Preimage theorem
In mathematics, particularly in the field of differential topology, the preimage theorem is a variation of the implicit function theorem concerning the preimage of particular points in a manifold under the action of a smooth map.Definition. Let be a smooth map between manifolds. We say that a point is a regular value of if for all the map is surjective. Here, and are the tangent spaces of and at the points and.
Theorem. Let be a smooth map, and let be a regular value of. Then is a submanifold of. If , then the codimension of is equal to the dimension of. Also, the tangent space of at is equal to.
