## Almost Hermitian manifold with flat Bochner tensor.(English)Zbl 1213.53037

Summary: Many researchers investigated the flat Bochner tensor on some kinds of almost Hermitian manifolds. In the present paper, the author studies this tensor on general almost Hermitian manifolds by using a new methodology which is called an adjoint G-structure space. Thus, this study generalizes the results which were 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 a 2-dimensional flat Ricci manifold or an $$n$$-dimensional flat scalar curvature tensor manifold. Also, 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. Then, equivalence of the classes $$R_2$$ and $$R_3$$ is investigated. Finally, we prove that, if $$M$$ is a flat manifold with flat Bochner tensor, then $$M$$ is an Einstein manifold with a cosmological constant.

### MSC:

 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C10 $$G$$-structures 53B35 Local differential geometry of Hermitian and Kählerian structures