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.


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