Spivak a comprehensive introduction to differential geometry

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Michael David Spivak is a mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. He is the author of the five-volume Comprehensive Introduction to Differential Geometry. He received a Ph.D. from Princeton University under the supervision of John Milnor in 1964.

His book Calculus takes a very rigorous and theoretical approach to

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Roughly:

• calculus,
• multivariable calculus (including differential forms, at the level of, say, Spivak's Calculus on Manifolds, althought that's not the best book to learn from),
• a strong background in linear algebra, and some multilinear algebra (at least comparable to that in Spivak's Calculus on Manifolds)
• perhaps a bit of abstract algebra, so that you know what a "group" is, although I didn't really know this when I first read the book
• you should probably have seen the existence-and-uniqueness theorem for ODEs at some point, too.
• A one-semester ugrad course on point-set topology is probably a Good Thing as well, although you won't need most of it.

Your calculus background should certainly involve real proofs of things like the intermediate value theorem, and the extreme value theorem. Your multivariable course should have proven the implicit and inverse function theorems. And if you'd heard of Sard's theorem (Milnor's Topology from the Differentiable Viewpoint might be a good reference), that'd do no harm either.

To be honest: I'd recommend reading (and doing most of the exercises) in Barrett O'Neil's book "Elementary Differential Geometry" as a first step. It's all for surfaces in 3-space, but it'll ground you in the main ideas so that much of Spivak will just seem like reasonably natural generalizations of what you've already learned.

Oh...and all this is for Volume 1. Later volumes certainly rely on a bit more abstract algebra.