\(N(k)\)-quasi Einstein manifolds satisfying certain curvature conditions (Q384161)

Let \((M,g)\) be an \(n\)-dimensional Riemannian manifold, denote by \(R\) its Riemannian curvature tensor and by \(S\) its Ricci tensor. If \(S \neq 0\) and if there exist smooth functions \(a,b\) and a unit vector field \(\xi\) on \(M\) such that \(S(X,Y) = ag(X,Y) + bg(X,\xi)g(Y,\xi)\) for all vector fields \(X,Y\) on \(M\), then \((M,g)\) is said to be a quasi-Einstein manifold. The \(k\)-nullity distribution \(N(k)\) of \((M,g)\) is defined by \(N_p(k) = \{Z \in T_pM \mid R(X,Y)Z = k(g(Y,Z)X - g(X,Z)Y)\;\text{for\;all}\;X,Y \in T_pM\}\). An \(N(k)\)-quasi-Einstein manifold is a quasi-Einstein manifold for which the vector field \(\xi\) is a section in \(N(k)\). In this case one has \(k = (a+b)/(n-1)\). Let \(\tilde{C}\) be the quasi-conformal curvature tensor of \((M,g)\). The authors derive some necessary and sufficient conditions for \(\tilde{C}(\xi,X) \cdot A = 0\), where \(A\) is the Ricci tensor of \((M,g)\), or the projective curvature tensor of \((M,g)\), or the concircular curvature tensor of \((M,g)\). The authors also show that a perfect fluid pseudo Ricci-symmetric space-time \((M,g)\) is an \(N(2r/9)\)-quasi Einstein manifold, where \(r\) is the scalar curvature of \((M,g)\).