On the canonical Hermitian connection in nearly Kähler manifolds

Abstract

In the present paper we prove that the Hermitian curvature tensor associated to a nearly Kähler metric g always satisfies the second Bianchi identity (Y, Z, ·, ·)=0 and that it satisfies the first Bianchi identity (X, Y, Z, ·)=0 if and only if g is a Kähler metric. Furthermore we characterize condition for to be parallel with respect to the canonical Hermitian connection in terms of the Riemann curvature tensor and in the last part of the paper we study the curvature of some generalizations of the nearly Kähler structure.