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

Abstract

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.

Keywords

• almost hermitian manifold
• vaisman — gray manifold
• al­most contact metric structure
• nearly cosymplectic structure
• totally ge­o­desic hypersurface
• cayley algebra