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Wikipedia gives the following formulation of the $\textbf{Local Submersion thoerem}$,

If $f: M \to N$ is a submersion at $p$ and $f(p)=q \in N$, then there exists an open neighborhood $U$ of $p$ in $M$, an open neighborhood $V$ of $q$ in $N$, and local coordinates $(x_1, \dots, x_m)$ at $p$ and $(x_1, \dots, x_n)$ at $q$ such that $f(U)=V$ and the map $f$ in these local coordinates is $$f(x_1, \dots, x_n, x_{n+1}, \dots, x_m) = (x_1, \dots, x_n)$$

Could I interpret the theorem as saying that for $p$ a regular point of $M$, there exists an open neighborhood of $p$ in $M$ for which all points in this neighborhood are regular points?

Yuugi
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1 Answers1

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If $f:M\to{N}$ is a smooth map of manifolds, and if $Df(p)$ has constant rank on $M$, then for any $q\in{f(M)}$, the inverse image $f^{-1}(q)\subset{M}$ is a regular submanifold.

Because there exist charts such that $\psi\circ{f}\circ\phi^{-1}:(x_1,...,x_m)\to{(x_1,..., x_k, 0,...,0)}$ and $f^{-1}(q)\cap{U}=\{x_1=\cdots=x_k=0\}$. Hence we obtain that $f^{-1}(q)$ is a codimension $k$ regular submanifold.

jimbo
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