Canonical immersion theorem

Let $f : M \to N$ be a immersion at $p \in M$ , then $m = \dim M \leq n = \dim N$ and there exist charts $(φ_{1}, U_{1}, V_{1})$ around $p$ and $(ψ_{1}, X_{1}, Y_{1})$ around $f (p)$ such that

ψ_{1} \circ f \circ φ_{1}^{- 1} = {ι |}_{V_{1}} .

being

{ι |}_{V_{1}} : (x^{1}, \dots, x^{m}) ⟼ (x^{1}, \dots, x^{m}, 0, \dots, 0)

In a sense is dual to canonical submersion theorem

The proof use the inverse function theorem.