On weakly symmetric spacetimes (Q2853297)

A non-flat semi-Riemannian manifold is called weakly symmetric if the curvature tensor \(R\) satisfies the condition \((\nabla_XR)(Y;Z)W=A(X)R(Y;Z)W+B(Y)R(X;Z)W+C(Z)R(Y;X)W+D(W)R(Y;Z)X+g(R(Y;Z)W;X)\rho\), where \(\nabla\) denotes the Levi-Civita connection on \((M^n;g)\) and \(A,B,C,D\) and \(\rho\) are 1-forms and a vector field respectively which are nonzero simultaneously. Such a manifold is denoted by \((\mathrm{WS})_n\).NEWLINENEWLINE From the conclusion: We show that a perfect fluid \((\mathrm{WS})_4\) spacetime having cyclic parallel Ricci tensor with the basic vector field as the velocity vector field is of Segre characteristic \([(111),1]\). We also prove that if in a \((\mathrm{WS})_4\) spacetime with cyclic parallel Ricci tensor, the matter distribution is a fluid with the basic vector field of \((\mathrm{WS})_4\) as the velocity vector field of the fluid, then such a fluid can not admit and heat flux. Next we show that a conformally flat perfect fluid \((\mathrm{WS})_4\) spacetime with cyclic parallel Ricci tensor is infinitesimally spatially isotropic to the unit time-like vector field \(\rho\). Finally, we construct an example of such type of spacetimes.