On Naturally Reductive $\boldsymbol{(\alpha_1,\alpha_2)}$-Metrics

TL;DR

This paper explores the conditions under which naturally reductive properties are inherited by $(oldsymbol{ ext{α}_1, ext{α}_2})$-metrics, examines geodesic vector fields on homogeneous spaces, and constructs specific metrics on tangent bundles of Lie groups.

Contribution

It proves a converse to the Tan-Xu theorem for $( ext{α}_1, ext{α}_2)$-metrics, analyzes geodesic vector fields, and constructs new invariant metrics on tangent bundles of Lie groups.

Findings

Converse of Tan-Xu theorem established under certain conditions.

Relationship between geodesic vector fields and homogeneous $( ext{α}_1, ext{α}_2)$-spaces clarified.

Constructed left-invariant $( ext{α}_1, ext{α}_2)$-metrics on tangent bundles of Lie groups.

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.