K-manifolds are normal metric globally framed f-manifolds whose Sasaki 2-form is closed. We introduce and study some subclasses of K-manifolds. We describe some examples and we also state local decomposition theorems.

We prove that any contact metric $(\kappa,\mu)$-space $M$ admits a canonical paracontact metric structure that verifies some compatibity conditions with the contact structure. We study this paracontact structure, proving that it satisfies a nullity condition and induces on the underlying contact manifold $M$ a sequence of compatible contact and paracontact metric structures satisfying nullity conditions. The behavior of that sequence is related to the Boeckx invariant $I_M$ and to the bi-Legendrian structure of $M$ associated to the original $(\kappa,\mu)$-structure. Finally we are able to define a canonical Sasakian structure on $M$ starting by the original metric $(\kappa,\mu)$-structure.

We consider a natural generalization of the metric almost contact manifolds that we call metric f.pk-manifolds. They are Riemannian manifolds with a compatible f-structure which admits a parallelizable kernel. With some additional conditions they are called S-manifolds. We give some examples and study some properties of harmonic 1-forms on such manifolds. We also study harmonicity and holomorphicity of vector fields on them.

We consider a natural generalization of the metric almost contact manifolds that we call metric f.pk-manifolds. They are Riemannian manifolds with a compatible f-structure which admits a parallelizable kernel. With some additional conditions they are called S-manifolds. We give some examples and study some properties of harmonic 1-forms on such manifolds. We also study harmonicity and holomorphicity of vector fields on them.

K-manifolds are normal metric globally framed f-manifolds whose Sasaki 2-form is closed. We introduce and study some subclasses of K-manifolds. We describe some examples and we also state local de- composition theorems.

Lorentz numbers are all couples $a + au b$ such that a , b are real numbers and $ au^ 2 = 1$. We study functions over Lorentz numbers and their diÆerentiability. We obtain basic properties about regularity, an extension result of functions on manifolds and an implicit function theorem in the case of one or more variables. Then, we consider manifolds modelled on Lorentz numbers and, as a particular case, we obtain paracomplex manifolds.

We are concerned with $\mathcal K$-manifolds which are a natural generalization of metric quasi-Sasakian ma\-ni\-folds. They are Riemannian manifolds with a compatible $f$-stru\-ctu\-re which admits a parallelizable kernel, have closed Sasaki 2-form and verify a certain normality condition. We study semi-invariant submanifolds of a $\cal K$-manifold and investigate the integrability of the various distributions involved. We also study the normality of semi-invariant submanifolds and present a significant example.