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Let $f:\mathbb{R}^n\to \mathbb{R}^m$ be a smooth function (or in general between two smooth manifolds). Then $p\in \mathbb{R}^n$ is a critical point if $df_p$ is not surjective. I feel confused about this definition. If $n<m$, then $df_p$ can never be surjective, so that every point in $\mathbb{R}^n$ is critical in this case?!

For instance, let $\alpha:\mathbb{R}\to\mathbb{R}^2$ be the curve $\alpha(t)=(\sin t,\cos t)$. Is every $t$ critical?

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  • $\begingroup$ Are you suuuuuure that's the definition of critical point? $\endgroup$ – Robin Goodfellow Mar 13 '15 at 22:09
  • $\begingroup$ @RobinGoodfellow Yes, I'm sure. $\endgroup$ – Vladimir Mar 13 '15 at 22:23
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    $\begingroup$ @DBS I think Sard says the set of critical values has measure zero, not necessarily the set of critical points. $\endgroup$ – Ben West Mar 14 '15 at 0:07
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    $\begingroup$ Typically, we define a point to be critical if and only if the pushforward is not injective. This coincides with the standard definition of critical point from elementary calculus (i.e. $f'(t)=0\iff t\text{ is a critical point}$). $\endgroup$ – Robin Goodfellow Mar 14 '15 at 0:35
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    $\begingroup$ Actually, here's a better link corroborating OP's definition. $\endgroup$ – André 3000 Mar 14 '15 at 4:02
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Yes, that's the definition! From a manifold of dimension smaller than the target all points are critical, and the regular values are the complement of the image. (For instante, the little Sard thm says that complement is residual.)

This may seem not natural in the circle example. There we are dealing with a different matter, that is a regular parametrization of the circle: when you consider that circle $S$ as a manifold (a curve), the mapping $\alpha:\mathbb R\to S$ has all points regular.

One main concern of regular points and values is to analyse properly level manifolds (inverses images that are smooth manifolds). The first instance of the notion ofregular point is the Implicit Functions Thm, the far reaching generalization is Thom's crucial notion of Transversality in Differential Topology.

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  • $\begingroup$ But can I ask why such a definition is counter-intuitive considering the circle example I give in the question? Why is defining critical point this way beneficial in that case? $\endgroup$ – Vladimir Mar 13 '15 at 22:40
  • $\begingroup$ I've edited a bit my answer. $\endgroup$ – Jesus RS Mar 13 '15 at 22:43

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