On pseudo-symmetric and pseudo Ricci symmetric \(K\)-contact manifolds (Q1915638)

If \((M,g)\) is a Riemannian manifold whose curvature tensor \(R\) satisfies the condition \[ (\nabla_X R_{YZ}) W = 2A(X) R_{YZ} W + A(Y) R_{XZ} W + A(Z) R_{YX} W + A(W) R_{YZ} X + g(R_{YZ} W,X )_, \] where a nonzero 1-form \(A\) and a vector field \(P\) are related by \(g(X,P) = A(X)\), then it is called a pseudo-symmetric manifold. If its Ricci tensor \(S\) satisfies \[ (\nabla_XS) (Y,Z) = 2A(X) S(Y,Z) + A(Y) S(X,Z) + A(Z) S(X,Y), \] then \((M,g)\) is said to be pseudo-Ricci symmetric. In this paper, the authors prove that pseudo-symmetric and pseudo-Ricci-symmetric manifolds cannot be \(K\)-contact (and Sasakian) manifolds.