2003/01/08 by Paul-Andi Nagy, Nagy, Paul-Andi
Mathematics · Physics and Astronomy · #53B20 #53C25 #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #math.DG #msc:53B20 #msc:53C25
paper · pdf · doi:10.48550/arxiv.math/0301069
latex 2e, 23 pages
arxiv created 2003/01/08 · openalex publication_date 2003/01/08 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study almost Kähler manifolds whose curvature tensor satisfies the second curvature condition of Gray (shortly \calAK2). This condition is interpreted in terms of the first canonical Hermitian connection. It turns out that this condition forces the torsion of this connection to be parallel in directions orthogonal to the Kähler nullity of the almost complex structure. We prove a local structure result for \calAK2 manifolds, showing that the basic pieces are manifolds with parallel torsion and special almost Kähler manifolds, a class generalizing, to some algebraic extent, the class of 4-dimensional \calAK2-manifolds. In the case of parallel torsion, the Einstein condition and the reducibility of the canonical Hermitian connection is studied.