Invariant $f$-structures on naturally reductive homogeneous spaces

We study invariant metric $f$-structures on naturally reductive homogeneous spaces and establish their relation to generalized Hermitian geometry. We prove a series of criteria characterizing geometric and algebraic properties of important classes of metric $f$-structures: nearly Kähler, Hermitian, Kähler, and Killing structures. It is shown that canonical $f$-structures on homogeneous $\Phi$-spaces of order $k$ (homogeneous $k$-symmetric spaces) play remarkable part in this line of investigation. In particular, we present the final results concerning canonical $f$-structures on naturally reductive homogeneous $\Phi$-spaces of order 4 and 5.