Flag curvature of invariant $(\alpha,\beta)$-metrics of type $\frac{(\alpha+\beta)^2}{\alpha}$

TL;DR

This paper derives a formula for the flag curvature of a specific class of invariant Finsler metrics on homogeneous spaces and Lie groups, focusing on cases where the Chern and Levi-Civita connections coincide.

Contribution

It provides a new explicit formula for flag curvature of invariant $(rac{( abla+eta)^2}{ abla})$-metrics on homogeneous spaces and Lie groups, especially under natural reductiveness.

Findings

Derived a formula for flag curvature of the specified metrics.

Analyzed cases of naturally reductive homogeneous spaces.

Examined implications for Lie groups.

Abstract

In this paper we study flag curvature of invariant $(\alpha,\beta)$-metrics of the form $\frac{(\alpha+\beta)^2}{\alpha}$ on homogeneous spaces and Lie groups. We give a formula for flag curvature of invariant metrics of the form $F=\frac{(\alpha+\beta)^2}{\alpha}$ such that $\alpha$ is induced by an invariant Riemannian metric $g$ on the homogeneous space and the Chern connection of $F$ coincides to the Levi-Civita connection of $g$. Then some conclusions in the cases of naturally reductive homogeneous spaces and Lie groups are given.