Riemann compatible tensors
  Derdzinski and Shen's theorem on the restrictions posed by a Codazzi tensor on the Riemann tensor holds more generally when a Riemann-compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity of the new "Codazzi deviation tensor" with a geometric significance. Examples are given of manifolds with Riemann-compatible tensors, in particular those generated by geodesic mapping. Compatibility is extended to generalized curvature tensors with an application to Weyl's tensor and general relativity. 
C. A. Mantica and L. G. Molinari
math.DG math-ph math.MP