Naturally reductive homogeneous $(\alpha,\beta)$-metric spaces

TL;DR

This paper investigates the conditions under which homogeneous $(ppa,eta)$-metric spaces are naturally reductive, proves the equivalence of two definitions, and computes their flag curvature.

Contribution

It provides necessary and sufficient conditions for natural reductiveness and establishes the equivalence of different definitions in this context.

Findings

Conditions for natural reductiveness are established.

Two definitions of naturally reductive homogeneous Finsler spaces are shown to be equivalent.

Flag curvature of these spaces is explicitly computed.

Abstract

In the present paper we study naturally reductive homogeneous $(\alpha,\beta)$-metric spaces. Under some conditions, we give some necessary and sufficient conditions for a homogeneous $(\alpha,\beta)$-metric space to be naturally reductive. Then we show that for such spaces the two definitions of naturally reductive homogeneous Finsler space, given in literature, are equivalent. Finally we compute the flag curvature of naturally reductive homogeneous $(\alpha,\beta)$-metric spaces.