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 $p$.

Proof

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Also see

• Local Normal Form for Submersions