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On weakly symmetric Riemannian manifolds. (English) Zbl 1136.53019
A non-flat Riemannian manifold $$(M^n,g)$$ $$(n>2)$$ is called weakly symmetric – and the denotes by $$(WS)_n$$ – if its curvature tensor $$R$$ of type $$(0,4)$$ satisfies the condition: \begin{aligned} & (\nabla_XR)(Y,Z,U,V)=A(X)\cdot R(Y,Z,U,V)+B(Y)\cdot R(X,Z,U,V)\\ &\qquad +C(Z)\cdot R(Y,X,U,V) + D(U)\cdot R(Y,Z,X,V)+E(V)\cdot R(Y,Z,U,X) \end{aligned} for all vector fields $$X,Y,Z,U,V\in \chi (M^n)$$, where $$A$$, $$B$$, $$C$$, $$D$$ and $$E$$ are 1-forms (non-zero simultaneously) and $$\nabla$$ is the operator of covariant differentiation with respect to $$g$$. The present note on $$(WS)_n$$ consists of 4 sections starting with “Introduction” and “Fundamental results of a $$(WS)_n$$ $$(n>2)$$”. In Section 3 on “Conformally flat $$(WS)_n$$” “ the authors show – among others – that a conformally flat $$(WS)_n$$ $$(n>3)$$ of non-zero scalar curvature is of hyper quasi-constant curvature (which generalizes the notion of quasi-constant curvature) and also such a manifold is a quasi-Einstein manifold (Theorems 6–9). Finally (Section 4), several examples of $$(WS)_n$$ of both zero and non-zero scalar curvature are obtained, in particular a manifold $$(WS)_n$$ $$(n\geq 4)$$ which is neither locally symmetric nor recurrent, the scalar curvature of which is vanishing (Theorem 12) or non-vanishing and non-constant (Theorem 14), respectively.

##### MSC:
 53B35 Local differential geometry of Hermitian and Kählerian structures 53B05 Linear and affine connections