This is an exercise from John Lee. I need to find a smooth map that is a topological submersion but not a smooth submersion. I am trying to work with the Euclidean space but cannot find a proper example. Could anyone please help me?
2026-03-25 08:54:40.1774428880
Smooth map that is a topological submersion but not a smooth submersion
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Consider the map from $\mathbb{R}^2$ to $\mathbb{R}$ sending $(x,y)$ to $x^3$. This is a topological submersion, as we can choose a chart on $\mathbb{R}$ by having $z = x^3$, and on this chart this map is the projection $(x,y) \mapsto x$. It is not a smooth submersion, as the derivative at $(0,0)$ is $0$.