Definition
Let M and N be differentiable manifolds and ƒ : M → N be a differentiable map between them. The map ƒ 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 ƒ, otherwise, p is a critical point. A point q ∈ N is a regular value of ƒ if all points p in the pre-image ƒ−1(q) are regular points. A differentiable map ƒ that is a submersion at each point is called a submersion. Equivalently, ƒ 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 ƒ 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.
Read more about this topic: Submersion (mathematics)
Famous quotes containing the word definition:
“One definition of man is “an intelligence served by organs.””
—Ralph Waldo Emerson (1803–1882)
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)