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On partially pseudo symmetric \(K\)-contact Riemannian manifolds. (English) Zbl 1017.53031
A Riemannian manifold \((M,g)\) is called pseudo-symmetric if \((R(X,Y)\circ R)(U,V,W) = f[((X\wedge Y)\circ R)(U,V,W)]\) is satisfied, where \(f\in C^\infty(M)\), and \(R\) is the curvature tensor of the manifold. It is called partially pseudo-symmetric if this relation is fulfilled by not all values of \(X,\dots, W\). In this paper partially pseudo-symmetric \(K\)-contact Riemannian manifolds are investigated. Even the case is considered when \(R\circ R \) is replaced by \(R\circ S\) (\(S\) being the Ricci tensor). The authors obtain sufficient conditions in order the manifold \(M\) be: (1) Sasakian of constant curvature 1 (in case of \(R\circ R \)); (2) an Einstein manifold (in case of \(R\circ S \)).

MSC:
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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