Almost Hermitian Manifold with Flat Bochner Tensor

Many researchers investigated the flat Bochner tensor on some kinds of almost Hermitian manifold. In the present paper the author studies this tensor on general class almost Hermitian manifold by using a new methodology which is called an adjoint G-structure space. Thus this study generalize the results which are found out by those researchers. It is proved that if M is an almost Hermitian manifold of class R-1 with flat Bochner tensor, then either M is 2-dimensional flat Ricci manifold or n-dimensional (n > 2) flat scalar curvature tensor manifold. As well, it is proved that if M is an almost Hermitian manifold with flat Bochner tensor, then M is a manifold of class R-3 if and only if M is a linear complex manifold. Later on, equivalently of classes R-2 and R-3 is investigated. Finally we prove that if M is flat manifold with flat Bochner tensor, then M is an Einstein manifold with a cosmological constant.