Almost Hermitian manifolds admitting holomorphically planar conformal vector fields

Abstract

We classify and characterize an almost Hermitian manifold M admitting a holomorphically planar conformal vector (HPCV) field (a generalization of a closed conformal vector field) V . We show that if V is nowhere vanishing and strictly non-geodesic, then it is homothetic and almost analytic. If, in addition,M satisfies Gray’s first condition, then M is Kaehler. For a semi-Kaehler manifold M admitting an HPCV field V we show that either V is closed, or M becomes almost Kaehler and V is homothetic and almost analytic.