In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential (or pushforward) is everywhere injective. Explicitly, f : M → N is an immersion if

${\displaystyle D_{p}f:T_{p}M\to T_{f(p)}N\,}$

is an injective function at every point p of M (where T_pX denotes the tangent space of a manifold X at a point p in X). Equivalently, f is an immersion if its derivative has constant rank equal to the dimension of M:

${\displaystyle \operatorname {rank} \,D_{p}f=\dim M.}$

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

A related concept is that of an embedding. A smooth embedding is an injective immersion f : M → N that is also a topological embedding, so that M is diffeomorphic to its image in N. An immersion is precisely a local embedding – that is, 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. For infinite dimensional manifolds, this is sometimes taken to be the definition of 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.