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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
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