Nullity conditions in paracontact geometry

Cappelletti Montano B., Erken I. K. , Murathan C.

DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS, vol.30, no.6, pp.665-693, 2012 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 30 Issue: 6
  • Publication Date: 2012
  • Doi Number: 10.1016/j.difgeo.2012.09.006
  • Page Numbers: pp.665-693
  • Keywords: Paracontact metric manifold, Para-Sasakian, Contact metric manifold, (kappa, mu)-manifold, Legendre foliation, CONTACT METRIC (KAPPA, MANIFOLDS


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 (kappa) over bar and (mu) over bar). 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 (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 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. (C) 2012 Elsevier B.V. All rights reserved.