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.

Statement of Theorem

Definition. Let f : X \to Y be a smooth map between manifolds. We say that a point y \in Y is a regular value of f if for all the map 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 \to Y be a smooth map, and let y \in Y be a regular value of f. Then f^{-1}(y) is a submanifold of X. If 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(df_x). There is also a complex version of this theorem: Theorem. Let X^n and Y^m be two complex manifolds of complex dimensions n > m. Let g : X \to Y be a holomorphic map and let be such that for all Then g^{-1}(y) is a complex submanifold of X of complex dimension n - m.