The exterior derivative of the Lee form of almost Hermitian manifolds

TL;DR

This paper investigates the exterior derivative of the Lee form in almost Hermitian manifolds, revealing its component structure and relations to curvature, advancing understanding of their geometric properties.

Contribution

It provides explicit expressions for the components of dθ in terms of intrinsic torsion and explores relations between the Lee form and curvature components.

Findings

The Rω-component of dθ is always zero.

Explicit formulas for other components of dθ are derived.

Relations between the Lee form and curvature components are established.

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.