# Submersion

A mapping $ f: M \rightarrow N $
from an $ m $-
dimensional manifold $ M $
into an $ n $-
dimensional manifold $ N $,
$ n \leq m $,
under which for any point $ p \in M $
it is possible to introduce local coordinates $ x _ {1} \dots x _ {m} $
on $ M $
near $ p $
and $ y _ {1} \dots y _ {n} $
on $ N $
near $ f( p) $
such that $ f $
is locally represented in terms of these coordinates by

$$ ( x _ {1} \dots x _ {m} ) \rightarrow ( x _ {1} \dots x _ {n} ). $$

If $ M $ and $ N $ possess the structure of a piecewise-linear, -analytic or -differentiable (of class $ C ^ {k} $) manifold and the local coordinates can be chosen piecewise-linear, -analytic or -differentiable (of class $ C ^ {l} $, $ l \leq k $), then the submersion is said to be piecewise-linear, -analytic or -differentiable (of class $ C ^ {l} $). A submersion can also be defined for a manifold with boundary (in topological problems it is advisable to impose an extra condition on the behaviour of the mapping close to the boundary, see [1]) and in the infinite-dimensional case (see [2]). The concept of a submersion in an informal sense is the dual of the concept of an immersion (cf. also Immersion of a manifold), and their theories are analogous.

#### References

[1] | V.A. Rokhlin, D.B. Fuks, "Beginner's course in topology. Geometrical chapters" , Springer (1984) (Translated from Russian) |

[2] | S. Lang, "Introduction to differentiable manifolds" , Interscience (1967) pp. App. III |

#### Comments

Submersions are classified by the induced mapping $ TM \rightarrow TN $ of tangent bundles, when $ M $ is an open manifold. See [a1].

#### References

[a1] | A. Phillips, "Submersions of open manifolds" Topology , 6 (1966) pp. 171–206 |

**How to Cite This Entry:**

Submersion.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Submersion&oldid=48895