Is there a compact Riemannian manifold $M^n$ and $p, q\in M^n$, such that $q$ is conjugate to $p$ along some geodesic $\gamma$, but $q$ is not conjugate to $p$ along another geodesic $\tilde{\gamma}$?
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Yes, take an oblate ellipsois and $p$ the south pole. The conjugate locus will be a sort of astroid with center at the north pole. See for example figure 7 here: https://projecteuclid.org/download/pdf_1/euclid.em/1087568023
Take now $\gamma$ to be a short meridian, in such a way that $\gamma(0)=p$ and $\gamma(\pi)$ is the north pole, and $\gamma$ remains a minimizing geodesic until there. After reaching the north pole the geodesic will cease to be optimal of course, and at some point $\gamma(\pi + \tau)$ will be conjugate to $p$ along $\gamma$ (this is precisely the point at which $\gamma$ intersects for the second time the astroid).
But then there exists another geodesic joining $p$ with $\gamma(\pi+\tau)$, which will be of course free of conjugate points.
