Immersion (mathematics)

In the differential geometry is meant by immersion a smooth map between manifolds and when the Push Forward this map is injective at each point. If, in addition, a topological embedding, then we speak of a ( smooth ) embedding. In this case, the image of the figure to a submanifold diffeomorphic of

The properties of the image in the general case are described in the entry Always Disseminated manifold.

Immersion in Euclidean space

There is the special case of a mapping between Euclidean spaces, it represents no more than the total discharge or the Jacobian matrix, wherein the Euclidean space are identified in a natural manner, with its tangent and a linear transformation using a matrix.

Generally, a differentiable map is an immersion if and only if for all the rank of the linear map is equal to the dimension of the manifold, so true