Submersion (mathematics)

In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential pushforward 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

:

M

→

N

{\displaystyle f\colon M\to N}

be a differentiable map between them. The map f is a submersion at a point

p

∈

M

{\displaystyle p\in M}

if its differential

D

f

p

:

T

p

M

→

T

f

(

p

)

N

{\displaystyle 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

∈

N

{\displaystyle q\in N}

is a regular value of f if all points p in the preimage

f

−

1

(

q

)

{\displaystyle f^{-1}(q)}

are regular points. A differentiable map f that is a submersion at each point

p

∈

M

{\displaystyle p\in M}

is called a submersion. Equivalently, f is a submersion if its differential

D

f

p

{\displaystyle Df_{p}}

has constant rank equal to the dimension of N.

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. However, 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).