Local Normal Form for Immersions

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

Let $f : \Omega \to \R^n$ be an immersion.

Let $p\in\Omega$.

Then $k\leq n$, and there exists a local diffeomorphism $\phi$ around $f(p)$ such that
 * $\phi\circ f (x) = (x, 0)$ for all $x$ in a neighborhood of $p$.

Also see

 * Local Normal Form for Submersions