Local classification and examples of an important class of paracontact metric manifolds

We study paracontact metric $(\kappa,\mu)$-spaces with $\kappa=-1$, equivalent to $h^2=0$ but not $h=0$. In particular, we will give an alternative proof of Theorem 3.2 of [11] and present examples of paracontact metric $(-1,2)$-spaces and $(-1,0)$-spaces of arbitrary dimension with tensor $h$ of every possible constant rank. We will also show explicit examples of paracontact metric $(-1, \mu)$-spaces with tensor $h$ of non-constant rank, which were not known to exist until now.Comment: 9 page

Paracontact metric manifolds without a contact metric counterpart

We study non-paraSasakian paracontact metric $(\kappa,\mu)$-spaces with $\kappa=-1$ (equivalent to $h^2=0$ but $h\neq0$). These manifolds, which do not have a contact geometry counterpart, will be classified locally in terms of the rank of $h$. We will also give explicit examples of every possible constant rank of $h$.Comment: 12 pages; several corrections have been made in this versio

Paracontact metric structures on the unit tangent sphere bundle

Starting from $g$-natural pseudo-Riemannian metrics of suitable signature on the unit tangent sphere bundle $T_1 M$ of a Riemannian manifold $(M,\langle,\rangle)$, we construct a family of paracontact metric structures. We prove that this class of paracontact metric structures is invariant under $\mathcal D$-homothetic deformations, and classify paraSasakian and paracontact $(\kappa,\mu)$-spaces inside this class. We also present a way to build paracontact $(\kappa,\mu)$-spaces from corresponding contact metric structures on $T_1 M$.Comment: 21 page

Some Non-Compactness Results for Locally Homogeneous Contact Metric Manifolds

We exhibit some sufficient conditions ensuring the non-compactness of a locally homogeneous, regular, contact metric manifold, under suitable assumptions on the Jacobi operator of the Reeb vector field. Mathematics Subject Classification. 53C25, 53C30.Universit´a degli Studi di Bari Aldo Mor

Generalized (κ, µ)-space forms and d-homothetic deformations

We study the Da-homothetic deformations of generalized (κ, µ)- space forms. We prove that the deformed spaces are again generalized (κ, µ)-space forms in dimension 3, but not in general, although a slight change in their definition would make them so. We give infinitely many examples of generalized (κ, µ)-space forms of dimension 3

The curvature tensor of almost cosymplectic and almost Kenmotsu (κ,μ,ν)-spaces

We study the Riemann curvature tensor of (κ,μ, ν)-spaces when they have almost cosymplectic and almost Kenmotsu structures, giving its writing explicitly. This leads to the definition and study of a natural generalisation of the contact metric (κ,μ, ν)-spaces. We present examples or obstruction results of these spaces in all possible cases.Ministerio de Educación, Cultura y DeporteJunta de Andalucí

Bochner and conformal flatness on normal complex contact metric manifolds

We will prove that normal complex contact metric manifolds that are Bochner flat must have constant holomorphic sectional curvature 4 and be Kähler. If they are also complete and simply connected, they must be isometric to the odd-dimensional complex projective space CP 2n+1(4) with the Fubini-Study metric. On the other hand, it is not possible for normal complex contact metric manifolds to be conformally flat