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Concept
Submersion (mathematics)
Summary
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion.
Let M and N be differentiable manifolds and be a differentiable map between them. The map f is a submersion at a point if its differential
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 is a regular value of f if all points p in the are regular points. A differentiable map f that is a submersion at each point is called a submersion. Equivalently, f is a submersion if its differential has constant rank equal to the dimension of N.
A word of warning: 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. But 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; e.g., in the formulation of Sard's theorem.
Given a submersion between smooth manifolds of dimensions and , for each there are surjective charts of around , and of around , such that restricts to a submersion which, when expressed in coordinates as , becomes an ordinary orthogonal projection. As an application, for each the corresponding fiber of , denoted can be equipped with the structure of a smooth submanifold of whose dimension is equal to the difference of the dimensions of and .
The theorem is a consequence of the inverse function theorem (see Inverse function theorem#Giving a manifold structure).
For example, consider given by The Jacobian matrix is
This has maximal rank at every point except for .
Official source
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