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$\mathbb{Z}_2^k$-symmetric space; flag manifolds; Riemannian metrics
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. The conditions for a metric adapted to the $\mathbb{Z}_2^2$-symmetric structure to be naturally reductive are detailed for the flag manifold $SO(5)/SO(2)\times SO(2) \times SO(1)$.
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