Immersion (math)

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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

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


  • 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 .