Submersion (mathematics)

Last updated February 29, 2024

In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion.

Definition

Let M and N be differentiable manifolds and $f\colon M\to N$ be a differentiable map between them. The map $f$ is a submersion at a point $p\in M$ if its differential

Df_{p}\colon T_{p}M\to T_{f(p)}N

is a surjective linear map.^[1] In this case $p$ is called a regular point of the map $f$ , otherwise, $p$ is a critical point. A point $q\in N$ is a regular value of $f$ if all points $p$ in the preimage $f^{-1}(q)$ are regular points. A differentiable map $f$ that is a submersion at each point $p\in M$ is called a submersion. Equivalently, $f$ is a submersion if its differential $Df_{p}$ has constant rank equal to the dimension of $N$ .

A word of warning: some authors use the term critical point to describe a point where the rank of the Jacobian matrix of $f$ at $p$ is not maximal.^[2] Indeed, this is the more useful notion in singularity theory. If the dimension of $M$ is greater than or equal to the dimension of $N$ then these two notions of critical point coincide. But if the dimension of $M$ is less than the dimension of $N$ , all points are critical according to the definition above (the differential cannot be surjective) but the rank of the Jacobian may still be maximal (if it is equal to dim $M$ ). The definition given above is the more commonly used; e.g., in the formulation of Sard's theorem.

Submersion theorem

Given a submersion between smooth manifolds $f\colon M\to N$ of dimensions $m$ and $n$ , for each $x\in M$ there are surjective charts $\phi :U\to \mathbb {R} ^{m}$ of $M$ around $x$ , and $\psi :V\to \mathbb {R} ^{n}$ of $N$ around $f(x)$ , such that $f$ restricts to a submersion $f\colon U\to V$ which, when expressed in coordinates as $\psi \circ f\circ \phi ^{-1}:\mathbb {R} ^{m}\to \mathbb {R} ^{n}$ , becomes an ordinary orthogonal projection. As an application, for each $p\in N$ the corresponding fiber of $f$ , denoted $M_{p}=f^{-1}(\{p\})$ can be equipped with the structure of a smooth submanifold of $M$ whose dimension is equal to the difference of the dimensions of $N$ and $M$ .

The theorem is a consequence of the inverse function theorem (see Inverse function theorem#Giving a manifold structure).

For example, consider $f\colon \mathbb {R} ^{3}\to \mathbb {R}$ given by $f(x,y,z)=x^{4}+y^{4}+z^{4}.$ The Jacobian matrix is

{\begin{bmatrix}{\frac {\partial f}{\partial x}}&{\frac {\partial f}{\partial y}}&{\frac {\partial f}{\partial z}}\end{bmatrix}}={\begin{bmatrix}4x^{3}&4y^{3}&4z^{3}\end{bmatrix}}.

This has maximal rank at every point except for $(0,0,0)$ . Also, the fibers

f^{-1}(\{t\})=\left\{(a,b,c)\in \mathbb {R} ^{3}:a^{4}+b^{4}+c^{4}=t\right\}

are empty for $t<0$ , and equal to a point when $t=0$ . Hence we only have a smooth submersion $f\colon \mathbb {R} ^{3}\setminus \{(0,0,0)\}\to \mathbb {R} _{>0},$ and the subsets $M_{t}=\left\{(a,b,c)\in \mathbb {R} ^{3}:a^{4}+b^{4}+c^{4}=t\right\}$ are two-dimensional smooth manifolds for $t>0$ .

Examples

Any projection $\pi \colon \mathbb {R} ^{m+n}\rightarrow \mathbb {R} ^{n}\subset \mathbb {R} ^{m+n}$
Local diffeomorphisms
Riemannian submersions
The projection in a smooth vector bundle or a more general smooth fibration. The surjectivity of the differential is a necessary condition for the existence of a local trivialization.

Maps between spheres

One large class of examples of submersions are submersions between spheres of higher dimension, such as

f:S^{n+k}\to S^{k}

whose fibers have dimension $n$ . This is because the fibers (inverse images of elements $p\in S^{k}$ ) are smooth manifolds of dimension $n$ . Then, if we take a path

\gamma :I\to S^{k}

and take the pullback

{\begin{matrix}M_{I}&\to &S^{n+k}\\\downarrow &&\downarrow f\\I&\xrightarrow {\gamma } &S^{k}\end{matrix}}

we get an example of a special kind of bordism, called a framed bordism. In fact, the framed cobordism groups $\Omega _{n}^{fr}$ are intimately related to the stable homotopy groups.

Families of algebraic varieties

Another large class of submersions are given by families of algebraic varieties ${\displaystyle \pi$ whose fibers are smooth algebraic varieties. If we consider the underlying manifolds of these varieties, we get smooth manifolds. For example, the Weierstrass family ${\displaystyle \pi$ of elliptic curves is a widely studied submersion because it includes many technical complexities used to demonstrate more complex theory, such as intersection homology and perverse sheaves. This family is given by

${\mathcal {W}}=\{(t,x,y)\in \mathbb {A} ^{1}\times \mathbb {A} ^{2}:y^{2}=x(x-1)(x-t)\}$

where $\mathbb {A} ^{1}$ is the affine line and $\mathbb {A} ^{2}$ is the affine plane. Since we are considering complex varieties, these are equivalently the spaces $\mathbb {C} ,\mathbb {C} ^{2}$ of the complex line and the complex plane. Note that we should actually remove the points $t=0,1$ because there are singularities (since there is a double root).

Local normal form

If $f : M \to N$ is a submersion at $p$ and $f (p) = q \in N$ , then there exists an open neighborhood $U$ of $p$ in $M$ , an open neighborhood $V$ of $q$ in $N$ , and local coordinates $(x 1, \dots, x m)$ at $p$ and $(x 1, \dots, x n)$ at $q$ such that $f (U) = V$ , and the map $f$ in these local coordinates is the standard projection

f(x_{1},\ldots ,x_{n},x_{n+1},\ldots ,x_{m})=(x_{1},\ldots ,x_{n}).

It follows that the full preimage $f -1 (q)$ in $M$ of a regular value $q$ in $N$ under a differentiable map $f : M \to N$ is either empty or is a differentiable manifold of dimension $dim M - dim N$ , possibly disconnected. This is the content of the regular value theorem (also known as the submersion theorem). In particular, the conclusion holds for all $q$ in $N$ if the map $f$ is a submersion.

Topological manifold submersions

Submersions are also well-defined for general topological manifolds.^[3] A topological manifold submersion is a continuous surjection $f : M \to N$ such that for all $p$ in $M$ , for some continuous charts $ψ$ at $p$ and $φ$ at $f(p)$ , the map $ψ -1 \circ f \circ φ$ is equal to the projection map from $R m$ to $R n$ , where $m = dim(M) \geq n = dim(N)$ .

Notes

↑ Crampin & Pirani 1994 , p. 243. do Carmo 1994 , p. 185. Frankel 1997 , p. 181. Gallot, Hulin & Lafontaine 2004 , p. 12. Kosinski 2007 , p. 27. Lang 1999 , p. 27. Sternberg 2012 , p. 378.
↑ Arnold, Gusein-Zade & Varchenko 1985.
↑ Lang 1999 , p. 27.

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