Spacetimes in which the Ricci equations characterize the Riemann tensor
The author considers a space-time \((M,g)\) admitting a tensor \(K^ a_{bcd}\) satisfying \[ K^ a_{bcd}=-K^ a_{bdc},\;K^ a_{bcd}+K^ a_{cdb}+K^ a_{dbc}=0,\;g_{ae}K^ e_{bcd}+g_{be}K^ e_{acd}=0, \] \[ K^ a_{bcd;ef}-K^ a_{bcd;fe}=-K^ m_{bcd}K^ a_{mef}+K^ a_{mcd}K^ m_{bef}+K^ a_{bmd}K^ m_{cef}+K^ a_{bcm}K^ m_{def} \] and asks under what conditions is \(K\) the Riemann tensor \(R^ a_{bcd}\) of the metric \(g\). The author has already provided a partial answer to this general problem [J. Geom. Phys. 7, 191-200 (1990; Zbl 0727.53031)] and a counterexample showing that \(K\) is not necessarily the Riemann tensor associated with \(g\) has been given by \textit{A. D. Rendall} [Mathematical Relativity, Proc. Conf., Canberra/Aust. 1988, Proc. Cent. Math. Anal. Aust. Natl. Univ. 19, 125-136 (1989; Zbl 0686.53013)]. Here the author gives a more general account of the situation by defining a tensor \(P^ a_{bcd}=R^ a_{bcd}-K^ a_{bcd}\) and proceeds with a study of \(K\) and \(P\) which involves the Petrov algebraic types of these tensors. A topological discussion of the generic nature of the results is also given using Whitney-type topological arguments.
scientific article; zbMATH DE number 167645
scientific article