## 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