On para-Sasakian manifold (Q1283610)

Let \(M\) be a Riemannian manifold with a \(P\)-Sasakian structure \(\Sigma=(\phi,\xi,\eta, g)\). Let \(R\), \(S\), and \(C\) be the curvature, Ricci, and conformal curvature tensor fields on \(M\), respectively. If \(R(X,Y)\cdot C=0\) (\(R(X,Y)\cdot R=0\), \(R(X,Y)\cdot S=0\)), then \(M\) is called \(C\)-symmetric (symmetric, Ricci-symmetric), where \(R(X,Y)\) is considered as a derivation of the tensor algebra at each point of \(M\) for tangent vectors \(X\) and \(Y\). In this paper, the authors show that a symmetric \(P\)-Sasakian manifold \(M\) is flat or the structure vector \(\xi\) is normal, and that a Ricci-symmetric \(P\)-Sasakian manifold \(M\) is Einstein and hence an \(SP\)-Sasakian manifold. Also, they show that a \(C\)-symmetric \(P\)-Sasakian manifold \(M\) is an \(SP\)-Sasakian manifold with constant scalar curvature.