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 $f:X\to Y$ 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, $T_{x}X$ 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 $y\in {\text{im}}(f)$, 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 $\ker(df_{x})$.