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Riemannian Geometry, by Manfredo Perdigao do Carmo
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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.
- Sales Rank: #83146 in Books
- Published on: 1992
- Original language: English
- Number of items: 1
- Dimensions: 9.21" h x .75" w x 6.14" l, 1.25 pounds
- Binding: Hardcover
- 300 pages
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
Language Notes
Text: English (translation)
Original Language: Portugese
About the Author
Francis Flaherty has worked for more than seventeen years at The New York Times. He has written for Harper's, Atlantic Monthly, Commonweal, and The Progressive and teaches journalism at NYU. He lives with his wife and two children in Brooklyn, New York.
Most helpful customer reviews
31 of 31 people found the following review helpful.
Best 1st semester Riemannian Geometry book after 1 semester DG
By Christina Sormani
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.
0 of 0 people found the following review helpful.
Excellent stepping stone to more advanced treatments
By Paul A. Bonyak
Though this text lacks a categorical flavor with commutative diagrams, pull-backs, etc. it is still at an intermediate to advanced level. Nevertheless, constructs are developed which are assumed in a categorical treatment. It does do Hopf-Rinow, Rauch Comparison, and the Morse Index Theorems which you would find in a text like Bishop-Crittendon. However, it does the Sphere Theorem, an advanced theorem dependent on the Morse Theory/calculus of variations methods in differential geometry. 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. Note in this construction we used the embedding in our interpretation, nevertheless this gradient/tangent hyperplane notion can be given an intrinsically defined method of getting the tangent space through the related notion of the directional derivative. Forging this to a linear tangent space is a key construct which the reader should grasp, one not available in Gauss's lifetime. 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. 56-57 of do Carmo in which you are to show how the affine connection and covariant derivative arise from parallel transport. Theorem 3.12 of Chapter VII in Boothby does this a bit too formally but you can find it in various forms on the web. 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."
I have certain quibbles like in defining the Riemannian metric as a bilinear symmetric form,i.e., his notation is a bit dated here and there but the text shines from chapter 5 on. So 5 stars.
P.S. There's a PDF entitled "An Introduction to Riemannian Geometry" by Sigmundur Gudmundsson which is free and short and is tailor made for do Carmo assuming only advanced calculus as in say rigorous proof of inverse function theorem or the first nine (or ten) chapters of Rudin's Principles 3rd. It does assume some familiarity with differential geometry in R^3 as in do Carmo's earlier text but you can probably fill this in from the web if you're not familiar from past coursework as in vector analysis. Differential manifold and tangent space are clearly developed without the topological detours-pretty much if you're familiar with the derivative as a linear map (as in Rudin), you're at the right level. Also Lang's "Introduction to Differentiable Manifolds" is available as a free PDF if you want to see the categorical treatment after you get through do Carmo-can also be used for reference concurrently, example-isomorphic linear spaces?
0 of 0 people found the following review helpful.
Whether To Read It
By Brook(LifeOfThePartyWoot)
:a full textbook of Riemannian geometry. it exists without linear difficulties that causes Euclidean geometry to eventually break down at points of movement.
:apparently the geometry that begins with an E and arrives from Greece is for the arrangement of numbers; thats why it looks at the sides of shapes. try and dont think its for music though.
:if youre looking for a full textbook of Riemannian geometry then look no further; and dont purchase the text by the name of Riemannian Geometry As A Field Over Another Geometry; that text is a doctoral thesis which has little or no connection to the desirable full textbook for sale on the site.
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