# Immersion (math)

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

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.

## 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

## Regular homotopy

Two immersions are **regularly called homotopic** if there is a homotopy with such that the mapping
for each

is an immersion again.

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

## See also

## 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 .