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On invariant Riemannian metrics on Ledger–Obata spaces

  • Y. Nikolayevsky
  • Yu. G. Nikonorov
Article

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.

Mathematics Subject Classification

53C30 53C25 17B20 

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsLa Trobe UniversityMelbourneAustralia
  2. 2.Southern Mathematical Institute of Vladikavkaz Scientific Centre of the Russian Academy of SciencesVladikavkazRussia

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