Three theorems on cosymplectic hypersurfaces of six-dimensional Hermitian submanifolds of the Cayley algebra

Ahmad Al-Othman,M Banaru

doi:10.1155/s0161171203206190

Abstract

It is proved that cosymplectic hypersurfaces of six-dimensional Hermitian submanifolds of the octave algebra are ruled manifolds. A necessary and sufficient condition for a cosymplectic hypersurface of a Hermitian submanifoldM6⊂Oto be a minimal submanifold ofM6is established. It is also proved that a six-dimensional Hermitian submanifoldM6⊂Osatisfying theg-cosymplectic hypersurfaces axiom is a Kählerian manifold.

Highlights

One of the most important properties of a hypersurface of an almost Hermitian manifold is the existence of such a hypersurface of an almost contact metric structure determined in a natural way
In the case the ambient manifold is Hermitian, comparatively little is known about the geometry of its hypersurfaces
The first group of the Cartan structural equations written in an A-frame of a hypersurface of a Hermitian manifold looks as follows [16]: dωa = ωab Λωb + Bcabωc Λωb + 2Bba3 + iσba ωbΛω iσ ab ωb Λω; dωa = −ωbaΛωb + Bacbωc Λωb + 2Bab3 − iσab ωbΛω iσab ωb Λω; dω = 2Bb3a − 2B3ab − 2iσba ωbΛωa + B33b + iσ3b ωΛωb

Summary

Introduction

One of the most important properties of a hypersurface of an almost Hermitian manifold is the existence of such a hypersurface of an almost contact metric structure determined in a natural way. Let N be an oriented hypersurface of a Hermitian submanifold M6 ⊂ O and let σ be the second fundamental of the immersion of N into M6 As it is well known [15, 17], the almost Hermitian structure on M6 induces an almost contact metric structure on N. The first group of the Cartan structural equations written in an A-frame of a hypersurface of a Hermitian manifold looks as follows [16]: dωa = ωab Λωb + Bcabωc Λωb + 2Bba3 + iσba ωbΛω. According to [6, definition (1)], the components of the Kirichenko tensors are connected to the components of ∇J: Bcab

Bac b

The same reasoning can be applied in reference to the condition