Third-order tensor potentials for the Riemann and Weyl tensors. II: Singular solutions
\textit{E. Udeschini Brinis} [Rend., Sci. Mat. Fis. Chim. Geol. 111, 466-475 (1977; Zbl 0414.53010)] conjectured that the Riemann tensor might be derivable as the covariant derivative of a third rank tensor suitably skewed and symmetrized. In 1983 the authors of this paper showed [Gen. Relativ. Gravitation 15, 375-386 (1983; Zbl 0508.53021)] that the integrability conditions prevent any ordinary (in the sense of Cartan) solutions. There can be singular solutions and in four dimensions these are not constrained by any integrability condition. (However the paper by \textit{E. Massa} and \textit{E. Pagani} announced below shows that one of the original conditions reduces, for solutions of the vacuum field equations with or without a cosmical term, to a severe condition on the space, which is not fulfilled by the Schwarzschild, Kerr or de Sitter solutions, nor in general for any metrics of Petrov type different from III, N or 0. It is to be hoped that this will be the end of the story.)
covariant derivative of a third rank tensor
scientific article; zbMATH DE number 3864023