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
We prove that any contact metric (κ, µ)-space (M, ϕ, ξ, η, g) admits a canonical paracontact metric structure that is compatible with the contact form η. We study this canonical paracontact structure, proving that it satisfies a nullity condition and induces on the underlying contact manifold (M, η) a sequence of compatible contact and paracontact metric structures satisfying nullity conditions. We then study the behavior of that sequence, which is related to the Boeckx invariant I M and to the bi-Legendrian structure of (M, ϕ, ξ, η, g). Finally we are able to define a canonical Sasakian structure on any contact metric (κ, µ)-space whose Boeckx invariant satisfies |I M | > 1.
We describe a contact metric manifold whose Reeb vector field belongs to the (κ, µ)-nullity distribution as a bi-Legendrian manifold and we study its canonical bi-Legendrian structure. Then we characterize contact metric (κ, µ)-spaces in terms of a canonical connection which can be naturally defined on them.2000 Mathematics subject classification: 53C12, 53C15, 53C25.
Dedicated to the memory of Late Professor Jerzy J. Konderak on the second anniversary of his departure.Abstract. Invariant submanifolds of contact (κ, μ)-manifolds are studied. Our main result is that any invariant submanifold of a non-Sasakian contact (κ, μ)-manifold is always totally geodesic and, conversely, every totally geodesic submanifold of a nonSasakian contact (κ, μ)-manifold, μ = 0, such that the characteristic vector field is tangent to the submanifold is invariant. Some consequences of these results are then discussed.2000 Mathematics Subject Classification. Primary 53C40; Secondary 53C25, 53D10, 53D15.
We correct the results in section 6 of [B. Cappelletti Montano, A. De Nicola, G. Dileo, 3-Quasi-Sasakian manifolds, Ann. Global Anal. Geom. 33 (2008), 397-409], concerning the corrected energy of the Reeb distribution of a compact 3-quasi-Sasakian manifold. The results are slightly different than what was originally claimed and they are obtained by using results in [B. Cappelletti Montano, A. De Nicola, G. Dileo, The geometry of a 3-quasi-Sasakian manifold, Int. J. Math., to appear, arXiv:0801.1818], where the geometry of these manifolds is more deeply investigated.
We present a compared analysis of some properties of 3-Sasakian and 3-cosymplectic manifolds. We construct a canonical connection on an almost 3-contact metric manifold which generalises the Tanaka-Webster connection of a contact metric manifold and we use this connection to show that a 3-Sasakian manifold does not admit any Darboux-like coordinate system. Moreover, we prove that any 3-cosymplectic manifold is Ricci-flat and admits a Darboux coordinate system if and only if it is flat
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