Canonical immersion theorem
Let $f:M \rightarrow N$ be a immersion at $p\in M$, then $m=\operatorname{dim} M \leq n=\operatorname{dim} N$ and there exist charts $\left(\varphi_{1}, U_{1}, V_{1}\right)$ around $p$ and $\left(\psi_{1}, X_{1}, Y_{1}\right)$ around $f(p)$ such that
$$ \psi_{1} \circ f \circ \varphi_{1}^{-1}=\left.\iota\right|_{V_{1}}. $$being
$$ \left.\iota\right|_{V_{1}}: \left(x^{1}, \ldots, x^{m}\right) \longmapsto \left(x^{1}, \ldots, x^{m}, 0, \ldots, 0\right) $$
In a sense is dual to canonical submersion theorem
The proof use the inverse function theorem.
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Author of the notes: Antonio J. Pan-Collantes
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