Showing papers on "Reeb vector field published in 2012"

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 \eqref{paranullity} below, for some real numbers $% \tilde\kappa$ and $\tilde\mu$). This class of pseudo-Riemannian manifolds, which includes para-Sasakian manifolds, was recently defined in \cite{MOTE}. 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 $% \mathcal{D}$-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.

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 κ ˜ 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 ( κ , μ ) -spaces. In particular, we prove that, under some natural assumption, any such paracontact metric manifold admits a compatible contact metric ( κ , 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.

Classification of homogeneous almost cosymplectic three-manifolds ☆

Abstract: The purpose of this paper is to classify all simply connected homogeneous almost cosymplectic three-manifolds. We show that each such three-manifold is either a Lie group G equipped with a left invariant almost cosymplectic structure or a Riemannian product of type R × N , where N is a Kahler surface of constant curvature. Moreover, we find that the Reeb vector field of any homogeneous almost cosymplectic three-manifold, except one case, defines a harmonic map.

Slant curves in 3-dimensional f-Kenmotsu manifolds

Abstract: The aim of this paper is to study slant curves of three-dimensional f-Kenmotsu manifolds. These curves are characterized through the scalar product between the normal at the curve and the Reeb vector field. The classification of slant curves in the hyperbolic 3-dimensional space is provided as well as some remarkable cases. Slant curves with proper mean curvature vector field are characterized.

The harmonicity of the Reeb vector field with respect to Riemannian g-natural metrics

Abstract: In this paper we show that a 3-dimensional non-Sasakian contact metric manifold [ M , ( η , ξ , ϕ , g ) ] is a ( κ , μ , ν ) -contact metric manifold with ν = const. , if and only if there exists a Riemannian g-natural metric G ˜ on T 1 M for which ξ : ( M , g ) ↦ ( T 1 M , G ˜ ) is a harmonic map. Furthermore, we give examples of 3-dimensional non-Sasakian contact metric manifolds [ M , ( η , ξ , ϕ , g ) ] such that the corresponding Reeb vector fields ξ : ( M , g ) ↦ ( T 1 M , G ˜ ) are harmonic maps, for suitable Riemannian g-natural metrics G ˜ on T 1 M which are not of Kaluza–Klein type. Finally, we prove that if ( M , g ) is an Einstein manifold and ( η ˜ , ξ ˜ , ϕ ˜ , G ˜ ) a g-natural contact metric structure on T 1 M , then the contact metric manifold [ T 1 M , ( η ˜ , ξ ˜ , ϕ ˜ , G ˜ ) ] is H-contact if and only if ( M , g ) is 2-stein.

On growth rate and contact homology

Abstract: It is a conjecture of Colin and Honda that the number of Reeb periodic orbits of universally tight contact structures on hyperbolic manifolds grows exponentially with the period, and they speculate further that the growth rate of contact homology is polynomial on non-hyperbolic geometries. Along the line of the conjecture, for manifolds with a hyperbolic component that fibers on the circle, we prove that there are infinitely many non-isomorphic contact structures for which the number of Reeb periodic orbits of any non-degenerate Reeb vector field grows exponentially. Our result hinges on the exponential growth of contact homology which we derive as well. We also compute contact homology in some non-hyperbolic cases that exhibit polynomial growth, namely those of universally tight contact structures non-transverse to the fibers on a circle bundle.

Entropy Form and the Contact Geometry of the Material Point Model

Abstract: In this work we investigate a material point model (MP-model) and exploit the geometrical meaning of the "entropy form" introduced by Coleman and Owen. We show that a modification of the thermodynamical phase space (studied and exploited in numerous works) is an appropriate setting for the development of the MP-model in different physical situations. This approach allows to formulate the MP-model and the corresponding entropy form in terms similar to those of homogeneous thermodynamics. Closeness condition of the entropy form is reformulated as the requirement that admissible processes curves belong to the (extended) constitutive surfaces foliating the extended thermodynamical phase space of the model over the space X of basic variables. Extended constitutive surfaces ΣS,κ are described as the Legendre submanifolds ΣS of the space shifted by the flow of Reeb vector field. This shift is controlled by the entropy production function κ. We determine which contact Hamiltonian dynamical systems ξK are tangent to the surfaces ΣS,κ, introduce conformally Hamiltonian systems μξK where conformal factor μ characterizes the increase of entropy along the trajectories. These considerations are then illustrated by applying them to the Coleman–Owen model of thermoelastic point.

Frankel Conjecture and Sasaki geometry

Abstract: We classify simply connected compact Sasaki manifolds of dimension $2n+1$ with positive transverse bisectional curvature In particular, the K\"ahler cone corresponding to such manifolds must be bi-holomorphic to $\C^{n+1}\backslash \{0\}$ As an application we recover the Mori-Siu-Yau theorem on the Frankel conjecture and extend it to certain orbifold version The main idea is to deform such Sasaki manifolds to the standard round sphere in two steps, both fixing the complex structure on the K\"ahler cone First, we deform the metric along the Sasaki-Ricci flow and obtain a limit Sasaki-Ricci soliton with positive transverse bisectional curvature Then by varying the Reeb vector field along the negative gradient of the volume functional, we deform the Sasaki-Ricci soliton to a Sasaki-Einstein metric with positive transverse bisectional curvature, ie a round sphere The second deformation is only possible when one treats simultaneously regular and irregular Sasaki manifolds, even if the manifold one starts with is regular(quasi-regular), ie K\"ahler manifolds(orbifolds)

Reeb periodic orbits after a bypass attachment

Abstract: On a 3-dimensional contact manifold with boundary, a bypass attachment is an elementary change of the contact structure consisting in the attachment of a thickened half-disc with a prescribed contact structure along an arc on the boundary. We give a model bypass attachment in which we describe the periodic orbits of the Reeb vector field created by the bypass attachment in terms of Reeb chords of the attachment arc. As an application, we compute the contact homology of a product neighbourhood of a convex surface after a bypass attachment, and the contact homology of some contact structures on solid tori.

Semi-parallelism of normal Jacobi operator for Hopf hypersurfaces in complex two-plane Grassmannians

Abstract: It is proved the non-existence of Hopf hypersurfaces in $G_{2}({\Bbb C}^{m+2})$, $m \geq 3$, whose normal Jacobi operator is semi-parallel, if the principal curvature of the Reeb vector field is non-vanishing and the component of the Reeb vector field in the maximal quaternionic subbundle ${\frak D}$ or its orthogonal complement ${\frak D}^{\bot}$ is invariant by the shape operator.