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Six theorems on six-dimensional Hermitian submanifolds of Cayley algebra. (English) Zbl 1031.53087

Let \(O\equiv\mathbb{R}^8\) be the Cayley algebra and \(M^6\subset O\) a 6-dimensional oriented submanifold of general type. The purpose of this paper is to present some results on a Hermitian submanifold \(M^6\): minimality, scalar curvature, para Kählerianity, bisectional holomorphic curvature and cosymplectic hypersurfaces of Hermitian \(M^6\). The author shows that the Hermitian submanifolds \(M^6\) are minimal submanifolds and their scalar curvature and bisectional holomorphic curvature are non-positive.
Theorem 1: The holomorphic bisectional curvature of six-dimensional Hermitian submanifolds of the Cayley algebra is nonpositive. Moreover, it vanishes precisely at geodesic points.
Theorem 2: Every cosymplectic hypersurface of a six-dimensional Hermitian submanifold of the Cayley algebra is a ruled manifold.
A six-dimensional Hermitian rule manifold \(M^6\) is a manifold of a zero holomorhpic bisectional curvature if and only if \(M^6\) is a part of a Kählerian flat manifold.
Theorem 3: Every six-dimensional Hermitian submanifold of the Cayley algebra is a para Kählerian manifold.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C55 Global differential geometry of Hermitian and Kählerian manifolds
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