Submersion (mathematics)
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 f : M → N be a differentiable map between them. The map f is a submersion at a point p ∈ M 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 q ∈ N is a regular value of f if all points p in the pre-image f−1(q) are regular points. A differentiable map f that is a submersion at each point p ∈ M is called a submersion. Equivalently, f is a submersion if its differential Dfp 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 more commonly used, e.g. in the formulation of Sard's theorem.
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