Integral manifolds of the first fundamental distribution $lcAC_S$-structure

In paper we consider aspects of the Hermitian geometry of $lcAC_S$structures. The effect of the vanishing of the Neyenhuis tensor and the associated tensors $N^{(1)}$, $N^{(2)}$, $N^{(3)}$, $N^{(4)}$ on the class of almost Hermitian structure induced on the first fundamental distribution of $lcAC_S$structures is investigated. It is proved that the almost Hermitian structure induced on integral manifolds of the first fundamental distribution: $lcAC_S $-manifolds is a structure of the class $W_2\oplus W_4$, and it will be almost Kähler if and only if $grad \ \sigma \subset L(\xi)$; an integrable $lcAC_S $-manifold is a structure of the class $W_4$; a normal $lcAC_S$-manifold is a Kähler structure; a $lcAC_S $-manifold for which $N^{(2)} (X,Y)=0$, or $N^{(3)} (X)=0$, or $N^{(4)} (X)=0$, is an almost Kähler structure in the Gray-Herwell classification of almost Hermitian structures.