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On the parallel transport of the Ricci curvatures. (English) Zbl 1169.53016
Summary: Geometrical characterizations are given for the tensor \(R\cdot S\), 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 \(\land_{g}\cdot S\), where the natural metrical operator \(\land_{g}\) 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.

MSC:
53C05 Connections (general theory)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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