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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.
A Customer on Nov 27, 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. Best 1st semester Riemannian Geometry book after 1 semester DG By Christina Sormani on Oct 26, 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.
Probably the best introduction to the subject. By A Reader on Mar 25, I had the pleasure of taking a course in Riemannian Geometry from the author himself, using the Portuguese version of this book. The fact is that the book is extremely well-written. 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. 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. 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.
The book does not take a unified approach to the subject that fits nicely with the full generality of the theory. Exposition of key concepts of RG affine connection, riemannian connection,geodesics, parallelism and sectional curvature, 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. 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. Excelent book. By Leandro on Sep 01, I studied the portuguese version of this book during the master degree program in mathematics at University of Brasilia, 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. 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. However, this is not for the absolute beginner. Let me explain what kinds of knowledge you should have before digging into this book.
You should already be familiar with basic smooth manifold theory found in first few chapters of books such as "Introduction to Smooth Manifolds" by John Lee. For example, the author assumes that you already know how to define tangent space using "derivation. He also assumes you know tensors. Also, you have to be able to understand his notation from the proof. He has his own set of notations without explanation. With all these prerequisite, reading should be smooth and fun.
Overall, great book to read on your own! I think the author deals very readily, and this dealing is suitable for beginners. Needs a table of symbols By Marvin J. Greenberg on Feb 04, 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. A good book and classical of riemannian geometry By Andy Lee on Jul 28, if you are study in riemannian geometry this book is a good textbook for you! Bonyak on Jun 13, Though this text lacks a categorical flavor with commutative diagrams, pull-backs, etc. Nevertheless, constructs are developed which are assumed in a categorical treatment.
Even "energy" is treated which is the kinetic energy functional integral used to determine minimal geodesics, reminiscent of the Maupertuis Principle in mechanics. The reader is assumed to be familiar with differentiable manifolds but a somewhat scant Chapter 0 is given which mostly collects results which will be needed later. The treatment is dominated by the "coordinate-free" approach so emphasis is on the tangent plane or space and properties intrinsic to the surface with only a brief section on tensor methods given.
Realize the tangent space has the same dimension as the surface to which it is tangent and this can be greater than 2. If you remember from advanced calculus, you took the gradient of a function of n variables the function maps to a constant as a sphere say does. The gradient defined the normal to the n-1 dimensional tangent hyperplane to the surface. The surface is also n-1 dimensional since given n-1 values to the variables the nth value is determined by the function equation implicitly.
The text by Boothby is more user-friendly here and is also available online as a free PDF. Boothby essentially covers the first five chapters of do Carmo including Chapter 0 filling in many of the gaps. Both in Boothby and do Carmo the affine connection makes appearance axiomatically and the covariant derivative results from imposed conditions in a theorem construct.
If this is a bit hard to chew it was for me there are exercises 1 and 2 on pp. Theorem 3. In particular there is a nice one where the tangent planes are related along the curve over which the parallel transport or propagation occurs resulting in a differential equation which gives both the affine connection and the covariant derivative. Just Google "parallel transport and covariant derivative. So 5 stars. By Erum Dost on Jul 28, The learning curve for this text is pretty steep if you do not have some decent exposure to differential geometry.
As mentioned in the text, Riemannian geometry was a natural development of the differential geometry of surfaces in R3. The author does recall some familiar topics in the preliminary exercises product manifold, embedding of the Klein bottle in R4, the orientable double covering etc. Note that the purpose of such exercises is to simply highlight some important results on differentiable manifolds that are necessary in order to proceed with the following chapters on Riemann geometry including Riemannian- metrics, connections, geodesics and curvature.
Jacobi fields are introduced in chapter 5 in order to formalize the velocity of the deviation of the geodesic in previous sections, you will see that the curvature determines how fast the geodesics spread apart. The chapter on isometric immersions is critical in understanding how the Gauss formula generalizes the fundamental theorem of Gauss, which the author describes as the point of departure for Riemannian geometry and even goes so far as to provide a geometric interpretation of the sectional curvature that is essentially the definition of curvature used by Riemann.
While a little over half the book discusses local properties of Riemannian manifolds, chapter 7 introduces the interplay that exists between the local and global properties of a Riemannian manifold. This section first defines these global properties i. The remainder of the book introduces more sophisticated notions in Riemannian geometry i. No doubt these chapters are indispensable and allow this text in particular to become a classical reference to the material.
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Manfredo do Carmo
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. A Customer on Nov 27, 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.
MANFREDO PERDIGAO DO CARMO RIEMANNIAN GEOMETRY PDF
Journal of Differential Geometry , v. Stable hypersurfaces with zero scalar curvature in Euclidean space. Arkiv for Matematik , v. Do Carmo, M.