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I am reading Road to Reality by Roger Penrose and I although I know about calculus, complex analysis, differential equations I do not know about manifolds, Riemann surfaces and so on. Which books can you recommend me in order to master the topics found on that book?

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James Munkres' "Analysis on Manifolds" and Michael Spivak's "Calculus on Manifolds" are nice accessible texts. – Bill Cook Nov 18 '11 at 0:36
It' can't be done. You need to know the material before you read the book, then you really wouldn't need to read the book unless you really wanted to know what Roger Penrose thoughts. – simplicity Nov 18 '11 at 0:48
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@simplicity I think the OP is looking for books to expand on what is discussed in the Road to Reality given he is currently reading it. – Bill Cook Nov 18 '11 at 1:07
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I will put in my standard warning about Calculus on Manifolds -- I found it really dense and unmotivated, and disliked it intensely. But people keep recommending it, so it obviously works for a lot of students. And it might work better mixed with Penrose, since he is verbose and extremely intuitive, but lacks the details you'll find in Spivak's book. – David Speyer Nov 18 '11 at 2:06
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abstrusegoose.com/272 (click on the comic). – Hans Lundmark Nov 18 '11 at 7:50
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