TITLE: What prerequisites do I need to read the book Ricci Flow and the Poincare Conjecture, published by CMI
QUESTION [14 upvotes]: As mentioned in the title, I want to understand the proof of Poincare Conjecture by Perelman, what prerequisites do I need?

REPLY [7 votes]: My humble advice for learning about Ricci flow generally, after obtaining some background in Riemannian geometry, would be to start with a book which gets you to important results quickly. An excellent book is the one by Peter Topping. (The only typo I observed there is the one regarding backwards uniqueness, which is now due to Brett Kotschwar.) After that, there are excellent books on the differentiable spherical space form theorem by Brendle and Andrews--Hopper; see also the original papers of Boehm--Wilking and Brendle--Schoen.
What is irreplaceable is to read and to reread the original works by the masters, i.e., Hamilton and Perelman. A collection of Ricci flow papers, mostly by Hamilton, is edited by H.D. Cao, etal.; this is a convenient place to get Hamilton's papers in one place. Perelman's papers are on arXiv. There are a number of excellent expositions of their work (focused on Perelman's work), which actually go beyond expositions and include various degrees of original work, namely in alphabetical order: Bessieres--Besson--Boileau--Maillot--Porti, Cao--Zhu, Kleiner--Lott, and Morgan--Tian.
The above remarks only pertain to Riemannian Ricci flow.