The Eisenhart problem of finding parallel tensors treated already in the framework of quasi-constant curvature manifolds in [15] is reconsidered for the symmetric case and the result is interpreted in terms of Ricci solitons. If the generator of the manifold provides a Ricci soliton then this is i) expanding on para-Sasakian spaces with constant scalar curvature and vanishing D-concircular tensor field and ii) shrinking on a class of orientable quasi-umbilical hypersurfaces of a real projective space=elliptic space form.2000 Math. Subject Classification: 53Cxx; 53C44; 53C21; 53C20; 53C25.
The weakly symmetry of the Sasakian lift G of a Riemannian metric g is characterized in terms of flatness for g and G. The cases of recurrent or pseudosymmetric G studied by Binh and Tamássy are obtained in particular.
The aim of this paper is to study the class of parallel tensor fields [Formula: see text] of [Formula: see text]-type in a Vaisman geometry [Formula: see text]. A sufficient condition for the reduction of such symmetric tensors [Formula: see text] to a constant multiple of [Formula: see text] is given by the skew-symmetry of [Formula: see text] with respect to the complex structure [Formula: see text]. As an application of the main result, we prove that certain vector fields on a [Formula: see text]-manifold turn out to be Killing. Also, we connect our main result with the Weyl connection of conformal geometry as well as with possible Ricci solitons in [Formula: see text] manifolds.
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