27
$\begingroup$

What is immersion and submersion at the intuitive level. What can be visually done in each case?

$\endgroup$
  • 1
    $\begingroup$ Both locally look like standard maps $\mathbb{R}^m \to \mathbb{R}^n$ $\endgroup$ – Yuri Vyatkin Feb 9 '13 at 8:56
  • 6
    $\begingroup$ they're conditions on induced maps of tangent spaces...so in an immersion $X\rightarrow Y$, $\dim X\leq\dim Y$, you're just allowing self-intersections of $X$ inside of $Y$, but you're not allowed to "compress" any of $X$ in such a way that the tangent space of a point goes to 0 (because the map must be injective on tangent spaces). For a submersion, you want the induced map to be surjective, so intuitively you're "crumpling up" $X$ to fit into $Y$ in such a way as to hit every possible curve in $Y$ with a curve in $X$. $\endgroup$ – gmoss Feb 9 '13 at 21:17
  • $\begingroup$ Related: math.stackexchange.com/questions/1214630 $\endgroup$ – Watson Mar 5 '17 at 11:27
20
$\begingroup$

First of all, note that if $f : M \to N$ is a submersion, then $\dim M \geq \dim N$, and if $f$ is an immersion, $\dim M \leq \dim N$.

The Rank Theorem may provide some insight into these concepts. The following statement of the theorem is taken from Lee's Introduction to Smooth Manifolds (second edition); see Theorem $4.12$.

Suppose $M$ and $N$ are smooth manifolds of dimensions $m$ and $n$, respectively, and $F : M \to N$ is a smooth map with constant rank $r$. For each $p \in M$ there exist smooth charts $(U, \varphi)$ for $M$ centered at $p$ and $(V, \psi)$ for $N$ centered at $F(p)$ such that $F(U) \subseteq V$, in which $F$ has a coordinate representation of the form $$\hat{F}(x^1, \dots, x^r, x^{r+1}, \dots, x^m) = (x^1, \dots, x^r, 0, \dots, 0).$$ In particular, if $F$ is a smooth submersion, this becomes $$\hat{F}(x^1, \dots, x^n, x^{n+1}, \dots, x^m) = (x^1, \dots, x^n),$$ and if $F$ is a smooth immersion, it is $$\hat{F}(x^1, \dots, x^m) = (x^1, \dots, x^m, 0, \dots, 0).$$

So a submersion locally looks like a projection $\mathbb{R}^n\times\mathbb{R}^{m-n} \to \mathbb{R}^n$, while an immersion locally looks like an inclusion $\mathbb{R}^m \to \mathbb{R}^m\times\mathbb{R}^{n-m}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.