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What do I need to learn to start on the rigorous study of differential geometry? I'm about to start my 3rd undergrad year at school, and have taken Cal 1-3, Linear Algebra, Elementary Number Theory, and ODEs.

If I want to take the most direct path to studying differential geometry, what should I start learning now? Should I start on Real Analysis or Topology? Should I just pick up Spivak's Intro to Differential Geometry? My multivariable calculus class did not cover differential forms, so should I read his Calculus on Manifolds or Munkres' Analysis on Manifolds first?

I'm pretty sure that I want to study differential geometry in grad school and I'd like to start on it before then (ideally, I'd like to be able to publish something -- not necessarily extremely deep -- in diff geo before applying to grad schools).

So what path should I take to self-study it?

NOTE: My school has 1 undergrad course in Differential Geometry available in the Spring, so I'll probably do that next year, but I'd rather start now and go well beyond whatever I'm going to learn in that class.

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  • $\begingroup$ I remember using a D.G. text by Boothby, thought it was very readable. $\endgroup$ – MPW Jun 15 '14 at 22:47
  • $\begingroup$ Ted recommended that, too (below). So I'll definitely be checking that out. $\endgroup$ – user157286 Jun 15 '14 at 22:55
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To do a standard undergraduate diff geo course, you don't need much beyond multivariable calculus and linear algebra. But if you want to pursue it at the graduate level, first and foremost, you need a solid foundation in multivariable analysis (with emphasis on the Inverse and Implicit Function Theorems and differential forms).

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  • $\begingroup$ So would that be Spivak's or Munkres' books? Or something else? $\endgroup$ – user157286 Jun 15 '14 at 22:35
  • $\begingroup$ Yes, those books will do fine. Munkres is a bit more readable — basically a rewrite of Spivak. $\endgroup$ – Ted Shifrin Jun 15 '14 at 22:36
  • $\begingroup$ And then is Spivak's Intro to Diff Geo a good follow-up? What exactly is the level of that series (undergrad or grad)? Will I need something else after that? $\endgroup$ – user157286 Jun 15 '14 at 22:37
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    $\begingroup$ Spivak's 5 volume book is way too wordy, but volume 1 is a good start. This is all graduate level. This is not easy stuff to "self-study." Boothby's introductory book covers things in a lot fewer pages and is a good starting point. But there's still plenty of time to learn this in graduate school. $\endgroup$ – Ted Shifrin Jun 15 '14 at 22:46
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    $\begingroup$ Very ambitious. But get started! :) $\endgroup$ – Ted Shifrin Jun 15 '14 at 23:36

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