Killing $f$-manifolds of constant type

The notion of constancy of type was introduced by Gray in the study of specific properties of the geometry of six-dimensional nearly Kahlerian manifolds, and has been investigated by many authors. This notion can be generalized in a natural manner to the case of metric $f$-manifolds with the Killing fundamental form (Killing $f$-manifolds). In this paper, the property of constancy of type is studied in the naturally arising class of so-called commutatively Killing $f$-manifolds, and some of their remarkable properties are investigated. An exhaustive description of commutatively Killing $f$-manifolds of constant type is obtained. In particular, it is proved that the constancy of type of commutatively Killing $f$-manifolds is tantamount to their local equivalence to the five-dimensional sphere $S^5$ endowed with the weakly cosymplectic structure induced by a special embedding of $S^5$ in the Cayley numbers.