Hello, I'm curious to what books etc. one could use to understand the basics of Ricci flow, what areas of math are needed and so? What areas should one specialize in? See it as a roadmap to understanding Ricci flow, or something.Say one wants to be able to read Perelman's proof of the Poincaré Conjecture.

Sorry if my english isn't that good and this seems a bit hurried, I'm on the run.

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To understand Perelman's proof of the Poincaré Conjecture, you need a solid background in Riemannian geometry. Many books can be used for an introduction to this field. There are two books I like on this subject: Riemannian Geometry, by Gallot, Hulin and Lafontaine and Riemannian Geometry by Petersen.

After, you can try to learn about Ricci flow, a good starting point is Chow and Knopff's "The Ricci Flow: an Introduction". It covers the basics of Ricci flow including Hamilton's theorem that on a compact 3-manifold with $Ric>0$, the (normalized) flow will converge to constant curvature.

Then, if you want to go into Perelman's work, there is the book "Ricci Flow and the Poincaré Conjecture" by Morgan and Tian. However you also have to understand Thurston's Geommetrization Conjecture, so you need a solid background in 3-manifold topology, I don't know the references for this part, maybe Thurston's lecture notes?

Another interesting road is to study the proof of the differentiable sphere theorem by Brendle and Schoen, a good reference is Brendle's "Ricci Flow and the Sphere Theorem".

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A phD student is supposed to make at least some subtle publicity to his advisor's work when such an opportunity occurs :) –  Benoît Kloeckner Sep 3 '10 at 12:25
I would also highly recommend Hamilton's 1995 survey "The formation of singularities in the Ricci flow". –  Robert Haslhofer Mar 27 '12 at 20:23

Another useful reference is Peter Topping's "Lectures on the Ricci Flow" which is freely available as a pdf at

Lectures on the Ricci Flow

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You might try Terence Tao's blog notes from his course on Perelman's proof. He assumes a basic understanding of Riemannian geometry (or at least goes over the requisite bits of it only very quickly) so you may also want to start with a book on Riemannian geometry (Tao himself was using Peter Petersen's book).

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In my view, the best place to start learning about Ricci flow is Hamilton's famous 1982 paper "Three-manifolds with positive Ricci curvature," modulo the short-time existence section. (DeTurck later came up with an easier way to prove short-time existence of solutions).

I like Hamilton's paper because it introduces the reader to the intense tensor computations involved in Ricci flow theory and requires only basic Riemannian geometry: Riemannian metrics, the Levi-Civita connection, covariant differentiation of tensor fields, parallel transport, geodesics, the exponential map, normal coordinates, curvature, the Hopf-Rinow theorem, variations of energy and Myers' theorem come to mind. Moreover, Hamilton proves the tensor maximum principle and illustrates the power of maximum principle techniques.

From there, you should be equipped to handle expository work on the Ricci flow. All of the sources mentioned above are great; I particularly like Simon Brendle's book "Ricci Flow and the Sphere Theorem" as a reference for convergence theory.

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The curvature evolution equation was not fully developed in 1982 and it simplifies presentation quite a bit. "Manifolds with positive curvature operators are space forms" by Boehm and Wilking is more readable and more general. –  Anton Petrunin Aug 9 '14 at 21:20

For the true Ricci flow part of Thomas' program, you can use the new B3MP (Bessières, Besson, Boileau, Maillot, Porti) book available at http://www-fourier.ujf-grenoble.fr/~besson/english_principal.pdf and to appear at EMS. It is aimed at explaining Ricci flow with surgery (or rather a variation called Ricci flow with bubbling-off) and the proof of geometrization to topologists and geometers, and the analysis of Ricci flow is mostly used as a blackbox, so that may suit you or not.

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In would recommend the book `The Ricci Flow in Riemannian Geometry' by Ben Andrews and Chris Hopper, which is available for download here:

http://maths.anu.edu.au/~andrews/publications.html

The book is suited to an honours/graduate student with a good background in Riemannian geometry. It develops Hamilton's Ricci flow from the ground up leading to Brendle and Schoen's proof of the differentiable sphere theorem and also provides a very good overview of the required geometry in the first chapter.

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Here is a list of literature which I compiled when I taught the course on Ricci flow.

Basic differential geometry:

Einstein Manifolds (Besse).
Riemannian geometry (Gallot S., Hulin D., Lafontaine J.)
Sign and geometric meaning of curvature (Gromov)
http://www.ihes.fr/~gromov/PDF/1%5B77%5D.pdf

Textbooks:
Lectures on the Ricci Flow (2006, 133 pp.) Topping P.
http://www.warwick.ac.uk/~maseq/RFnotes.html
Hamilton's Ricci Flow (Chow B., Lu P., Ni L.)

Standard texts:
http://en.wikipedia.org/wiki/Solution_of_the_Poincar%C3%A9_conjecture

Perelman, Grisha (November 11, 2002).
The entropy formula for the Ricci flow and its geometric applications.

Perelman, Grisha (March 10, 2003).
Ricci flow with surgery on three-manifolds.
Perelman, Grisha (July 17, 2003).
Finite extinction time for the solutions to the Ricci flow on certain three-manifolds.

Bruce Kleiner, John Lott. Notes on Perelman's papers
Huai-Dong Cao, Xi-Ping Zhu. Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture.
John W. Morgan, Gang Tian. Ricci Flow and the Poincaré Conjecture

It's very obsolete (2007), and does not contain much on short-term existence of solutions of Ricci flow.

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