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.
