On the type constancy of some six-dimensional planar submanifolds of Cayley algebra

Abstract

The notion of type constancy was introduced by Alfred Gray for nearly Kählerian manifolds and later generalized by Vadim F. Kirichenko and Irina V. Tret’yakova for all Gray — Hervella classes of almost Her­mitian manifolds. In the present note, we consider the notion of type con­stancy for some six-dimensional almost Hermitian planar submanifolds of Cayley algebra. The almost Hermitian structure on such six-dimensional submanifolds is induced by means of so-called Brown — Gray three-fold vector cross products in Cayley algebra. We select the case when six-dimensional submanifolds of Cayley algebra are locally symmetric. It is proved that six-dimensional locally symmetric submanifolds of Ricci type of Cayley algebra are almost Hermitian manifolds of zero con­stant type. This result means that six-dimensional locally symmetric sub­manifolds of Ricci type of Cayley algebra possess a property of six-dimensional Kählerian submanifolds of Cayley algebra. However, there exist non-Kählerian six-dimensional locally symmetric submanifolds of Ricci type in Cayley algebra.

Keywords

• almost hermitian structure
• type constancy
• six-dimen­sional hermitian planar submanifold of cayley algebra
• submani­fold of ricci type