Submersion (mathematics)

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. 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. 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 $$ \phi : U \to \R^m $$ of $$ M $$ around $$x$$, and $$\psi : V \to \R^n$$ of $$ N $$ around $$f(x) $$, such that $$f $$ restricts to a submersion $$f \colon U \to V$$ which, when expressed in coordinates as $$\psi \circ f \circ \phi^{-1} : \R^m \to \R^n $$, 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).

For example, consider $$f\colon \R^3 \to \R$$ 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
 * $$f^{-1}(\{t\}) = \left\{(a,b,c)\in \R^3 : a^4 + b^4 + c^4 = t\right\}$$

are empty for $$t < 0$$, and equal to a point when $$t = 0$$. Hence we only have a smooth submersion $$f\colon \R^3\setminus \{(0,0,0)\}\to \R_{>0},$$ 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 $$\pi\colon \R^{m+n} \rightarrow \R^n\subset\R^{m+n}$$
 * 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 $$\pi:\mathfrak{X} \to S$$ whose fibers are smooth algebraic varieties. If we consider the underlying manifolds of these varieties, we get smooth manifolds. For example, the Weierstrass family $$\pi:\mathcal{W} \to \mathbb{A}^1$$ 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 $f: M → N$ is a submersion at $p$ and $f(p) = q ∈ N$, then there exists an open neighborhood $U$ of $p$ in $M$, an open neighborhood $V$ of $q$ in $N$, and local coordinates $(x_{1}, …, x_{m})$ at $p$ and $(x_{1}, …, x_{n})$ 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 $f: M → N$ 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. A topological manifold submersion is a continuous surjection $f : M → N$ such that for all $p$ in $M$, for some continuous charts $ψ$ at $p$ and $φ$ at $f(p)$, the map $ψ^{−1} ∘ f ∘ φ$ is equal to the projection map from $R^{m}$ to $R^{n}$, where $m = dim(M) ≥ n = dim(N)$.