On special weakly symmetric Riemannian manifolds (Q2754951)

Weakly symmetric Riemannian manifolds \((WS)_n\) introduced by \textit{T. Q. Binh} and \textit{L. Tamássy} [in: Differential geometry and its applications, ed. by J. Szenthe et. al., Colloq. Math. Soc. János Bolyai 56, 663--670 (1992; Zbl 0791.53021)] are generalizations of locally symmetric manifolds requiring that \(\nabla R\) is a linear expression in \(R\) which is less than \(\nabla R=0\). At weakly conharmonically symmetric manifolds \((WNS)_n\,\) \(\;R\) is replaced by the conharmonic curvature tensor \(N\). ``Special'' means that the coefficients of the linear expression in \(N\) are special: \((SWNS)_n\).NEWLINENEWLINEIt is proved that: 1) the scalar curvature \(r\) of a \((SWNS)_n\) is constant; 2) if an \((SWNS)_n\) admits a parallel unit vectorfield, then it is a \((WS)_n\); 3) if \(r\) of a projectively flat \((SWNS)_n\) is constant, then the Ricci tensor vanishes. Relations to Einstein manifolds are also discussed.