*(English)*Zbl 1136.53019

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

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 ${\left(WS\right)}_{n}$ $(n>2)$”. In Section 3 on “Conformally flat ${\left(WS\right)}_{n}$” “ the authors show – among others – that a conformally flat ${\left(WS\right)}_{n}$ $(n>3)$ 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 ${\left(WS\right)}_{n}$ of both zero and non-zero scalar curvature are obtained, in particular a manifold ${\left(WS\right)}_{n}$ $(n\ge 4)$ 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 | Hermitian and Kählerian structures (local differential geometry) |

53B05 | Linear and affine connections |