We study properties of pseudo-Riemannian metrics corresponding to Monge-Ampère structures on four dimensional . We describe a family of Ricci flat solutions, which are parametrized by six coefficients satisfying the Plücker embedding equation. We also focus on pullbacks of the pseudo-metrics on two dimensional , and describe the corresponding Hessian structures.
We give a classification of pseudo-Riemannian weakly symmetric manifolds in dimensions and , based on the algebraic approach of such spaces through the notion of a pseudo-Riemannian weakly symmetric Lie algebra. We also study the general symmetry of reductive -dimensional pseudo-Riemannian weakly symmetric spaces and particularly prove that a -dimensional reductive -fold symmetric pseudo-Riemannian manifold must be globally symmetric.
In this paper we classify pseudosymmetric and Ricci-pseudosymmetric -contact metric manifolds in the sense of Deszcz. Next we characterize Weyl-pseudosymmetric -contact metric manifolds.
A trans-Sasakian 3-manifold is pseudo-symmetric if and only if it is η-Einstein. In particular, a quasi-Sasakian 3-manifold is pseudo-symmetric if and only if it is a coKähler manifold or a homothetic Sasakian manifold. Some examples of non-Sasakian pseudo-symmetric contact 3-manifolds are exhibited.