Immersion (mathematics)

Immersion (mathematics)

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

is an injective map at every point p of M (where the notation T_pX represents the tangent space of X at the point p). Equivalently, f is an immersion if it has constant rank equal to the dimension of M:

The map f itself need not be injective, only its derivative.

A related concept is that of an embedding. A 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. An immersion is precisely a local embedding – i.e. for any point x ∈ M there is a neighbourhood, U ⊂ M, of x such that f : U → N is an embedding, and conversely a local embedding is an immersion.

If M is compact, an injective immersion is an embedding, but if M is not compact then injective immersions need not be embeddings; compare to continuous bijections versus homeomorphisms.