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.

Statement of Theorem

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

There is also a complex version of this theorem:

Theorem. Let and be two complex manifolds of complex dimensions Let be a holomorphic map and let be such that for all Then is a complex submanifold of of complex dimension

See also

• Fiber (mathematics) – Set of all points in a function's domain that all map to some single given point
• Level set – Subset of a function's domain on which its value is equal