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Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 18, No. 1, pp. 19-25 (2002)
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On partially pseudo symmetric $K$-contact Riemannian manifolds

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T. Q. Binh, U. C. De, L. Tamássy

Debrecen University, Hungary; University Kalyani, India

**Abstract:** A Riemannian manifold $(M,g)$ is semi-symmetric if

$(R(X,Y)\circ R) (U,V,W)=0$. It is called pseudo-symmetric if $R\circ R= \frak F$, $\frak F$ being a given function of $X,\dots,W$ and $g$. It is called partially pseudo-symmetric if this last relation is fulfilled by not all values of $X,\dots,W$. Such manifolds were investigated by several mathematicians: I.Z. Szabo, S. Tanno, K. Nomizu, R. Deszcz and others.

In this paper we investigate $K$-contact Riemannian manifolds. In these manifolds the structure vector field $\xi$ plays a special role, and this motivates our interest in the partial pseudo-symmetry of these manifolds. We also investigate the case when $R\circ R$ is replaced by $R\circ S$ ($S$ being the Ricci tensor). We obtain conditions in order that our manifold be: (1) Sasakian or Sasakian of constant curvature 1 (in case of $R\circ R$); (2) an Einstein manifold (in case of $R\circ S$). - Our investigation is closely related to the results of S. Tanno.

**Keywords:** pseudo symmetric Riemann manifolds, $K$-contact Riemann manifolds

**Classification (MSC2000):** 53C15

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