On Riemannian non symmetric spaces and flag manifolds

Details On Riemannian non symmetric spaces and flag manifolds On Riemannian non symmetric spaces and flag manifolds

2006-09-28

arxiv.org/abs/math/0609790v1

Mathematics

Differential Geometry

16 pages

Scientific paper

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.

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