On the canonical Hermitian connection in nearly Kähler manifolds

Abstract

In the present paper we prove that the Hermitian curvature tensor $\tilde{R}$ associated to a nearly Kähler metric g always satisfies the second Bianchi identity $\mathfrak{S}(\tilde{\nabla}_X\tilde{R})$ (Y, Z, ·, ·)=0 and that it satisfies the first Bianchi identity $\mathfrak{S}\tilde{R}$(X, Y, Z, ·)=0 if and only if g is a Kähler metric. Furthermore we characterize condition for $\tilde{R}$ to be parallel with respect to the canonical Hermitian connection $\tilde{\nabla}$ 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.