Local Normal Form for Submersions

Theorem
Let $\Omega\subset\R^n$ be open.

Let $f : \Omega \to \R^k$ be an submersion.

Let $p\in\Omega$.

Then $n\geq k$, and there exists a local diffeomorphism $\phi$ around $f(p)$ such that
 * $\phi\circ f (x, y) = x$ for all $(x, y)$ in a neighborhood of $p$.

Also see

 * Local Normal Form for Immersions