On Naturally Reductive $(\hat{\mathbf{I}}{\pm }_1,\hat{\mathbf{I}}{\pm }_2)$-Metrics

Abstract

In this paper, we investigate the converse of the Tan-Xu theorem, which states that the naturally reductive property of a Riemannian metric is inherited by a naturally reductive $(\hat{I}{\pm }_1,\hat{I}{\pm }_2)$-metric, and we show that, under certain conditions, the converse also holds. We also examine the relationship between geodesic vector fields on homogeneous Riemannian spaces and homogeneous $(\hat{I}{\pm }_1,\hat{I}{\pm }_2)$-spaces. Finally, we construct left-invariant $(\hat{I}{\pm }_1,\hat{I}{\pm }_2)$-metrics on the tangent bundle of Lie groups using left-invariant Randers metrics on the base Lie group, and study their geometric relations.