On generalised Ricci-recurrent Sasakian manifolds (Q2716737)

An almost contact metric manifold \((M,\xi,\eta,g)\) is called a \(K\)-contact Riemannian manifold if \(\xi\) is a Killing vector field. A \(K\)-contact Riemannian manifold is Sasakian if \((\nabla_X\varphi)(Y)=g(X,Y)\xi- \eta(Y)X\). A non-flat Riemannian manifold \((M,g)\) is called a generalized Ricci-recurrent manifold if its Ricci tensor \(S\) satisfies the condition \((\nabla_X S)(Y,Z)=A(X)S(Y,Z) +B(X)g(Y,Z)\), where \(A\) and \(B\) are 1-forms on \(M\) such that \(g(X,P)=A(X)\) and \(g(X,Q)=B(X)\), where \(P\), \(Q\) are two vector fields. In this paper, the author studies properties of these manifolds and shows that in a generalized Ricci-recurrent Sasakian manifold \(M\) the vectors \(P\) and \(Q\) have the same direction with opposite orientations. Also, it is shown that if the Ricci tensor \(S\) of a generalized Ricci-recurrent Sasakian manifold \(M\) satisfies \((\nabla_X S)(Y,Z) +(\nabla_Y S)(Z,X)+(\nabla_Z S)(X,Y) =0\), then \(M\) is Einstein.