A non-flat Riemannian manifold is called weakly symmetric – and the denotes by – if its curvature tensor of type satisfies the condition:
for all vector fields , where , , , and are 1-forms (non-zero simultaneously) and is the operator of covariant differentiation with respect to . The present note on consists of 4 sections starting with “Introduction” and “Fundamental results of a ”. In Section 3 on “Conformally flat ” “ the authors show – among others – that a conformally flat 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 of both zero and non-zero scalar curvature are obtained, in particular a manifold 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.