Preimage theorem

Definition. Let ${\displaystyle f:X\to Y}$ be a smooth map between manifolds. We say that a point ${\displaystyle y\in Y}$ is a regular value of ${\displaystyle f}$ if for all ${\displaystyle x\in f^{-1}(y)}$ the map ${\displaystyle df_{x}:T_{x}X\to T_{y}Y}$ is surjective. Here, ${\displaystyle T_{x}X}$ and ${\displaystyle T_{y}Y}$ are the tangent spaces of ${\displaystyle X}$ and ${\displaystyle Y}$ at the points ${\displaystyle x}$ and ${\displaystyle y}$.

Theorem. Let ${\displaystyle f:X\to Y}$ be a smooth map, and let ${\displaystyle y\in Y}$ be a regular value of ${\displaystyle f}$. Then ${\displaystyle f^{-1}(y)}$ is a submanifold of ${\displaystyle X}$. If ${\displaystyle y\in {\text{im}}(f)}$, then the codimension of ${\displaystyle f^{-1}(y)}$ is equal to the dimension of ${\displaystyle Y}$. Also, the tangent space of ${\displaystyle f^{-1}(y)}$ at ${\displaystyle x}$ is equal to ${\displaystyle \ker(df_{x})}$.