Torsion in almost Kaehler geometry

Abstract

We study almost K\"ahler manifolds whose curvature tensor satisfies the second curvature condition of Gray (shortly ${\cal{AK}}_2$). 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 K\"ahler nullity of the almost complex structure. We prove a local structure result for ${\cal{AK}}_2$ manifolds, showing that the basic pieces are manifolds with parallel torsion and special almost K\"ahler manifolds, a class generalizing, to some algebraic extent, the class of 4-dimensional ${\cal{AK}}_2$-manifolds. In the case of parallel torsion, the Einstein condition and the reducibility of the canonical Hermitian connection is studied.