  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 $\Gamma $-symmetric space when $\Gamma $ is a general
abelian finite group, the symmetric case corresponding to $\Gamma =\Z_2$. We
describe and study all the riemannian metrics on $SO(2n+1)/SO(r_1)\times
SO(r_2)\times SO(r_3)\times SO(2n+1-r_1-r_2-r_3)$ 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.
