On Riemannian non symmetric spaces and flag manifolds

Abelkader Bouyakoub; Michel Goze; Elisabeth Remm

paper

On Riemannian non symmetric spaces and flag manifolds

arXiv:math/0609790

Abstract

In this work we study riemannian metrics on flag manifolds adapted to the symmetries of these homogeneous nonsymmetric spaces. We first introduce the notion of riemannian $Γ$ -symmetric space when $Γ$ is a general abelian finite group, the symmetric case corresponding to $Γ = Z_{2}$ . We describe and study all the riemannian metrics on $S O (2 n + 1) / S O (r_{1}) \times S O (r_{2}) \times S O (r_{3}) \times S O (2 n + 1 - r_{1} - r_{2} - r_{3})$ for which the symmetries are isometries. We consider also the lorentzian case and give an example of a lorentzian homogeneous space which is not a symmetric space.

16 pages