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The definition of submersion is:

Let $f:U\to\mathbb{R}^n$ a differentiable function defined in the open $U\subset\mathbb{R}^m$, if for all $x\in U$, $f'(x)$ is an surjective linear transformation then $f$ is a submersion.

I have to find a open function $f:\mathbb{R}^2\to\mathbb{R}$ of class $C^\infty$, so that $f$ is not a submersion.

I thought in the function, $f(x,y)=x^3$.

If the open $U$ contains any point of the form $(0,y)$, we will have that $f'(0,y)$ is not surjective. But if $U=\{(a,b) : a>1 \}$ we will have that $f$ will be a submersion.

I was wondering if it would be possible to construct a open function $f:\mathbb{R}^2\to\mathbb{R}$ of class $C^\infty$ so that for every open $U\subset\mathbb{R}^2$, $f$ it would not be a submersion.

For another example see this link.

Mrcrg
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1 Answers1

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This is not possible, as a consequence of Sard's theorem. In fact, given a smooth open map $f:\mathbb{R}^2\rightarrow \mathbb{R}$, and given any non-empty $U\subseteq \mathbb{R}^2$, there is an open subset $V\subseteq U$ for which $f|_V$ is a submersion.

To see this, suppose $f:\mathbb{R}^2\rightarrow \mathbb{R}$ is a smooth open map and let $U\subseteq \mathbb{R}^2$ be any open subset.

Then $f(U)$ is open, so by Sard's theorem, it contains a regular value $r\in f(U)$. Pick $x\in U$ with $f(x) = r$. Saying $r$ is a regular value means that $(\nabla f)(x)\neq \langle 0,0\rangle$. Since the partial derivatives $f_x$ and $f_y$ are continuous, there is some neighborhood $V$ of $x$ (which we can take to be a subset of $U$ if we wish) for which $\|(\nabla f)(y)\| \geq \|(\nabla f)(x)\|/2 > 0$ for any $y\in V$.

Because $\nabla f$ is non-zero on $V$, $f|_V$ is a submersion.

Jason DeVito - on hiatus
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