IT is well known that by contraction one can deduce from the Bianchi identities in Riemannian geometry the following result: From this it follows that if: which is the usual condition for an Einstein space, then: This implies: for the conformal and protective curvature tensors in Eisenhart's1 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 context2.
Eisenhart, L. P., “Riemannian Geometry”, 82, 91 and 135 (Princeton, 1926).
Narlikar, V. V., and Singh, K. P., Phil. Mag., 41, 152 (1950).
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NARLIKAR, V. Results of Gravitational Significance in Riemannian Geometry. Nature 177, 1138 (1956). https://doi.org/10.1038/1771138a0
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