# 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${\ displaystyle F \ colon M \ rightarrow N}$ ${\ displaystyle M}$ ${\ displaystyle N}$ ${\ displaystyle F _ {\ ast p} \ colon T_ {p} M \ to T_ {F (p)} N}$ ${\ displaystyle p \ in M}$ ${\ displaystyle F}$ ${\ displaystyle M}$ ${\ displaystyle N.}$ 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 ${\ displaystyle F: M \ rightarrow N}$ ${\ displaystyle p \ in M}$ ${\ displaystyle F _ {\ ast}}$ ${\ displaystyle M}$ ${\ displaystyle \ operatorname {rank} F_ {p} = \ dim (\ operatorname {image} (F _ {\ ast p})) = \ dim M.}$ ## 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 .