Motivated by Zirnbauer [Zir 1996], we develop a theory of Riemannian supermanifolds up to a definition of Riemannian symmetric superspaces. Various fundamental concepts needed for the study of these spaces both from the Riemannian and the Lie theoretical viewpoint are introduced, e.g. geodesics, isometry groups and invariant metrics on Lie supergroups and homogeneous superspaces.

Pseudo-Riemannian geometry and Hilbert–Schmidt norms are two important fields of research in applied mathematics. One the main goals this paper will be to find a link between these fields. In respect, present paper, we introduce analyze quantities pseudo-Riemannian geometry, namely H-distorsion and, respectively, Hessian χ-quotient. This second quantity investigated using Frobenius (Hilbert–Sch...