Immersion (Mathematics)
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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 to T_{f(p)}N, is an injective map at every point p of M (where the notation TpX 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 t...
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 to T_{f(p)}N, is an injective map at every point p of M (where the notation TpX 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 precisely a local embedding i.e. for any point xin M there is a neighbourhood, Usubset M, of x such that f:Uto N is an embedding, and conversely a local embedding is an immersion. An injectively immersed submanifold that is not an embedding. 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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