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 rankequal 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 but the rank of the Jacobian may still be maximal. The definition given above is the more commonly used; e.g., in the formulation of Sard's theorem.
Given a submersion between smooth manifolds the fibers of, denoted can be equipped with the structure of a smooth manifold. This theorem coupled with the Whitney embedding theorem implies that every smooth manifold can be described as the fiber of a smooth map. For example, consider given by The Jacobian matrix is This has maximal rank at every point except for. Also, the fibers are empty for, and equal to a point when. Hence we only have a smooth submersion and the subsets are two-dimensional smooth manifolds for.
If is a submersion at and, then there exists an open neighborhood of in, an open neighborhood of in, and local coordinates at and at such that, and the map in these local coordinates is the standard projection It follows that the full preimage in of a regular value in under a differentiable map is either empty or is a differentiable manifold of dimension, possibly disconnected. This is the content of the regular value theorem. In particular, the conclusion holds for all in if the map is a submersion.
Submersions are also well-defined for general topological manifolds. A topological manifold submersion is a continuous surjection such that for all in, for some continuous charts at and at, the map is equal to the projection map from to, where.
