A non-injective immersion: R
→ R 2
↦ ( t 2
- 1, t
( t 2
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
Two immersions are regularly called homotopic if there is a homotopy with such that the mapping
is an immersion again.
The Hirsch-Smale theory deals with the regular homotopy classes of immersions .
- 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 .