On six-dimensional Vaisman — Gray submanifolds of the octave algebra

The W1 W4 class of almost Hermitian manifolds (in accordance with the Gray — Hervella classification) is usually named as the class of Vaisman — Gray manifolds. This class contains all Kählerian, nearly Kählerian and locally conformal Kählerian manifolds. As it is known, Vaisman — Gray manifolds are invariant under the conformal transformations of the metric. A criterion in the terms of the configuration tensor for an arbitrary six-dimensional submanifold of Cayley algebra to belong to the Vaisman — Gray class of almost Hermitian manifolds is established. The Cartan structural equations of the almost contact metric structures induced on oriented hypersurfaces of six-dimensional Vaisman — Gray submanifolds of the octave algebra are obtained. It is proved that totally geodesic hypersurfaces of six-dimensional Vaisman — Gray submanifolds of Cayley algebra admit nearly cosymplectic structures (or Endo structures). This result is a generalization of the previously proved fact that totally geodesic hypersurfaces of nearly Kählerian manifolds also admit nearly cosymplectic structures.