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