# 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.

## Definition

Let *M* and *N* be differentiable manifolds and *f* : *M* → *N* be a differentiable map between them. The map *f* is a **submersion at a point** *p* ∈ *M* if its differential

is a surjective linear map.^{[1]} In this case *p* is called a **regular point** of the map *f*, otherwise, *p* is a critical point. A point *q* ∈ *N* is a **regular value** of *f* if all points *p* in the pre-image *f*^{−1}(*q*) are regular points. A differentiable map *f* that is a submersion at each point *p* ∈ *M* is called a **submersion**. Equivalently, *f* is a submersion if its differential *Df*_{p} has constant rank equal to the dimension of *N*.

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 more commonly used, e.g. in the formulation of Sard's theorem.

## Examples

- Any projection
- 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.

## Local normal form

If *f*: *M* → *N* is a submersion at *p* and *f*(*p*) = *q* ∈ *N* then there exist 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 pre-image *f*^{−1}(*q*) in *M* of a regular value *q* ∈ *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* ∈ *N* if the map *f* is a submersion.

## Topological manifold submersions

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* ∈ *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*).

## See also

## Notes

- ↑ Crampin & Pirani 1994, p. 243. do Carmo 1994, p. 185. Frankel 1997, p. 181. Gallot, Hulin & Lafontaine 2004, p. 12. Kosinski 2007, p. 27. Lang 1999, p. 27. Sternberg 2012, p. 378.
- ↑ Arnold, Gusein-Zade & Varchenko 1985.
- ↑ Lang 1999, p. 27.

## References

- Arnold, V. I.; Gusein-Zade, S. M.; Varchenko, A. N. (1985).
*Singularities of Differentiable Maps: Volume 1*. Birkhäuser. ISBN 0-8176-3187-9. - Bruce, J. W.; Giblin, P. J. (1984),
*Curves and Singularities*, Cambridge University Press, ISBN 0-521-42999-4 - Crampin, Michael; Pirani, Felix Arnold Edward (1994).
*Applicable differential geometry*. Cambridge, England: Cambridge University Press. ISBN 978-0-521-23190-9. - do Carmo, Manfredo Perdigao (1994).
*Riemannian Geometry*. ISBN 978-0-8176-3490-2. - Frankel, Theodore (1997).
*The Geometry of Physics*. Cambridge: Cambridge University Press. ISBN 0-521-38753-1. - Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (2004).
*Riemannian Geometry*(3rd ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-20493-0. - Kosinski, Antoni Albert (2007) [1993].
*Differential manifolds*. Mineola, New York: Dover Publications. ISBN 978-0-486-46244-8. - Lang, Serge (1999).
*Fundamentals of Differential Geometry*. Graduate Texts in Mathematics. New York: Springer. ISBN 978-0-387-98593-0. - Sternberg, Shlomo Zvi (2012).
*Curvature in Mathematics and Physics*. Mineola, New York: Dover Publications. ISBN 978-0-486-47855-5.