On invariant Riemannian metrics on Ledger-Obata spaces

We study invariant metrics on Ledger-Obata spaces $F^m/\mathrm{diag}(F)$. We give the classification and an explicit construction of all naturally reductive metrics, and also show that in the case $m=3$, any invariant metric is naturally reductive. We prove that a Ledger-Obata space is a geodesic orbit space if and only if the metric is naturally reductive. We then show that a Ledger-Obata space is reducible if and only if it is isometric to the product of Ledger-Obata spaces (and give an effective method of recognising reducible metrics), and that the full connected isometry group of an irreducible Ledger-Obata space $F^m/\mathrm{diag}(F)$ is $F^m$. We deduce that a Ledger-Obata space is a geodesic orbit manifold if and only if it is the product of naturally reductive Ledger-Obata spaces.