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On weakly Ricci symmetric spacetime manifolds. (English) Zbl 1088.53009

A non-flat Riemannian manifold \((M^n,g)\), \(n>2\), is called weakly Ricci symmetric if its Ricci tensor \(S\) is not identically zero and satisfies the condition \(\nabla_X S(Y,Z) = A(X)S(Y,Z) + B(Y)S(Z,X) +D(Z)S(X,Y) \), where \(A,B,C\) are three non-zero 1-forms. Such an \(n\)-dimensional manifold is denoted by \((WRS)_n\) [L. Tamassy and T. Q. Binh, Tensor, New Ser. 53, 140–148 (1993; Zbl 0849.53038)]. If \(\delta = B-D \neq 0,\) the Ricci tensor is of the form \(S(X,Y) =rT(X)T(Y)\), where \(T\) is a non-zero 1-form defined by \(T(x) = g(X,\rho )\), \(r\) is the scalar curvature and \(\rho\) is called the basic vector field of \((WRS)_n\).
The authors prove: Theorem 1. If in a weakly Ricci symmetric space-time of non-zero constant scalar curvature the matter distribution is that of a perfect fluid whose velocity vector field is the basic vector field \(\rho \) of the space-time, the acceleration vector of the fluid must be zero and the expansion scalar also. Moreover, the cosmological constant \(\lambda\) satisfies \(\lambda \in (\frac{r}{6}, \frac{3r}{2})\).
Theorem 2. \((WRS)_4\) space-time can not admit a heat flux.
Theorem 3. In a conformally flat \((WRS)_4\) space-time the vector field \(\rho\) is non-rotational and its integral curves are geodesics.
Theorem 4. A conformally flat perfect fluid \((WRS)_4\) space-time obeying the Einstein equation without cosmological constant and having the basic field \(\rho\) as the velocity vector field has the following property: All planes perpendicular to \(\rho\) have sectional curvature \(\frac{r}{2}\) and all planes containing \(\rho\) have sectional curvature \(0\).

MSC:

53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)

Citations:

Zbl 0849.53038
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