Results of gravitational significance in Riemannian geometry

Abstract

It is well known that by contraction one can deduce from the Bianchi identities in Riemannian geometry the following result: R^h_ijk;h=R_if;k-R_ik;f (1)From this it follows that if: R_if=λg_if (2) which is the usual condition for an Einstein space, then: R^h_ijk;h=0 (3) This implies: (i)C^h_ijk;h=0 (ii) W^h_ikj;h=0 (4) for the conformal and protective curvature tensors in Eisenhart's notation. The gravitational significance of (3) and (4) does not appear to have been examined so far, although (4i) has received some attention in a particular context.