We observe that the class of metric f -K-contact manifolds, which naturally contains that of K-contact manifolds, is closed under forming mapping tori of automorphisms of the structure. We show that the de Rham cohomology of compact metric f -K-contact manifolds naturally splits off an exterior algebra, and relate the closed leaves of the characteristic foliation to its basic cohomology.2010 Mathematics Subject Classification. Primary 53C25, 53C15, Secondary 53D10, 32V05.

We show that any compact metric f-K-contact, respectively S-manifold is obtained from a compact K-contact, respectively Sasakian manifold by an iteration of constructions of mapping tori, rotations, and type II deformations.

We classify locally the contact metric (k,μ)‐spaces whose Boeckx invariant is ⩽−1 as tangent hyperquadric bundles of Lorentzian space forms.

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We present a classification of the complete, simply connected, contact metric (κ, µ)-spaces as homogeneous contact metric manifolds, by studying the base space of their canonical fibration. According to the value of the Boeckx invariant, it turns out that the base is a complexification or a paracomplexification of a sphere or of a hyperbolic space. In particular, we obtain a new homogeneous representation of the contact metric (κ, µ)-spaces with Boeckx invariant less than −1. (2000): Primary 53C25, 53D10; Secondary 53C35, 53C30. Mathematics Subject Classification