TITLE: Prerequisite for Petersen's Riemannian Geometry
QUESTION [13 upvotes]: A professor recently told me that if I can cover the chapters on curvature in Petersen's Riemannian geometry book (linked here) within the next few months then I can work on something with him. However, before I plan on reading this book I need to pick up some manifold theory. I plan on reading Lee's Smooth Manifolds book. 
Assuming I don't know anything covered in Lee's book and that I want to read it just to be able to read Petersen's can somebody recommend me what chapters I can skip. I know it'll be beneficial to read the whole book but I'll cover the whole book in a course next year and I just want to be able to read the rudiments of Riemannian geometry before then. Or if one can suggest a book better suited for this goal that would be appreciated too. Thank you.

REPLY [15 votes]: Petersen's book is challenging, but very clear and thorough. If you want to learn the prerequisites quickly-as I'm sure all graduate students who want to begin research do-then John Lee's books aren't really the best option for you. They're wonderfully lucid and comprehensive texts,but thier sheer length means they're really best for self study when you have several months to invest. If you need a faster introduction, Loring Tu's An Introduction To Manifolds is a better choice. Unfortunately it avoids basic topology, so you're going to need to supplement it with a topology text. John McCleary's A First Course In Topology is short, beautifully written and probably has everything you need. Lastly, I'd be remiss if I didn't call to your attention Petersen's own notes on basic manifold theory, which he's made available at his website: http://www.math.ucla.edu/~petersen/manifolds.pdf 
They're more sophisticated then Tu and they'll make wonderful collateral reading. 
That should get you started and ready for your advisor by summer. Good luck!