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Inspired by this question about Jacobi equation. conjugate points and limit cycle theory we ask the following question:

Is there a geodesible 1 dimensional foliation $\mathcal{F}$ on a manifold $M$, prefreably on the punctured plane, such that a closed leaf has two different index with respect to two different $\mathcal{F}$-compatible Riemannian metric $g_1,g_2$ on $M$. By $\mathcal{F}$-compatible Riemannian metric $g$ we mean a metric such that all leaves of $\mathcal{F}$ are geodesics for the metric.

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    $\begingroup$ I do not see how there can be any conjugate points at all when a two-dimensional manifold is foliated by geodesics. $\endgroup$
    – Tom Goodwillie
    Commented May 24, 2023 at 11:22
  • $\begingroup$ @TomGoodwillie dear Prof. Goodwillie thank you very much foryour attention to my question. Is not possible two conjugate points p and q on a closed geodesics(closed leaf of the foliation) woul be jointe to eh other via amilly of geodesics which are not necessarilly leaves of the foliation? Why does foliation by geodesics imply no conugate points? $\endgroup$
    – Ali Taghavi
    Commented May 24, 2023 at 15:10
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    $\begingroup$ Suppose that all leaves are geodesics. Then of course there will be geodesics that are not leaves, but such a geodesic can never be tangent to a leaf at any point. Now suppose that $C$ is a leaf and $p\in C$ has a conjugate point on the leaf $C$. So the geodesic $C$ can be slightly perturbed, yielding a new geodesic $D$ which starts at $p$, runs along close to $C$ for a while, and then hits $C$ again. But then, wherever $D$ begins coming back to $C$, it must be tangent to a nearby leaf. (This is not a formal proof, but I find it convincing.) $\endgroup$
    – Tom Goodwillie
    Commented May 24, 2023 at 15:59

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