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Would I need anything else after Rudin? I've heard that the first volume in Spivak's Differential Geometry series covers calculus on manifolds, so would it be necessary to first read his smaller book Calculus on Manifolds?

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Further, you need to be familiar with things like metric spaces. Familiarity with topology is even better.

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I would comment about Spivak's differential geometry; but you seem to consider only Spivak from your OP. I personally find Spivak too verbose (I ended up not learning diff. geom. from him), and preferred Subrovin/Fomenko/Novikov's modern geometry for a physics' approach. Otherwise Lee's books (Intro to topological manifolds/smooth manifolds) or Guillemin & Pollacks' are as good.

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From what you and Redsummers are saying it sounds like I will be fine with just single variable analysis and familiarity with multivariable calculus. This comes as music to my ears.

The reason I'm so set on Spivak's Differential Geometry books is that my first physics professor told me to that they are true gems and should be digested thoroughly.

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micromass, would the topology in Rudin be sufficient? I was scared after reading the first Amazon review (of Spivak's DG 1 text) which listed differential topology as well as abstract algebra as prerequisites. Also, could you recommend a book for multivariable calculus? Do you mean something like Stewart?

Calculus on manifolds is actually a multivariable calc text. But it's a little compact.

Stewart would of course also be good, but it's not really rigorous.

As for the prerequisities of diff topology and abstract algebra. It's certainly not needed. But as with everything in math: the more math you're familiar with, the easier it will be for you. So if you know abstract algebra, then it will likely be a bit easier. But don't learn abstract algebra just to read Spivak, that's overkill.

From what you and Redsummers are saying it sounds like I will be fine with just single variable analysis and familiarity with multivariable calculus. This comes as music to my ears.

Yeah, but you'll need more than familiarity with multivariable calculus. You need to know it inside out. Differential geometry is actually a generalization of multivariable calculus to more abstract spaces (=manifolds). So you must know the former really, really well.

As for analysis: you need to be acquainted with metric spaces and the topology of metric spaces. If you insist on using Rudin (which I think is a horrible book, but other people think it's a gem), you will need to know chapter 2 really well. The other chapters are things you will probably remember from calculus, but it's good to go through to them anyway. Chapter 9-11 are horrible in Rudin, so I would suggest another book for that

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