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
This paper is a study of three-dimensional paracontact metric [Formula: see text]-manifolds. Three-dimensional paracontact metric manifolds whose Reeb vector field [Formula: see text] is harmonic are characterized. We focus on some curvature properties by considering the class of paracontact metric [Formula: see text]-manifolds under a condition which is given at Definition 3.1. We study properties of such manifolds according to the cases [Formula: see text] [Formula: see text] and construct new examples of such manifolds for each case. We also show the existence of paracontact metric [Formula: see text] spaces with dimension greater than 3, such that [Formula: see text] but [Formula: see text]
MSC:primary 53B30 53C15 53C25 secondary 53D10 Keywords:Almost paracontact metric manifold Almost paracosymplectic manifold Almost para-Kenmotsu manifold Para-Kaehler manifold a b s t r a c t This paper is a complete study of almost α-paracosymplectic manifolds. Basic properties of such manifolds are obtained and general curvature identities are proved. The manifolds with para-Kaehler leaves are characterized. It is proved that, for dimensions greater than 3, almost α-paracosymplectic manifolds are locally conformal to almost paracosymplectic manifolds and locally D-homothetic to almost para-Kenmotsu manifolds. Furthermore, it is proved that characteristic (Reeb) vector field ξ is harmonic on almost α-para-Kenmotsu manifold if and only if it is an eigenvector of the Ricci operator. It is showed that almost α-para-Kenmotsu (κ, µ, ν)-space has para-Kaehler leaves. 3-dimensional almost α-paraKenmotsu manifolds are classified. As an application, it is obtained that 3-dimensional almost α-para-Kenmotsu manifold is (κ, µ, ν)-space on an every open and dense subset of the manifold if and only if Reeb vector field is harmonic. Furthermore, examples are constructed.
Abstract. The purpose of this paper is to study anti-invariant Riemannian submersions from Kenmotsu manifolds onto Riemannian manifolds. Several fundamental results in this respect are proved. The integrability of the distributions and the geometry of foliations are investigated. We proved that there do not exist (anti-invariant) Riemannian submersions from Kenmotsu manifolds onto Riemannian manifolds such that characteristic vector field ξ is a vertical vector field. We gave a method to get horizontally conformal submersion examples from warped product manifolds onto Riemannian manifolds. Furthermore, we presented an example of anti-invariant Riemannian submersions in the case where the characteristic vector field ξ is a horizontal vector field and an anti-invariant horizontally conformal submersion such that ξ is a vertical vector field.
We introduce anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We survey main results of anti-invariant Riemannian submersions defined on cosymplectic manifolds. We investigate necessary and sufficient condition for an anti-invariant Riemannian submersion to be totally geodesic and harmonic. We give examples of anti-invariant submersions such that characteristic vector field ξ is vertical or horizontal. Moreover we give decomposition theorems by using the existence of anti-invariant Riemannian submersions.
Abstract. This is an expository paper, which provides a first approach to nearly Kenmotsu manifolds. The purpose of this paper is to focus on nearly Kenmotsu manifolds and get some new results from it. We prove that for a nearly Kenmotsu manifold is locally isometric to warped product of real line and nearly Kähler manifold. Finally, we prove that there exist no nearly Kenmotsu hypersurface M 2n+1 of nearly Kähler manifold N 2n+2 . It is shown that a normal nearly Kenmotsu manifold is Kenmotsu manifold.
Abstract. In this paper, we introduce slant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We obtain some results on slant Riemannian submersions of a cosymplectic manifold. We also give examples and inequalities between the scalar curvature and squared mean curvature of fibres of such slant submersions in the cases where the characteristic vector field is vertical or horizontal.
The purpose of the paper is to study Yamabe solitons on three-dimensional para-Sasakian, paracosymplectic and para-Kenmotsu manifolds. Mainly, we proved that •If the semi-Riemannian metric of a three-dimensional para-Sasakian manifold is a Yamabe soliton, then it is of constant scalar curvature, and the flow vector field V is Killing. In the next step, we proved that either manifold has constant curvature −1 and reduces to an Einstein manifold, or V is an infinitesimal automorphism of the paracontact metric structure on the manifold.•If the semi-Riemannian metric of a three-dimensional paracosymplectic manifold is a Yamabe soliton, then it has constant scalar curvature. Furthermore either manifold is η-Einstein, or Ricci flat.• If the semi-Riemannian metric on a three-dimensional para-Kenmotsu manifold is a Yamabe soliton, then the manifold is of constant sectional curvature −1, reduces to an Einstein manifold. Furthermore, Yamabe soliton is expanding with λ = −6 and the vector field V is Killing.Finally, we construct examples to illustrate the results obtained in previous sections.2010 Mathematics Subject Classification. 53C25, 53C21, 53C44, 53D15.
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