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On weakly symmetric Riemannian manifolds. (English) Zbl 1136.53019

A non-flat Riemannian manifold $\left({M}^{n},g\right)$ $\left(n>2\right)$ is called weakly symmetric – and the denotes by ${\left(WS\right)}_{n}$ – if its curvature tensor $R$ of type $\left(0,4\right)$ satisfies the condition:

$\begin{array}{cc}& \left({\nabla }_{X}R\right)\left(Y,Z,U,V\right)=A\left(X\right)·R\left(Y,Z,U,V\right)+B\left(Y\right)·R\left(X,Z,U,V\right)\hfill \\ & \phantom{\rule{2.em}{0ex}}+C\left(Z\right)·R\left(Y,X,U,V\right)+D\left(U\right)·R\left(Y,Z,X,V\right)+E\left(V\right)·R\left(Y,Z,U,X\right)\hfill \end{array}$

for all vector fields $X,Y,Z,U,V\in \chi \left({M}^{n}\right)$, 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 ${\left(WS\right)}_{n}$ consists of 4 sections starting with “Introduction” and “Fundamental results of a ${\left(WS\right)}_{n}$ $\left(n>2\right)$”. In Section 3 on “Conformally flat ${\left(WS\right)}_{n}$” “ the authors show – among others – that a conformally flat ${\left(WS\right)}_{n}$ $\left(n>3\right)$ 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 ${\left(WS\right)}_{n}$ of both zero and non-zero scalar curvature are obtained, in particular a manifold ${\left(WS\right)}_{n}$ $\left(n\ge 4\right)$ 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 Hermitian and Kählerian structures (local differential geometry) 53B05 Linear and affine connections