Naturally reductive $(\alpha_1, \alpha_2)$ metrics

TL;DR

This paper characterizes naturally reductive $( ext{α}_1, ext{α}_2)$ metrics on reductive homogeneous manifolds, linking their natural reductiveness to local $f$-products, curvature properties, and providing explicit flag curvature formulas.

Contribution

It introduces a characterization of natural reductiveness as a local $f$-product and establishes equivalences among curvature properties for these metrics, with explicit curvature formulas.

Findings

Natural reductiveness characterized as local $f$-product.

Equivalence established among properties related to mean Berwald and S-curvature.

Explicit formula derived for flag curvature in naturally reductive cases.

Abstract

Let $F$ be a homogeneous $(\alpha_1,\alpha_2)$ metric on the reductive homogeneous manifold $G/H$. Firstly, we characterize the natural reductiveness of $F$ as a local $f$-product between naturally reductive Riemannian metrics. Secondly, we prove the equivalence among several properties of $F$ for its mean Berwald curvature and S-curvature. Finally, we find an explicit flag curvature formula when $F$ is naturally reductive.