Hermitian geometry of 6-dimensional submanifolds of the Cayley algebra

Orientable 6-dimensional submanifolds (of general type) of the Cayley algebra are investigated on which the 3-fold vector cross products in the octave algebra induce a Hermitian structure. It is shown that such submanifolds of the Cayley algebra are minimal, non-compact, and para-Kähler, their holomorphic bisectional curvature is positive and vanishes only at the geodesic points.
It is also proved that cosymplectic hypersurfaces of 6-dimensional Hermitian submanifolds of the octave algebra are ruled. A simple test for the minimality of such surfaces is obtained. It is shown that 6-dimensional submanifolds of the Cayley algebra satisfying the axiom of $g$-cosymplectic hypersurfaces are Kähler manifolds.