# zbMATH — the first resource for mathematics

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.)
Full Text: