Nullity conditions in paracontact geometry

CAPPELLETTI MONTANO, BENIAMINO;Kupeli Erken I;Murathan C.

2012-01-01

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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.