Inmathematics, asubmersion is adifferentiable map betweendifferentiable manifolds whosedifferential is everywheresurjective. It is a basic concept indifferential topology, dual to that of animmersion.

LetM andN bedifferentiable manifolds, and let${\displaystyle f:M\to N}$ be adifferentiable map between them. The mapf is asubmersion at a point${\displaystyle p\in M}$ if itsdifferential

${\displaystyle D{f}_p:T_{p}M\to T_{f(p)}N}$

is asurjectivelinear map.^[1] In this case,p is called aregular point of the mapf; otherwise,p is acritical point. A point${\displaystyle q\in N}$ is aregular value off if all pointsp in thepreimage${\displaystyle f^{-1}(q)}$ are regular points. A differentiable mapf that is a submersion at each point${\displaystyle p\in M}$ is called asubmersion. Equivalently,f is a submersion if its differential${\displaystyle D{f}_p}$ hasconstant rank equal to the dimension ofN.

Some authors use the termcritical point to describe a point where therank of theJacobian matrix off atp is not maximal.:^[2] Indeed, this is the more useful notion insingularity theory. If the dimension ofM is greater than or equal to the dimension ofN, then these two notions of critical point coincide. However, if the dimension ofM is less than the dimension ofN, 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 dimM). The definition given above is the more commonly used one, e.g., in the formulation ofSard's theorem.

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

This theorem is a consequence of theinverse function theorem (seeInverse function theorem#Giving a manifold structure).

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

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

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

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

• Any projection${\displaystyle \pi :{\mathbb{R}}^{m+n}\to {\mathbb{R}}^n\subset {\mathbb{R}}^{m+n}}$
• Local diffeomorphisms
• Riemannian submersions
• The projection in a smoothvector bundle or a more general smoothfibration. The surjectivity of the differential is a necessary condition for the existence of alocal trivialization.

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

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

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

${\displaystyle \gamma :I\to S^k}$

and take thepullback

we get an example of a special kind ofbordism, called aframed bordism. In fact, the framed cobordism groups${\displaystyle {\mathrm{\Omega }}_n^{fr}}$ are intimately related to thestable homotopy groups.

Another large class of submersions is given by families ofalgebraic varieties${\displaystyle \pi :\mathfrak{X}\to S}$ whose fibers are smooth algebraic varieties. If we consider the underlying manifolds of these varieties, we get smooth manifolds. For example, the Weierstrass family${\displaystyle \pi :\mathcal{W}to{\mathbb{A}}^1}$ ofelliptic curves is a widely studied submersion because it includes many technical complexities used to demonstrate more complex theory, such asintersection homology andperverse sheaves. This family is given by

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

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

Iff:M →N is a submersion atp andf(p) =q ∈N, then there exists anopen neighborhoodU ofp inM, an open neighborhoodV ofq inN, and local coordinates(x₁, …,x_m) atp and(x₁, …,x_n) atq such thatf(U) =V, and the mapf in these local coordinates is the standard projection

${\displaystyle f(x_1,\dots ,x_n,x_{n+1},\dots ,x_m)=(x_1,\dots ,x_n).}$

It follows that the full preimagef⁻¹(q) inM of a regular valueq inN under a differentiable mapf:M →N is either empty or a differentiable manifold of dimensiondimM − dimN, possiblydisconnected. This is the content of theregular value theorem (also known as thesubmersion theorem). In particular, the conclusion holds for allq inN if the mapf is a submersion.

Submersions are also well-defined for generaltopological manifolds.^[3] A topological manifold submersion is acontinuous surjectionf :M →N such that for allp inM, for some continuous chartsψ atp andφ atf(p), the mapψ⁻¹ ∘ f ∘ φ is equal to theprojection map fromR^m toRⁿ, wherem = dim(M) ≥n = dim(N).

• Ehresmann's fibration theorem

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

• Maps of manifolds
• Smooth functions

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