Suppose that $\langle M,g \rangle$ is a complete, simply connected Riemannian symmetric space. The surface geodesically generated by a vector $\xi$ in $T_pM$ is the set of points lying on geodesics passing through $p$ that are orthogonal to $\xi$.
Suppose that $\xi$ and $\xi^\prime$ are vectors in $T_pM$ and $T_{p^\prime}M$ respectively that geodesically generate the same surface.
Does it follow that the vector obtained by parallel transporting $\xi$ along the geodesic connecting $p$ and $p^\prime$ is proportional to $\xi^\prime$?