On Partially Pseudo Symmetric K-contact Riemannian Manifolds

A Riemannian manifold (M, g) is semi-symmetric if (R(X,Y ) ◦ R)(U, V,W ) = 0. It is called pseudo-symmetric if R ◦ R = F, F being a given function of X, . . . ,W and g. It is called partially pseudosymmetric if this last relation is fulfilled by not all values of X, . . . ,W . Such manifolds were investigated by several mathematicians: I.Z. Szabó, S. Tanno, K. Nomizu, R. Deszcz and others. In this paper we investigate K-contact Riemannian manifolds. In these manifolds the structure vector field ξ plays a special role, and this motivates our interest in the partial pseudo-symmetry of these manifolds. We also investigate the case when R◦R is replaced by R◦S (S being the Ricci tensor). We obtain conditions in order that our manifold be: (1) Sasakian or Sasakian of constant curvature 1 (in case of R ◦ R); (2) an Einstein manifold (in case of R ◦ S). – Our investigation is closely related to the results of S. Tanno.