= Critical point of a smooth map
{title2=$\operatorname{rank}df_x<\dim Y$}
= Critical points of a smooth map
{synonym}
Imagine folding a sheet along its horizontal axis by the <smooth map> $F:\mathbb R^2\to\mathbb R^2$, $F(x,y)=(x,y^2)$. The inputs $(a,r)$ and $(a,-r)$ have the same output. Along $y=0$, horizontal motion still changes the output, but vertical motion produces no first-order change: $F(a,\varepsilon)-F(a,0)=(0,\varepsilon^2)$. These inputs are <critical points of a smooth map> even though their <differential> is not the zero <linear map>. The missing direction in the target <tangent space> is what matters.
\Image[/critical-points-values-fold.png]
{title=The fold $F(x,y)=(x,y^2)$: critical inputs, critical outputs, and fibers}
{id=critical-points-values-fold}
Precisely, let $f:X^m\to Y^n$ be a <smooth map> between <smooth manifolds> without boundary. An input $p\in X$ is a <critical point of a smooth map> when its <differential>
$$
df_p:T_pX\longrightarrow T_{f(p)}Y
$$
is not <surjective>, equivalently when its <rank of a linear map> is less than $n$. An input where it is <surjective> is a regular point; a map with no <critical points of a smooth map> is a <submersion>. The threshold is the target <dimension>, not $\min(m,n)$. In <manifold charts>, compute the <matrix rank> of the $n\times m$ <Jacobian matrix>; changes of charts preserve that <matrix rank>. https://www.math.toronto.edu/mein/teaching/LectureNotes/Man.pdf[Eckhard Meinrenken, Differentiable Manifolds, Definitions 1.8 and 1.10]{ref}
For the fold,
$$
DF_{(x,y)}=\begin{pmatrix}1&0\\0&2y\end{pmatrix},\qquad
\operatorname{rank}DF_{(x,y)}=\begin{cases}2&y\ne0,\\1&y=0.\end{cases}
$$
Thus its set of <critical points of a smooth map> is the entire horizontal axis. Away from that axis, its <differential> can produce every infinitesimal target displacement. At the axis, it can produce only horizontal ones. This diagnoses where the two inverse branches in the picture merge.
For a scalar <smooth function> $f:X\to\mathbb R$, the target <dimension> is one. A <linear map> to $\mathbb R$ fails to be <surjective> exactly when it is zero, so here a <critical point of a smooth map> means $df_p=0$. In <Euclidean space> this is precisely the familiar <critical point> condition $\nabla f(p)=0$: every first <partial derivative> vanishes. On a <Riemannian manifold>, it is equivalently $\nabla_gf(p)=0$ for the chosen <Riemannian metric>; the <Riemannian gradient> depends on that metric, but whether it vanishes does not. See https://www.math.toronto.edu/~jeffrey/matd67/tu.pdf[Loring W. Tu, An Introduction to Manifolds, Proposition 8.23] for the scalar coordinate criterion. The scalar <smooth function> $f(t)=t^3$ has a <critical point> at zero without a <local maximum> or <local minimum>.
A source of lower <dimension> gives a different warning. For the <smooth map> $G:\mathbb R\to\mathbb R^2$, $G(t)=(t,t^2)$, the <Jacobian matrix> is the column $(1,2t)^T$, of <matrix rank> one at every input. Every input is a <critical point of a smooth map>, since the target <dimension> is two, even though this <parametric curve> has nonzero <velocity> everywhere and is an <immersion>. More generally, if $m<n$, no <differential> $T_pX\to T_{f(p)}Y$ can be <surjective>, so every input is critical. The terminology measures the ability to fill target directions, rather than whether the parametrized <curve> looks smooth.
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