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:$D\_pf\; :\; T\_p\; M\; o\; T\_\{f(p)\}N,$is an injective map at every point "p" of "M" (where the notation $T\_p\; X$ 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"::$operatorname\{rank\},f\; =\; dim\; M.$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 a local embedding (i.e. for any point $xin\; M$ there is a neighbourhood, $Usubset\; M$, of "x" such that $f:U\; o\; N$ is an embedding.)

ee also

*Submersion
*Immersed submanifold