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:
 * $k \le n$

and there exists a local diffeomorphism $\phi$ around $\map f p$ such that:
 * $\phi \circ \map f x = \tuple {x, 0}$

for all $x$ in a neighborhood of $p$.

Also see

 * Local Normal Form for Submersions