Regular value

Let $M$ and $N$ be smooth manifolds , and let $f:M\to N$ be a smooth map .

Definition

A point $y\in N$ is called a regular value of $f$ if for every $x\in f^{-1}(y)$, the differential

is surjective (with tangent spaces understood in the usual sense).

Equivalently: $y$ is regular if $f$ is a submersion at every point of the fiber $f^{-1}(y)$. Points of $N$ that are not regular values are called critical values.

Regular value theorem

If $y\in N$ is a regular value of $f:M^m\to N^n$, then the fiber $f^{-1}(y)$ is an embedded smooth submanifold of $M$ of codimension $n$ (and hence of dimension $m-n$). Moreover, for each $x\in f^{-1}(y)$,

This identifies the tangent space to the level set with the kernel of the differential at points on that level set.

Examples

1. Distance-squared on the plane. For $f:\mathbb{R}^2\to \mathbb{R}$, $f(x,y)=x^2+y^2$, every $r>0$ is a regular value: along $f^{-1}(r)$ the gradient $(2x,2y)$ is never zero, so the differential is surjective. The value $0$ is not regular, since the differential vanishes at $(0,0)\in f^{-1}(0)$.

2. Height on the sphere. Let $f:S^2\to \mathbb{R}$ be the height function $f(x,y,z)=z$. Then every $c\in(-1,1)$ is a regular value and $f^{-1}(c)$ is a circle (a “latitude”). The values $c=\pm 1$ are not regular, because the fiber consists of a single pole where the differential cannot be surjective.

3. A map that is everywhere regular. If $f:M\to N$ is a smooth submersion , then every $y\in N$ is a regular value. For instance, for the projection $\pi_M:M\times F\to M$, all fibers $\pi_M^{-1}(m)=\{m\}\times F$ are smooth submanifolds.