2011/10/28 by G. Bande, Gianluca Bande, D. E. Blair +6
Mathematics · Physics and Astronomy · #53B35 #53C12 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #msc:53B20 #msc:53B35 #msc:53C12 #msc:53C25 #primary 53C25 #secondary 53B20
paper · pdf · doi:10.48550/arxiv.1110.6278
arxiv created 2011/10/28 · openalex publication_date 2011/10/28 · arxiv updated 2011/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider manifolds endowed with metric contact pairs for which the two characteristic foliations are orthogonal. We give some properties of the curvature tensor and in particular a formula for the Ricci curvature in the direction of the sum of the two Reeb vector fields. This shows that metrics associated to normal contact pairs cannot be flat. Therefore flat non-Kähler Vaisman manifolds do not exist. Furthermore we give a local classification of metric contact pair manifolds whose curvature vanishes on the vertical subbundle. As a corollary we have that flat associated metrics can only exist if the leaves of the characteristic foliations are at most three-dimensional.