In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential pushforward is everywhere surjective. It is a basic concept in differential topology, dual to that of an immersion.
Definition Let M and N be differentiable manifolds, and let f : M → N {\displaystyle f\colon M\to N} be a differentiable map between them. The map f is a submersion at a point p ∈ M {\displaystyle p\in M} if its differential
D f p : T p M → T f ( p ) N {\displaystyle Df_{p}\colon T_{p}M\to T_{f(p)}N}
is a surjective linear map. In this case, p is called a regular point of the map f; otherwise, p is a critical point. A point q ∈ N {\displaystyle q\in N} is a regular value of f if all points p in the preimage f − 1 ( q ) {\displaystyle f^{-1}(q)} are regular points. A differentiable map f that is a submersion at each point p ∈ M {\displaystyle p\in M} is called a submersion. Equivalently, f is a submersion if its differential D f p {\displaystyle Df_{p}} has constant rank equal to the dimension of N. Some authors use the term critical point to describe a point where the rank of the Jacobian matrix of f at p is not maximal.: Indeed, this is the more useful notion in singularity theory. If the dimension of M is greater than or equal to the dimension of N, then these two notions of critical point coincide. However, if the dimension of M is less than the dimension of N, all points are critical according to the definition above (the differential cannot be surjective), but the rank of the Jacobian may still be maximal (if it is equal to dim M). The definition given above is the more commonly used one, e.g., in the formulation of Sard's theorem.
Submersion theorem Given a submersion f : M → N {\displaystyle f\colon M\to N} between smooth manifolds of dimensions m {\displaystyle m} and n {\displaystyle n} , for each x ∈ M {\displaystyle x\in M} there exist surjective charts ϕ : U → R m {\displaystyle \phi :U\to \mathbb {R} ^{m}} of M {\displaystyle M} around x {\displaystyle x} , and ψ : V → R n {\displaystyle \psi :V\to \mathbb {R} ^{n}} of N {\displaystyle N} around f ( x ) {\displaystyle f(x)} , such that f {\displaystyle f} restricts to a submersion f : U → V {\displaystyle f\colon U\to V} which, when expressed in coordinates as ψ ∘ f ∘ ϕ − 1 : R m → R n {\displaystyle \psi \circ f\circ \phi ^{-1}:\mathbb {R} ^{m}\to \mathbb {R} ^{n}} , becomes an ordinary orthogonal projection. As an application, for each p ∈ N {\displaystyle p\in N} the corresponding fiber of f {\displaystyle f} , denoted M p = f − 1 ( p ) {\displaystyle M_{p}=f^{-1}({p})} can be equipped with the structure of a smooth submanifold of M {\displaystyle M} whose dimension equals the difference of the dimensions of N {\displaystyle N} and M {\displaystyle M} . This theorem is a consequence of the inverse function theorem (see Inverse function theorem#Giving a manifold structure). For example, consider f : R 3 → R {\displaystyle f\colon \mathbb {R} ^{3}\to \mathbb {R} } given by f ( x , y , z ) = x 4 + y 4 + z 4 . {\displaystyle f(x,y,z)=x^{4}+y^{4}+z^{4}.} . The Jacobian matrix is
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