The exterior derivative of the Lee form of almost Hermitian manifolds

Abstract

The exterior derivative $d \theta$ of the Lee form $\theta$ of almost Hermitian manifolds is studied. If $\omega$ is the K\"ahler two-form, it is proved that the $\mathbb{R}\omega$-component of $d\theta$ is always zero. expressions for the other components, in $[\lambda_0^{1,1}]$ and in $[[ \lambda^{2,0} ]]$, of $d\theta$ are also obtained. They are given in terms of the intrinsic torsion. Likewise, it is described some interrelations between the Lee form and $U(n)$-components of the Riemannian curvature tensor.