On the parallel transport of the Ricci curvatures

Geometrical characterizations are given for the tensor R⋅SR⋅S, where SS is the Ricci tensor   of a (semi-)Riemannian manifold (M,g)(M,g) and RR denotes the curvature operator   acting on SS as a derivation, and of the Ricci Tachibana tensor  ∧g⋅S∧g⋅S, where the natural metrical operator  ∧g∧g also acts as a derivation on SS. As a combination, the Ricci curvatures   associated with directions on MM, of which the isotropy determines that MM is Einstein, are extended to the Ricci curvatures of Deszcz   associated with directions and planes on MM, and of which the isotropy determines that MM is Ricci pseudo-symmetric in the sense of Deszcz.