On the parallel transport of the Ricci curvatures

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