Flag Curvature of Invariant Randers Metrics on Homogeneous Manifolds

TL;DR

This paper reviews recent findings on the flag curvature of invariant Randers metrics on homogeneous manifolds, corrects an existing formula with a counterexample, and provides an explicit formula for naturally reductive cases.

Contribution

It corrects a previous formula for flag curvature and derives an explicit formula for naturally reductive homogeneous manifolds with invariant Randers metrics.

Findings

Previous formula for flag curvature was incorrect.

Counterexample demonstrated the flaw in existing formula.

New explicit formula for flag curvature on naturally reductive manifolds.

Abstract

In this article we review the recent results about the flag curvature of invariant Randers metrics on homogeneous manifolds and by using a counter example we show that the formula which obtained for the flag curvature of these metrics is incorrect. Then we give an explicit formula for the flag curvature of invariant Randers metrics on the naturally reductive homogeneous manifolds $(G/H,g)$, where the Randers metric induced by the invariant Riemannian metric $g$ and an invariant vector field $\tilde{X}$ which is parallel with $g$.