On the Reissner-Nordström-de Sitter type spacetimes
Let \((M,g)\), \(n\geq 3\), be a semi-Riemannian manifold and \(T\) be a (0,4)-tensor on \(M\) satisfying \[ T =\alpha{\overline A} + \beta g\wedge A +\gamma G, \tag{1} \] where \(\alpha\), \(\beta\), \(\gamma\) are functions on \(M\) and \(A\) a symmetric (0,2)-tensor on \(M\). Clearly, \(T\) is a generalized curvature tensor. It is known that if we have \(T= {\overline A} +\gamma G\), where \(\gamma\) is a function on \(M\) and \(A\) a symmetric (0,2)-tensor, then \[ T\cdot T=\mathcal Q( \text{Ric}(T), T)-(n- 2)\gamma \mathcal Q(g, \text{Weyl}(T)). \] The author proves a generalization of this result. Namely, if (1) holds on \(M\) then at all points of \(M\) at which \(\alpha\) is nonzero we have \[ T\cdot T =\mathcal Q(\text{Ric}(T), T) + L_2\mathcal Q(g, \text{Weyl}(T)),\quad L_2 = (n-2)\left(\frac{\beta^2}{\alpha}-\gamma\right). \] In particular, if \(A = \text{Ric}(T)\) then (1) takes the form \[ T=\alpha{\overline{\text{Ric}(T)}}+\beta g\wedge \text{Ric}(T)+\gamma G.\tag{2} \] The author presents a family of new curvature conditions of pseudosymmetry type. Such conditions are fulfilled on a semi-Riemannian manifolds \((M,g)\), \(n\geq 4\), admitting a generalized curvature tensor \(T\) such that (2) holds on \(U_{\text{Ric}(T)}\cap U_{\text{Weyl}(T)}\subset M\). It is shown that the curvature tensors of some Reissner-Nordström-de Sitter type space-times satisfy these conditions. An example of a family of warped product space-times satisfying (2) for \(T=R\), a Roter type tensor, is given.
curvature conditions of pseudosymmetry type
scientific article; zbMATH DE number 5142560