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I'm an undergraduate getting ready to take a graduate course in general relativity next quarter. I purchased Wald's General Relativity (who incidentally will be teaching the class) in order to get a head start on some of the material. However, he sprints through all of the differential geometry at break neck pace, and is extremely concise. So, I got Lee's Introduction to Smooth Manifolds, which covers what Wald covers in 50 pages in about 300.

I was just wondering if anyone could offer some suggestions on how to study the material. I'm trying to proceed systematically through Lee, but don't think I'll have time to read each page in depth. However, sometimes I find that if I jump ahead, I will have missed some important notation or concept. Any alternative book or maybe just a list of topics to focus on would be extremely helpful. Thanks all.

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Personal Opinion: Boothby-Introduction to Differentiable Manifolds and Morita-Geometry of Differential Forms are books that can be read without following it page by page like Lee –  Test123 Dec 28 '13 at 22:52

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I recommend using Lee as a reference (it has a very thorough index which is helpful if you are going to use it as a reference book). If there is a concept in the course, or even in your own preparation, that you are struggling with due to Wald's concise nature, look it up in Lee's book for a very detailed explanation. Another benefit of Lee is that many lemmas, propositions, and theorems are actually proved rather than left as exercises. This allows you to see how arguments involving these new foreign objects should look.

I cannot recommend Lee enough and I think it would be a shame for you not to use it. In addition to what I said above, Lee has plenty of good problems from which you can learn a lot.

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It is not clear to me whether you are an aspiring physicist or mathematician but - besides the fine references given by Michael Albanese et al.- I would suggest another one which might be more in line with the immediate needs and style of a physicist.

Differential geometry and Lie groups for physicists by Marian Fecko

This book will give you what you need to get a headstart in GR (general relativity) and some QFT (quantum field theory) using an easy, conversational style.

Pleasee note that Wald employs Penrose's abstract index notation, which is similar, yet different than the usual index notation used by most/all mathematics books

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Thank you so much for the reference. I will be taking QFT next year and some mathematics courses that will involve differential geometry so this book may help me in more ways than one! I'll be sure to look into it. PS. I am actually neither! I think I'm going to medical school. –  BRayhaun Dec 30 '13 at 2:56
    
Then you will need to learn differential...diagnostic and integral... nutrition :-) –  magma Dec 30 '13 at 10:25

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