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Let $\Sigma$ be a closed surface smoothly embedded in $\mathbb R^3$. For any Morse function $h:\mathbb R^3 \to \mathbb R$, can we isotope $\Sigma$ so that the restriction of $h$ on $\Sigma$ is also a Morse function?

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  • $\begingroup$ No, of course not. For example, what if $h$ is constant? $\endgroup$
    – Ryan Budney
    Commented Jul 25, 2021 at 3:54
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    $\begingroup$ @RyanBudney I forgot to add that $h$ is a Morse function in $\mathbb R^3$. $\endgroup$
    – Zhiqiang
    Commented Jul 25, 2021 at 4:19
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    $\begingroup$ Then I think the answer is yes. Step (1) perturb the surface to be disjoint from the critical points of $h$. Step (2) Use that $h$ is a submersion in a neighbourhood of the surface. Specifically, $h$ restricted to the surface may not be Morse, but a small perturbation of it is. Using the local triviality of $h$ near the surface allows you to perturb the surface, using the flow of a vector field, to mimic the perturbation of a non-Morse function to a Morse function. $\endgroup$
    – Ryan Budney
    Commented Jul 25, 2021 at 4:31

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