# 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 ) and in the infinite-dimensional case (see ). 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.