)
2 star:		(0)
1 star:		(0)

 

 

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.

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.

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.

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.

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.

this book covers basic definitions of submanifold, riema mannian metrics, curvature, geodesics and morse theory, sphere theorem, which is the main content in riemannian geometry. I think the author deals very readily, and this dealing is suitable for beginners .

I studied the portuguese version of this book during the master degree program in mathematics at University of Brasilia, 1999. The book is very well written with beautiful results. Manfredo is an excellent mathematician, a great professor, and I had the chance to be present in many colloquiuns where he was the speaker. This is an very good exposition for those interested to learn more about the subject.

This is another well-written text by Do Carmo. I browsed through it and found I could not understand several passages because I did not know what the special symbols meant and there was no table of symbols. I plead with the publisher to add such a table to the next edition or printing.