Immersion

In mathematics, a immersion is a differentiable map between differentiable manifolds whose derivative is everywhere injective. Explicitly, f : M → N is a immersion if

is a injective map at every point p of M. Equivalently, f is a immersion if it has constant rank equal to the dimension of M:

The map f itself need not be injective.

A related concept is that of an embedding. An smooth embedding is an injective immersion f : M → N which is also a topological embedding, so that M is diffeomorphic to its image in N.