Immersion (mathematics)

In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective. Explicitly, $f : M \to N$ is an immersion if

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

is an injective function at every point $p$ of $M$ (where $T p X$ 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 $M$ :

\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 $f : M \to N$ 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 $x \in M$ there is a neighbourhood, $U \subseteq M$ , of $x$ such that $f : U \to N$ 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.

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