Submersion (mathematics)

Template:Short descriptionTemplate:Redirect 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 <math>f\colon M\to N</math> be a differentiable map between them. The map Template:Math is a submersion at a point <math>p \in M</math> if its differential

<math>Df_p \colon T_p M \to T_{f(p)}N</math>

is a surjective linear map.<ref>Template:Harvnb. Template:Harvnb. Template:Harvnb. Template:Harvnb. Template:Harvnb. Template:Harvnb. Template:Harvnb.</ref> In this case, Template:Math is called a regular point of the map Template:Math; otherwise, Template:Math is a critical point. A point <math>q \in N</math> is a regular value of Template:Math if all points Template:Math in the preimage <math>f^{-1}(q)</math> are regular points. A differentiable map Template:Math that is a submersion at each point <math>p \in M</math> is called a submersion. Equivalently, Template:Math is a submersion if its differential <math>Df_p</math> has constant rank equal to the dimension of Template:Math.

Some authors use the term critical point to describe a point where the rank of the Jacobian matrix of Template:Math at Template:Math is not maximal.:<ref>Template:Harvnb.</ref> Indeed, this is the more useful notion in singularity theory. If the dimension of Template:Math is greater than or equal to the dimension of Template:Math, then these two notions of critical point coincide. However, if the dimension of Template:Math is less than the dimension of Template:Math, 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 Template:Math). The definition given above is the more commonly used one, e.g., in the formulation of Sard's theorem.

Submersion theorem

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

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

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

<math>\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}.</math>

This has maximal rank at every point except for <math>(0,0,0)</math>. Also, the fibers

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

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

Examples

• Any projection <math>\pi\colon \mathbb{R}^{m+n} \rightarrow \mathbb{R}^n\subset \mathbb{R}^{m+n}</math>
• 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

<math>f:S^{n+k} \to S^k</math>

whose fibers have dimension <math>n</math>. This is because the fibers (inverse images of elements <math>p in S^k</math>) are smooth manifolds of dimension <math>n</math>. Then, if we take a path

<math>\gamma: I \to S^k</math>

and take the pullback

<math>\begin{matrix}

M_I & \to & S^{n+k} \\ \downarrow & & \downarrow f \\ I & x\rightarrow{\gamma} & S^k \end{matrix}</math> we get an example of a special kind of bordism, called a framed bordism. In fact, the framed cobordism groups <math>\Omega_n^{fr}</math> are intimately related to the stable homotopy groups.

Families of algebraic varieties

Another large class of submersions is given by families of algebraic varieties <math>\pi:\mathfrak{X} \to S</math> whose fibers are smooth algebraic varieties. If we consider the underlying manifolds of these varieties, we get smooth manifolds. For example, the Weierstrass family <math>\pi:\mathcal{W} to \mathbb{A}^1</math> 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

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

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

Local normal form

If Template:Math is a submersion at Template:Math and Template:Math, then there exists an open neighborhood Template:Math of Template:Math in Template:Math, an open neighborhood Template:Math of Template:Math in Template:Math, and local coordinates Template:Math at Template:Math and Template:Math at Template:Math such that Template:Math, and the map Template:Math in these local coordinates is the standard projection

<math>f(x_1, \ldots, x_n, x_{n+1}, \ldots, x_m) = (x_1, \ldots, x_n).</math>

It follows that the full preimage Template:Math in Template:Math of a regular value Template:Math in Template:Math under a differentiable map Template:Math is either empty or a differentiable manifold of dimension Template:Math, possibly disconnected. This is the content of the regular value theorem (also known as the submersion theorem). In particular, the conclusion holds for all Template:Math in Template:Math if the map Template:Math is a submersion.

Topological manifold submersions

Submersions are also well-defined for general topological manifolds.<ref>Template:Harvnb.</ref> A topological manifold submersion is a continuous surjection Template:Math such that for all Template:Math in Template:Math, for some continuous charts Template:Math at Template:Math and Template:Math at Template:Math, the map Template:Math is equal to the projection map from Template:Math to Template:Math, where Template:Math.

See also

• Ehresmann's fibration theorem

References

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Further reading

• https://mathoverflow.net/questions/376129/what-are-the-sufficient-and-necessary-conditions-for-surjective-submersions-to-b?rq=1