Abstract
We study invariant metrics on Ledger–Obata spaces \(F^m/{\text {diag}}(F)\). 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 \(F^m/{\text {diag}}(F)\) 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.
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The first named author was partially supported by ARC Discovery Grant DP130103485.
The second named author was partially supported by Grant 1452/GF4 of Ministry of Education and Sciences of the Republic of Kazakhstan for 2015–2017.
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Nikolayevsky, Y., Nikonorov, Y.G. On invariant Riemannian metrics on Ledger–Obata spaces. manuscripta math. 158, 353–370 (2019). https://doi.org/10.1007/s00229-018-1029-9
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DOI: https://doi.org/10.1007/s00229-018-1029-9