Summary: A class of manifolds which admit an $f$-structure with $s$-dimensional parallelizable kernel is introduced and studied. Such manifolds are Kenmotsu manifolds if $s=1$, and carry a locally conformal Kähler structure of Kashiwada type when $s=2$. The existence of several foliations allows to state some local decomposition theorems. The Ricci tensor together with Einstein-type conditions and $f$-sectional curvatures are also considered. Furthermore, each manifold carries a homogeneous Riemannian structure belonging to the class

$${\mathcal{T}}_{1}\oplus {\mathcal{T}}_{2}$$

of the classification stated by Tricerri and Vanhecke, provided that it is a locally symmetric space.

##### MSC:

53C15 | Differential geometric structures on manifolds |

53D15 | Almost contact and almost symplectic manifolds |

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |