2012/08/20 by Florin Belgun, Belgun, Florin · 1 citation
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1208.4021
openalex publication_date 2012/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a class of hermitian metrics with \em Lee potential, that generalize the notion of l.c.K. metrics with potential introduced in \citeov and show that in the classical examples of Calabi and Eckmann of complex structures on S2p+1\x S2q+1, the corresponding hermitian metrics are of this type. These examples satisfy, actually, a stronger differential condition, that we call \em generalized Calabi-Eckmann, condition that is satisfied also by the \em Vaisman metrics (previously also refered to as \em generalized Hopf manifolds). This condition means that, in addition to being with Lee potential, the torsion of the \em characteristic (or Bismut) connection is parallel. We give a local geometric characterization of these generalized Calabi-Eckmann metrics, and, in the case of a compact threefold, we give detailed informations about their global structure. More precisely, the cases which can not be reduced to Vaisman structures can be obtained by deformation of locally homogenous hermitian manifolds that can be described explicitly.