Nullity conditions in paracontact geometry

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 (1.2), for some real numbers and k and μ. This class of pseudo-Riemannian manifolds, which includes para-Sasakian manifolds, was recently defined in Cappelletti Montano (2010) [13]. In this paper we show in fact that there is a kind of duality between those manifolds and contact metric (k,μ)-spaces. In particular, we prove that, under some natural assumption, any such paracontact metric manifold admits a compatible contact metric (k,μ)-structure (eventually Sasakian). Moreover, we prove that the nullity condition is invariant under 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

Yamabe solitons on three-dimensional normal almost paracontact metric manifolds

“…The importance of paracontact geometry, and in particular of para-Sasakian geometry, has been pointed out especially in the last years by several papers highlighting the interplays with the theory of para-Kähler manifolds and its role in semi-Riemannian geometry and mathematical physics (cf. e.g., [5] , [13]).…”

GENERALIZED (Κ˜ ≠ −1, Μ˜)-Paracontact METRIC MANIFOLDS WITH Ξ(µ˜) = 0

“…This new class of pseudo-Riemannian manifolds was introduced in [6]. In [7], the authors showed that while the values ofκ andμ change the form of (1.1) remains unchanged under D-homothetic deformations. There are differences between a contact metric (κ, µ)-space (M 2n+1 , ϕ, ξ, η, g) and a paracontact metric (κ,μ)-space (M 2n+1 ,φ, ξ, η,g).…”

Ricci almost solitons and gradient Ricci almost solitons in $(k,\mu)$-paracontact geometry

“…In [6], the authors construct an example of a 5-dimensional (k, µ)-paracontact metric manifold. With the help of that example we construct a new example as follows:…”

“…Let g be the Lie algebra of a Lie group G admits a basis {e 1 , e 2 , e 3 , e 4 , e 5 } such that [6] [e 1 We consider the metric such that g(e 1 , e 1 ) = g(e 4 , e 4 ) = g(e 5 , e 5 ) = 1, g(e 2 , e 2 ) = g(e 3 , e 3 ) = −1 and g(e i , e j ) = 0, for i = j.…”

“…Paracontact metric manifolds have been studied by Cappelletti-Montano et al [6,7], Martin-Molina [16,17] and many others. According to Cappelletti-Montano et al [6] we have the following definition.…”