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