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What is an example of a non vanishing smooth vector field on a manifold $M$ whose corresponding foliation is a Riemannian foliation with respect to no Riemannian metric on $M$

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    $\begingroup$ A closed leaf in a Riemannian foliation cannot have an asymptote. $\endgroup$
    – Anton Petrunin
    Commented Feb 8, 2023 at 21:53
  • $\begingroup$ @AntonPetrunin Dear Anton Thank you for your kind attention to my question and your very helpful answer. BTW in the reference list of the book of Toendeur I had found a title of a paper some things like "on stability of Riemannian foliation" But I did not read the paper. Then I post this question here. Is your idea parallel to this paper mentioned in the (very long) bibliography of Tondeur book? How does orthogonality of leaf-geodesics relate to non existence of limit cycle? $\endgroup$
    – Ali Taghavi
    Commented Feb 9, 2023 at 8:07

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