Riemannian symmetries in flag manifolds

TL;DR

This paper explores the Riemannian symmetric structures on flag manifolds, focusing on metrics compatible with their $ ext{Z}_2^k$-symmetry and conditions for natural reductiveness, especially in specific cases.

Contribution

It characterizes Riemannian metrics on flag manifolds with $ ext{Z}_2^k$-symmetry and details conditions for natural reductiveness in particular examples.

Findings

Flag manifolds admit $ ext{Z}_2^k$-symmetric structures.

Conditions for adapted metrics to be naturally reductive are identified.

Specific analysis for $SO(5)/SO(2) imes SO(2) imes SO(1)$ case.

Abstract

Flag manifolds are in general not symmetric spaces. But they are provided with a structure of $\mathbb{Z}_2^k$-symmetric space. We describe the Riemannian metrics adapted to this structure and some properties of reducibility. We detail for the flag manifold $SO(5)/SO(2)\times SO(2) \times SO(1)$ what are the conditions for a metric adapted to the $\mathbb{Z}_2^2$-symmetric structure to be naturally reductive.