submersion

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. Definition Let M and N be differentiable manifolds and be a differentiable map between them. The map is a submersion at a point if its differential is a surjective linear map. In this case is called a regular point of the map , otherwise, is a critical point. A point is a regular value of if all points in the preimage are regular points. A differentiable map that is a submersion at each point is called a submersion. Equivalently, is a submersion if its differential has constant rank equal to the dimension of . A word of warning: some authors use the term critical point to describe a point where the rank of the Jacobian matrix of at is not maximal. Indeed, this is the more useful notion in singularity theory. If the dimension of is greater than or equal to the dimension of then these two notions of critical point coincide. But if the dimension of is less than the dimension of , 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 ). The definition given above is the more commonly used; e.g., in the formulation of Sard's theorem.