Prerequisite for Petersen's Riemannian Geometry 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.
 A: I really liked "Riemannian Geometry" by Manfredo do Carmo - http://www.amazon.com/dp/0817634908/ref=rdr_ext_sb_ti_sims_1 (a PDF copy can be found by googling "Riemannian geometry do Carmo" and looking at the first few search results)
Chapter 0 discusses the preliminaries from smooth manifold theory and subsequent chapters immediately begin in Riemannian geometry (Chapter 0 is only roughly 30 pages in length and the entire book is roughly 300 pages in length). Of course, one should be warned that Chapter 0 is quite terse and I think it is better to have some familiarity with smooth manifolds beforehand. However, with enough mathematical maturity, it should be possible to learn Riemannian geometry from do Carmo without any background in smooth manifold theory, beginning with Chapter 0.
Also, another alternative is "An Introduction to Differentiable Manifolds and Riemannian Geometry" by William Boothby - http://www.amazon.com/Introduction-Differentiable-Manifolds-Riemannian-Mathematics/dp/0121160513 
I really liked this book - it covers both smooth manifold theory (at roughly the level of Lee but in the space of 300, rather than 500 pages) and also covers Riemannian geometry in two chapters. The depth of coverage in Riemannian geometry is not very much but the coverage of smooth manifold theory is quite efficient compared to Lee. You could just read chapters 1 - 5 (roughly 200 pages with relatively large font) and skip chapter 6 (chapters 7 and 8 are on introductory Riemannian geometry, which you can read if you like, or move onto a more specialised textbook in the subject such as Peterson or do Carmo).
Take a look and let me know what you think!
A: 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! 
A: First a disclaimer: I have never read Petersen's book, so I'll answer based on having skimmed through the contents and first chapters and assuming it is not much different from other books of the same level. Therefore take this with a grain of salt.
I really like Lee's Smooth Manifolds, but since you want a brief introduction so you can study Riemmanian geometry, and you will take a course in manifolds, I would strongly suggest considering another book. Since Lee's is very complete you would have to read the first 16 chapters (possibly omitting cp. 7 and 13) so you cover basic structures, bundles and differential forms. That's over 300 hundred pages.
I'll second Amitesh Datta on Boothby's book, which is great. If you want an even shorter introduction I think is hard to do better than "Geometry of Manifolds" by Richard Bishop and Richard Crittenden: http://www.amazon.com/Geometry-Manifolds-AMS-Chelsea-Publishing/dp/0821829238
The first 6 chapters (~100 pages) gets you everything you need to start doing Riemmanian geometry, and if you like the book you can go on with it too. Although I don't recommend as the place to start, when I first studied manifolds I found very useful to have nearby Kobayashi and Nomizu's "Foundations of Differential geometry vol. 1": http://www.amazon.com/Foundations-Differential-Geometry-Classics-Library/dp/0471157333. The first 4 chapters (~150 pages) have what you need, so maybe you'll give it a shot when something is eluding you.
