Parallel second-order tensors on Vaisman manifolds

Abstract

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.