Common · Knowledge
Immersion (mathematics)
Differentiable function whose derivative is everywhere injective
In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective. Explicitly, f : M → N is an immersion if D p f : T p M → T f ( p ) N {\displaystyle D_{p}f:T_{p}M\to T_{f(p)}N\,} is an injective function at every point p of M (where TpX denotes the tangent space of a manifold X at a point p in X and Dp f is the derivative (pushforward) of the map f at point p).
From Wikipedia
In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective. Explicitly, f : M → N is an immersion if D p f : T p M → T f ( p ) N {\displaystyle D_{p}f:T_{p}M\to T_{f(p)}N\,} is an injective function at every point p of M (where TpX denotes the tangent space of a manifold X at a point p in X and Dp 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: rank D p f = dim M . {\displaystyle \operatorname {rank} \,D_{p}f=\dim M.} The function f itself need not be injective, only its derivative must be.
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