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On a complete Riemann manifold $(M,g)$, we know if $v \in T_pM$ is a cut locus of $p$, i.e. $\exp_p(sv)$ is length minimizing for $s \in [0,1]$ and is not length minimizing for $s \in [0,1+\epsilon]$, then either there are two distinct length minimizing geodesics connecting $p$ to $q=\exp_p(v)$, or $q$ is the first conjugate point of $p$ along the geodesic $(s \mapsto \exp_p(sv))$.

Also if we have two disctinct length minimizing geodesics connecting $p$ to $q$, then $v$ is a cut locus of $p$.

My question is, if $q$ is the first conjugate point of $p$ along $(s \mapsto \exp_p(sv))$, is $v$ necessarily a cut locus of $p$?

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No, this is false. Take the quotient of the round $S^3$ by a finite cyclic group of isometries acting freely and having order $n$ sufficiently high. Then for any pair of conjugate points their distance along the geodesic is $\pi$, which is greater than the injectivity radius (if $n$ is high enough, but maybe even $n=3$ will suffice).

Edit: Indeed, $n=2$ actually suffices. The easiest example is the projective plane $RP^2$ with the constant curvature ($+1$) metric. Then for each unit speed geodesic $\gamma(t)$ on $RP^2$, the distance along $\gamma$ between consecutive conjugate points is $\pi$, while the distance between points connected by a nonunique geodesic is $\pi/2$. Hence, for the constant curvature projective plane, the first conjugate point always occurs after the cut locus point.

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