On invariant Riemannian metrics on Ledger-Obata spaces

TL;DR

This paper classifies and constructs invariant metrics on Ledger-Obata spaces, showing that such spaces are geodesic orbit if and only if they are naturally reductive, and characterizing their reducibility and isometry groups.

Contribution

It provides a complete classification and explicit construction of naturally reductive invariant metrics on Ledger-Obata spaces, and characterizes geodesic orbit spaces within this class.

Findings

All invariant metrics on Ledger-Obata spaces are classified and explicitly constructed.

A Ledger-Obata space is geodesic orbit if and only if the metric is naturally reductive.

The isometry group of an irreducible Ledger-Obata space is $F^m$.

Abstract

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.