Many courses called Differential Geometry (at least in Germany; at least as far as I know!) solely deal with manifolds and not classical/elementary Differential Geometry (curves, curvature, fundamental forms, ...).

So my simple question is: Should you study Elementary Differential Geometry before having a serious go at manifolds?

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    $\begingroup$ Should? No. Can? If you want. You should have some (basic) familiarity with the tools of analyis however, such as the derivative of a map $\Bbb R^n\to \Bbb R^m$. $\endgroup$ – s.harp Aug 1 at 19:56
  • $\begingroup$ About basic analysis tools: Sure. I don't know why anybody would be interested in manifolds before knowing basic analysis. $\endgroup$ – Qi Zhu Aug 1 at 20:11
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    $\begingroup$ Before you have a go at manifolds, not necessarily. I'm personally fond of Guillemin and Pollack's differential topology text, which works exclusively with submanifolds of $\Bbb R^n$. However, if you're going to take a sophisticated course in differential geometry (connections and curvature), then by all means get some intuition for it by first studying (curves and) surfaces. Otherwise, it's pretty sophisticated and often unmotivated. $\endgroup$ – Ted Shifrin Aug 1 at 20:15
  • $\begingroup$ For geometry, say, the classical groups are really basic, like $SO(n)$ for example. Then one should know what a group is, and also what a Lie group is (hence what a manifold is). Actually, I encountered $SO(3)$ even before basic analysis with maps $\Bbb R^n\rightarrow \Bbb R^m$. $\endgroup$ – Dietrich Burde Aug 1 at 20:32

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