# 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.

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.^{[2]} 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.

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

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

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.

Submersions are also well-defined for general topological manifolds.^{[3]} 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*).