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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.2011-11-18
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    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.2011-11-18
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    A really excellent book; I happen to be reading it right now as well!2011-11-18
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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.2011-11-18
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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.2011-11-18
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    http://abstrusegoose.com/272 (click on the comic).2011-11-18
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    For learning about Riemann surfaces, I like Forster's book. As far as I recall, you shouldn't have to know about manifolds and differential geometry in order to start reading it.2011-11-18

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