A note on Gray problem

Abstract

We consider posed in 1960s Alfred Gray problem on the existence of a six-dimensional non-Kählerian almost Kählerian manifold. We study six-dimensional almost Hermitian locally symmetric sub­manifolds of Ricci type of Cayley algebra (the notion of such six-dimensional submanifolds of the octave algebra was introduced by Vadim Feodorovich Kirichenko). Our main result is the following: it is proved that a six-dimensional almost Hermitian locally symmetric submanifold of Ricci type of Cayley algebra does not admit a non-Kählerian almost Kählerian structure.

Keywords

• gray problem
• almost hermitian structure
• almost kählerian manifold
• six-dimensional almost hermitian locally symmetric submanifold of ricci type of cayley algebra