Generalized Hermitian geometry on a manifold with \(f\)-structures (Q2386973)

From the standpoint of \textit{V. F. Kirichenko} in [Izv. Vyssh. Uchebn. Zaved., Mat. 1984, No. 7 (266), 50--58 (1984; Zbl 0553.53021); J. Sov. Math. 42, No. 5, 1885--1919 (1988); translation from Itogi Nauki Tekh., Ser. Probl. Geom. 18, 25--71 (1986; Zbl 0715.53033)] referring to the notion of generalized almost Hermitian structure of rank \(r=1\) on a manifold and its associated \(Q\)-algebra, the author shows how to construct such structures of rank greater than 1, by the use of two commuting \(f\)-structures. Furthermore, he applies the results on homogeneous \(\Phi\)-spaces of finite order studied by \textit{V. V. Balashchenko} and \textit{N. A. Stepanov} [Sb. Math. 186, No. 11, 1551--1580 (1995); translation from Mat. Sb. 186, No. 11, 3--34 (1995; Zbl 0872.53025)].