Submersion (mathematics)

{{Short description|Differential map between manifolds whose differential is everywhere surjective}}{{redirect|Regular point|"regular point of an algebraic variety"|Singular point of an algebraic variety}}

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

Definition

Let M and N be differentiable manifolds, and let $f\colon M\to N$ be a differentiable map between them. The map {{math|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.{{harvnb|Crampin|Pirani|1994|page=243}}. {{harvnb|do Carmo|1994|page=185}}. {{harvnb|Frankel|1997|page=181}}. {{harvnb|Gallot|Hulin|Lafontaine|2004|page=12}}. {{harvnb|Kosinski|2007|page=27}}. {{harvnb|Lang|1999|page=27}}. {{harvnb|Sternberg|2012|page=378}}. In this case, {{math|p}} is called a regular point of the map {{math|f}}; otherwise, {{math|p}} is a critical point. A point $q \in N$ is a regular value of {{math|f}} if all points {{math|p}} in the preimage $f^{-1}(q)$ are regular points. A differentiable map {{math|f}} that is a submersion at each point $p \in M$ is called a submersion. Equivalently, {{math|f}} is a submersion if its differential $Df_p$ has constant rank equal to the dimension of {{math|N}}.

Some authors use the term critical point to describe a point where the rank of the Jacobian matrix of {{math|f}} at {{math|p}} is not maximal.:{{harvnb|Arnold|Gusein-Zade|Varchenko|1985}}. Indeed, this is the more useful notion in singularity theory. If the dimension of {{math|M}} is greater than or equal to the dimension of {{math|N}}, then these two notions of critical point coincide. However, if the dimension of {{math|M}} is less than the dimension of {{math|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 {{math|M}}). The definition given above is the more commonly used one, e.g., in the formulation of Sard's theorem.

Submersion theorem

Given a submersion $f\colon M\to N$ between smooth manifolds of dimensions $m$ and $n$ , for each $x \in M$ there exist 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 equals the difference of the dimensions of $N$ and $M$ .

This 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 =

A 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 & x\rightarrow{\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 is 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

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

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 {{math|f: M → N}} is a submersion at {{math|p}} and {{math|f(p) {{=}} q ∈ N}}, then there exists an open neighborhood {{math|U}} of {{math|p}} in {{math|M}}, an open neighborhood {{math|V}} of {{math|q}} in {{math|N}}, and local coordinates {{math|(x₁, …, x_m)}} at {{math|p}} and {{math|(x₁, …, x_n)}} at {{math|q}} such that {{math|f(U) {{=}} V}}, and the map {{math|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 {{math|f⁻¹(q)}} in {{math|M}} of a regular value {{math|q}} in {{math|N}} under a differentiable map {{math|f: M → N}} is either empty or a differentiable manifold of dimension {{math|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 {{math|q}} in {{math|N}} if the map {{math|f}} is a submersion.

Topological manifold submersions

Submersions are also well-defined for general topological manifolds.{{harvnb|Lang|1999|page=27}}. A topological manifold submersion is a continuous surjection {{math|f : M → N}} such that for all {{math|p}} in {{math|M}}, for some continuous charts {{math|ψ}} at {{math|p}} and {{math|φ}} at {{math|f(p)}}, the map {{math|ψ⁻¹ ∘ f ∘ φ}} is equal to the projection map from {{math|R^m}} to {{math| Rⁿ}}, where {{math|m {{=}} dim(M) ≥ n {{=}} dim(N)}}.

Notes

References

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