Summary: The notion of type constancy was introduced by A. Gray, [Tôhoku Math. J. (2) 28, 601–612 (1976; Zbl 0351.53040)] for nearly Kählerian manifolds and later generalized by V. F. Kirichenko and I. V. Tret’yakova, [Math. Notes 68, No. 5, 569–575 (2000; Zbl 1056.53045); translation from Mat. Zametki 68, No. 5, 668–676 (2000)] for all Gray-Hervella classes of almost Hermitian manifolds. In the present note, we consider the notion of type con-stancy 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.