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Let $M$ be a smooth manifold, $N\subset M$ be a smooth closed hypersurface not bounding a compact submanifold, and $X$ be a smooth nowhere-zero vector field on $M$. I would like to learn what is known about the following

Question. When can $N$ be moved by an isotopy to be nowhere-tangent to $X$?

A necessary condition for existence of such an isotopy is that $X$ be homotopic through nowhere-zero vector fields to one nowhere-tangent to $N$. Are there invariants that can be used to imply that the desired isotopy does not exist even when this necessary condition is satisfied?

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The simplest case is when $M$ is a compact manifold with connected boundary $N$. If $N$ is nowhere tangent to $X$ then, by replacing $X$ by $-X$ if necessary, we can assume $X$ points outwards at points of $N$. Then, since $X$ has no zeros the Poincaré-Hopf theorem implies the Euler characteristic vanishes: $\chi(M)=0$.

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    $\begingroup$ Thanks - good point. I edited the question to clarify that the main case I’m interested in is $M$ noncompact and boundaryless. $\endgroup$
    – Matthew Kvalheim
    Commented Nov 8, 2022 at 18:48
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    $\begingroup$ Does $N$ separate $M$ into two pieces, one of which is compact? If so, the Euler characteristic may still be of use. $\endgroup$
    – Joel Fine
    Commented Nov 8, 2022 at 19:07
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    $\begingroup$ In my cases of interest $N$ separates $M$ into two or more pieces which are all noncompact. $\endgroup$
    – Matthew Kvalheim
    Commented Nov 8, 2022 at 19:09

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