On nearly Kaehler geometry

Abstract

We consider complete nearly K\"ahler manifolds with the canonical Hermitian connection. We prove some metric properties of strict nearly K\"ahler manifolds and give a sufficient condition for the reducibility of the canonical Hermitian connection. A holonomic condition for a nearly K\"ahler manifold to be a twistor space over a quaternionic K\"ahler manifold is given. This enables us to give classification results in 10-dimensions.