On a certain application of Patterson's curvature identity (Q2714381)

Let \((M,g)\) be a connected \(n\)-dimensional, \(n\geq 3\), semi-Riemannian manifold of class \(C^\infty\), and denote by \(R\), \(C\), \(S\) and \(\kappa\) the corresponding Riemann-Christoffel curvature tensor, the Weyl conformal curvature tensor, the Ricci tensor and the scalar curvature of \((M,g)\), respectively. For a \((0, k)\)-tensor field \(T\), \(k\geq 1\), and a \((0,2)\)-tensor field \(A\) on \((M,g)\) the tensors \(R\cdot T\) and \(Q(A,T)\) are defined. The article then deals with arbitrary 4-dimensional semi-Riemannian manifolds fulfilling a relation (1) \(R\cdot C=L\cdot Q(S,C)\), for some ``associated'' function \(L\). The case \(Q(S, C)=0\) implies \(R\cdot R=0\) (semisymmetric manifold) and \(\kappa=0\) on \((M,g)\). Every 4-dimensional manifold \((M,g)\) satisfying (1) with \(L\neq 0\) is proved to be pseudosymmetric, fulfilling a relation \(R\cdot R=K\cdot Q(g,R)\) with \(K=\kappa/12\neq 0\), and the relation (1) with \(L=1/3\). Finally, the authors describe a non-trivial example of a manifold realizing all these conditions.