Submersion

In differential topology , a differentiable mapping between two differentiable manifolds is called submersion , if their differential is surjective at every point . A special class of submersions are the Riemannian submersions considered in differential geometry .

Points at which the differential is not surjective are called critical or singular .

Important example of a submersion is the projection of the first coordinates in the Euclidean space . In fact, each submersion can be represented locally in the form of such a projection by a suitable choice of maps . ${\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{m}\;;\;(x_{1};x_{2};\cdots ;x_{m};\cdots ;x_{n})\mapsto (x_{1};x_{2};\cdots ;x_{m})}$ ${\displaystyle n\geq m}$ ${\displaystyle m}$

If the target space is the real straight line , then a differentiable function is a submersion if and only if its differential does not vanish anywhere identical 0. ${\displaystyle \mathbb {R} }$

Leaves and bundles of fibers

If there is a submersion, then the level sets form a foliation of . This follows from the theorem of the implicit function . ${\displaystyle f\colon M\rightarrow B}$ ${\displaystyle f^{-1}(b),b\in B}$ ${\displaystyle M}$

If compact and is a submersion, then is a fiber bundle with the level quantities as fibers. That is the statement of Ehresmann's theorem . ${\displaystyle M}$ ${\displaystyle f\colon M\rightarrow B}$ ${\displaystyle f\colon M\rightarrow f(M)}$

2-dimensional reeb foliage

An example of a submersion whose level sets form a foliage but not a fiber bundle is

${\displaystyle f\colon \left[-1,1\right]\times {\mathbb {R} }\rightarrow {\mathbb {R} }}$

${\displaystyle f\left(x,y\right)=\left(x^{2}-1\right)e^{y}}$.

The image on the right shows the projection of this scroll using the identification . ${\displaystyle \left[-1,1\right]\times S^{1}}$ ${\displaystyle S^{1}=\mathbb {R} /\mathbb {Z} }$

See also

• Immersion

literature

• John M. Lee: Introduction to Smooth Manifolds (= Graduate Texts in Mathematics 218). Springer, New York NY et al. 2003, ISBN 0-387-95448-1 .
• R. Abraham, Jerrold E. Marsden, T. Ratiu: Manifolds, Tensor Analysis and Applications (= Applied Mathematical Sciences 75). 2nd edition. Springer, New York NY et al. 1988, ISBN 0-387-96790-7 .

Web links

Wiktionary: Submersion  - explanations of meanings, word origins, synonyms, translations