Preimage theorem

In mathematics, particularly in differential topology, the preimage theorem is a 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 $x\in f^{-1}(y)$ the map $df_{x}:T_{x}X\to T_{y}Y$ is surjective. Here, $T_{x}X{\text{ and }}T_{y}Y$ 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 $\{x:x\in f^{-1}(y)\}$ is a submanifold of X. Further, the codimension of this manifold in X is equal to dim(X)-dim(Y).