HOMOGENEOUS SASAKI AND VAISMAN MANIFOLDS OF UNIMODULAR LIE GROUPS

A Vaisman manifold is a special kind of locally conformally Kähler manifold, which is closely related to a Sasaki manifold. In this paper, we show a basic structure theorem of simply connected homogeneous Sasaki and Vaisman manifolds of unimodular Lie groups, up to holomorphic isometry. For the case of unimodular Lie groups, we obtain a complete classification of simply connected Sasaki and Vaisman unimodular Lie groups, up to modification.

Parallel second-order tensors on Vaisman manifolds

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.