Curvature copies and the uniqueness of Bianchi-type equations: the symmetric connection case

Abstract

It is shown, in general circumstances in a four-dimensional spacetime, that (i) the symmetric metric connection of a Riemann tensor cannot be a curvature copy of a symmetric non-metric connection, (ii) the only Bianchi-type equation (in a symmetric connection) obeyed by a Riemann tensor is its own Bianchi equation. Further, these results for Riemann tensors (of metric connections) are generalized naturally to curvature tensors of Weyl connections.