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Let $M$ be a smooth manifold( if necessary one can assume it is closed), $V$ be a non-vanishing vector field of $TM$.

Q: Under what condition, we can say that $\overline{\exp(tV)}$, i.e. the closure of the one-parameter group in $Diff(M)$, is compact?

Is there paper or research to show such a thing?

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  • $\begingroup$ I suspect that the keyword to search for is "almost periodic": en.wikipedia.org/wiki/… (see the definition of Bochner) $\endgroup$
    – Giuseppe Negro
    Commented Oct 18, 2018 at 8:18
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    $\begingroup$ This happens if an only if there exists a Riemannian metric on $M$ invariant under $V$. $\endgroup$
    – Dmitri Panov
    Commented Dec 11, 2018 at 23:06

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