The paper is a complete study of paracontact metric manifolds for which the Reeb vector field of the underlying contact structure satisfies a nullity condition (the condition \eqref{paranullity} below, for some real numbers $% \tilde\kappa$ and $\tilde\mu$). This class of pseudo-Riemannian manifolds, which includes para-Sasakian manifolds, was recently defined in \cite{MOTE}. In this paper we show in fact that there is a kind of duality between those manifolds and contact metric $(\kappa,\mu)$-spaces. In particular, we prove that, under some natural assumption, any such paracontact metric manifold admits a compatible contact metric $(\kappa,\mu)$-structure (eventually Sasakian). Moreover, we prove that the nullity condition is invariant under $% \mathcal{D}$-homothetic deformations and determines the whole curvature tensor field completely. Finally non-trivial examples in any dimension are presented and the many differences with the contact metric case, due to the non-positive definiteness of the metric, are discussed.

Nullity conditions in paracontact geometry

Sasaki–Einstein and paraSasaki–Einstein metrics from (κ,μ)-structures

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 $h$.

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Paracontact metric manifolds without a contact metric counterpart

This paper is a complete study of three-dimensional paracontact metric $(\tilde{\kappa},\tilde{\mu},\tilde{\nu})$-manifolds. Three dimensional paracontact metric manifolds whose Reeb vector field $\xi $ is harmonic are characterized. We focus on some curvature properties by considering the class of paracontact metric $(\tilde{\kappa},\tilde{\mu},\tilde{\nu})$-manifolds under a condition which is given at Definition 3.1. In Theorem 3.9, it is shown that this condition is meaningless in dimension higher than three and hence we study three dimensional case. We study properties of such manifolds according to the cases $\tilde{\kappa}>-1,$ $\tilde{\kappa}=-1, \tilde{\kappa}<-1$ and construct new examples of such manifolds for each case.

A Complete Study of Three-Dimensional Paracontact $(\kappa ,\mu ,\nu) $-SPACES

The object of the present paper is to study the symmetric and skew-symmetric properties of a second order parallel tensor on paracontact metric (k, µ)-spaces and almost β-para-Kenmotsu (k, µ)-spaces. In this paper, we prove that if there exists a second order symmetric parallel tensor on a paracontact metric (k, µ)-space M, then either M is locally isometric to a product of a flat n + 1-dimensional manifold and an n-dimensional manifold of constant sectional curvature −4, or the second order parallel tensor is a constant multiple of the associated metric tensor g of M2n+1. If there is a second order parallel tensor on an almost β-para-Kenmotsu (k, µ)-space with k ≠ 0, then it is a constant multiple of the associated metric tensor g of M2n+1.

Second order parallel tensors on some paracontact manifolds

An example of a three dimensional flat paracontact metric manifold with respect to Levi-Civita connection is constructed. It is shown that no such manifold exists for odd dimensions greater than or equal to five.

Fakultet Po Matematika I Informatika

Papers

Non-existence of flat paracontact metric structures in dimension greater than or