Immersion (math)

A non-injective immersion: R  →  R ² , t  ↦ ( t ²  - 1,  t  ( t ²  - 1))

In differential topology , immersion is understood to be a smooth mapping between manifolds and when the push forward of this mapping is injective at every point . If there is also a topological embedding , one speaks of a (smooth) embedding . In this case the image of the map is too diffeomorphic a submanifold of ${\displaystyle F\colon M\rightarrow N}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle F_{\ast p}\colon T_{p}M\to T_{F(p)}N}$ ${\displaystyle p\in M}$ ${\displaystyle F}$ ${\displaystyle M}$ ${\displaystyle N.}$

The properties of the image in the general case are described in the entry Immersed Manifold .

Immersion in Euclidean space

If there is the special case of a mapping between Euclidean spaces, then it represents nothing other than the total derivative or the Jacobi matrix , whereby the Euclidean space is identified in a natural way with its tangential space and a linear mapping with a matrix. ${\displaystyle F:\mathbb {R} ^{m}\rightarrow \mathbb {R} ^{n}}$ ${\displaystyle F_{\ast }:T_{p}\mathbb {R} ^{m}\rightarrow T_{F(p)}\mathbb {R} ^{n}}$ ${\displaystyle DF(p)\colon \mathbb {R} ^{m}\rightarrow \mathbb {R} ^{n}}$

Immersion in manifolds

In general, a differentiable mapping is an immersion if and only if the rank of the linear mapping is equal to the dimension of the manifold for all of them , that is, it applies ${\displaystyle F:M\rightarrow N}$ ${\displaystyle p\in M}$ ${\displaystyle F_{\ast }}$ ${\displaystyle M}$

${\displaystyle \operatorname {rang} F_{p}=\dim(\operatorname {Bild} (F_{\ast p}))=\dim M.}$

Regular homotopy

Two immersions are regularly called homotopic if there is a homotopy with such that the mapping for each ${\displaystyle F_{0},F_{1}\colon M\to N}$ ${\displaystyle F\colon M\times \left[0,1\right]\to N}$ ${\displaystyle F(m,0)=F_{0}(m),F(m,1)=F_{1}(m)\forall m\in M}$ ${\displaystyle t\in \left[0,1\right]}$

${\displaystyle F_{t}\colon M\to N}$

${\displaystyle F_{t}(m)=F(m,t)}$

is an immersion again.

The Hirsch-Smale theory deals with the regular homotopy classes of immersions .

See also

• Submersion

literature

• John M. Lee: Introduction to Smooth Manifolds (= Graduate Texts in Mathematics 218). Springer-Verlag, New York NY et al. 2003, ISBN 0-387-95448-1 .