Nullity conditions in paracontact geometry

TL;DR

This paper thoroughly investigates paracontact metric manifolds satisfying a nullity condition, revealing dualities with contact metric spaces, invariance under deformations, and providing explicit examples and curvature characterizations.

Contribution

It establishes a duality between nullity conditions in paracontact and contact metric manifolds, and shows that such structures admit compatible contact structures and are invariant under certain deformations.

Findings

Nullity condition determines the entire curvature tensor.

Existence of compatible contact metric structures under natural assumptions.

Explicit non-trivial examples in all dimensions.

Abstract

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.