Immersion (mathematics)

In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective.^[1] 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 and D_p f is the derivative (pushforward) of the map f at point p). Equivalently, f is an immersion if its derivative has constant rank equal to the dimension of M:^[2]

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

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