Riemannian Geometry: Theory & Applications (Mathematics: Theory & Applications)

Riemannian geometry was the first topic I considered 'difficult' when I was at university. I didn't study it at undergraduate level but needed a background in riemannian geometry for a Master's level module on Morse Theory. So I essentially had to teach myself Riemannian geometry. I used Do Carmo's Differential Geometry of Curves and Surfaces at undergraduate level: I found it very long-winded, dry and seemed very much proof and proof and proof which gave the text a disconnected feel and quite painful to follow after a while. But it's counterpart, this text, I found much easier to follow for independent study. It's probably the best text on the subject but it does lack diagrams and the explanations of concepts are often quite brief. This text does have some excellent problems for you to attempt, but there are no solutions and there should be, that is my main complaint about the text. So in summary, a bit more detailed explanation and a lot more diagrams to support these details would massively improve the text. If you're studying this topic at university then this book is ideal as you'll have all the explanation in lectures and this text will serve as very concisely written mathematics.

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?

Este libro es la traducción en los primeros setenta de un texto en portugués. Su principal interés reside en el hincapié que hace en destacar el papel que juegan las distintas hipótesis y condiciones en los teoremas de la geometria diferencial. Si alguien quiere entender lo que está haciendo al manipular símbolos este es el libro.

Habe es für mein Studium gebraucht und fand es sehr hilfreich. Hat mein Verständnis erweitert, die VL unterstützt. Wer sich deutsche Mathebücher kaufen will, sollte sich endlich daran gewöhnen, dass die wirklichen Referenzen auf Englisch sind. Dieses Buch ist so eine Referenz. Klare Empfehlung, darf in keinem Bücherregal fehlen.

Puntuale, come da descrizione

Do Carmo's book it's a sucessfully introduction to Riemannian geometry, Un my personal point oficina view it gives a general Outlook about this subject so people who doesn't know anything about Riemannian geometry I recommend this book to introduce un this awesome wolrd of geometry with a previous good calculs Andrea linear algebra knowledge.