View source for Submersion

A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s0909401.png" /> from an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s0909402.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s0909403.png" /> into an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s0909404.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s0909405.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s0909406.png" />, under which for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s0909407.png" /> it is possible to introduce local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s0909408.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s0909409.png" /> near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s09094010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s09094011.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s09094012.png" /> near <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s09094013.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s09094014.png" /> is locally represented in terms of these coordinates by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s09094015.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s09094016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s09094017.png" /> possess the structure of a piecewise-linear, -analytic or -differentiable (of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s09094018.png" />) manifold and the local coordinates can be chosen piecewise-linear, -analytic or -differentiable (of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s09094019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s09094020.png" />), then the submersion is said to be piecewise-linear, -analytic or -differentiable (of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s09094021.png" />). 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 [[#References|[1]]]) and in the infinite-dimensional case (see [[#References|[2]]]). The concept of a submersion in an informal sense is the dual of the concept of an [[Immersion|immersion]] (cf. also [[Immersion of a manifold|Immersion of a manifold]]), and their theories are analogous.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.A. Rokhlin,   D.B. Fuks,   "Beginner's course in topology. Geometrical chapters" , Springer  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lang,   "Introduction to differentiable manifolds" , Interscience  (1967)  pp. App. III</TD></TR></table>



====Comments====
Submersions are classified by the induced mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s09094022.png" /> of tangent bundles, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090940/s09094023.png" /> is an open manifold. See [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A. Phillips,   "Submersions of open manifolds"  ''Topology'' , '''6'''  (1966)  pp. 171–206</TD></TR></table>