In mathematics, particularly in 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 f : X → Y be a smooth map between manifolds. We say that a point y ∈ Y is a regular value of f if for all x ∈ f − 1 ( y ) the map d f x : T x X → T y Y is surjective. Here, T x X and T y Y are the tangent spaces of X and Y at the points x and y.
Theorem. Let f : X → Y be a smooth map, and let y ∈ Y be a regular value of f; then f − 1 ( y ) is a submanifold of X. If y ∈ im ( f ) , then the codimension of f − 1 ( y ) is equal to the dimension of Y. Also, the tangent space of f − 1 ( y ) at x is equal to ker ( d f x ) .
