The signature in actions of semisimple Lie groups on pseudo-Riemannian manifolds

we study the relationship between the signature of a semisimple lie group and a pseudoriemannian manifold on wich the group acts topologically transitively and isometrically. we also provide a description of the bi-invariant pseudo-riemannian metrics on a semisimple lie group over r in terms of the complexification of the lie algebra associated to the group, and then we utilize it to prove a remark of gromov.