Immersion (mathematics)

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


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