On Naturally Reductive $\boldsymbol{(\alpha_1,\alpha_2)}$-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 $(\alpha_1,\alpha_2)$-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 $(\alpha_1,\alpha_2)$-spaces. Finally, we construct left-invariant $(\alpha_1,\alpha_2)$-metrics on the tangent bundle of Lie groups using left-invariant Randers metrics on the base Lie group, and study their geometric relations.