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Reeb vector field
About: Reeb vector field is a research topic. Over the lifetime, 254 publications have been published within this topic receiving 4118 citations.
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TL;DR: In this paper, the authors investigated the full and partial integrability condition of the entropy form for the model of thermoelastic point and for the deformable ferroelectric crystal media point.
Abstract: In this work we investigate the material point model and exploit the geometrical meaning of the "entropy form" introduced by B.Coleman and R.Owen. We analyze full and partial integrability (closeness) condition of the entropy form for the model of thermoelastic point and for the the deformable ferroelectric crystal media point. We show that the thermodynamical phase space (TPS) introduced by R.Hermann and widely exploited by R. Mrugala with his collaborators and other researchers, extended possibly by time, with its canonical contact structure is an appropriate setting for the development of material point models in different physical situations. This allows us to formulate the model of a material point and the corresponding entropy form in terms similar to those of the homogeneous thermodynamics. Closeness condition of the entropy form is reformulated as the requirement that the admissible processes curves belongs to the constitutive surface S of the model.
Our principal result is the description of the constitutive surfaces S of the material point model as the Legendre submanifolds of the TPS shifted by the flow of Reeb vector field. This shift is controlled, at the points of Legendre submanifold by the entropy production function $\sigma$.
TL;DR: In this article, a volume-preserving fieldX on a 3-manifold is the one that satisfies LXΩ ≡ 0 for some volume Ω, while the Reeb vector field of a contact form is of volume preserving, but not conversely.
Abstract: Volume–preserving fieldX on a 3–manifold is the one that satisfies LXΩ ≡ 0 for some volume Ω. The Reeb vector field of a contact form is of volume–preserving, but not conversely. On the basis of Geiges–Gonzalo’s parallelization results, we obtain a volume–preserving sphere, which is a triple of everywhere linearly independent vector fields such that all their linear combinations with constant coefficients are volume–preserving fields. From many aspects, we discuss the distinction between volume–preserving fields and Reeb–like fields. We establish a duality between volume–preserving fields and h–closed 2–forms to understand such distinction. We also give two kinds of non–Reeb–like but volume–preserving vector fields to display such distinction.
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TL;DR: In this paper, real hypersurfaces in complex Grassmannians of rank two were studied and the nonexistence of mixed foliate real hypersursus is proven. And the Reeb principal curvature is constant along integral curves of Reeb vector field.
Abstract: In this paper, we study real hypersurfaces in complex Grassmannians of rank two. First, the nonexistence of mixed foliate real hypersurfaces is proven. With this result, we show that for Hopf hypersurfaces in complex Grassmannians of rank two, the Reeb principal curvature is constant along integral curves of the Reeb vector field. As a result the classification of contact real hypersurfaces is obtained. We also introduce the notion of $q$-umbilical real hypersurfaces in complex Grassmannians of rank two and obtain a classification of such real hypersurfaces.
TL;DR: In this article, it was shown that if the contact form is degenerate, the convergence of a finite-energy pseudoholomorphic curve will be asymptotic at each of its non removable punctures to a single periodic orbit of the Reeb vector field.
Abstract: It's known from from work of Hofer, Wysocki, and Zehnder [1996] and Bourgeois [2002] that in a contact manifold equipped with either a nondegenerate or Morse-Bott contact form, a finite-energy pseudoholomorphic curve will be asymptotic at each of its non removable punctures to a single periodic orbit of the Reeb vector field and that the convergence is exponential. We provide examples here to show that this need not be the case if the contact form is degenerate. More specifically, we show that on any contact manifold $(M, \xi)$ with cooriented contact structure one can choose a contact form $\lambda$ with $\ker\lambda=\xi$ and a compatible complex structure $J$ on $\xi$ so that for the associated $\mathbb{R}$-invariant almost complex structure $\tilde J$ on $\mathbb{R}\times M$ there exist families of embedded finite-energy $\tilde J$-holomorphic cylinders and planes having embedded tori as limit sets.
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TL;DR: In this article, the transverse Kahler holonomy groups on Sasaki manifolds were studied and their stability properties under transverse holomorphic deformations of the characteristic foliation by the Reeb vector field were shown.
Abstract: We study the transverse Kahler holonomy groups on Sasaki manifolds $(M,{\oldmathcal S})$ and their stability properties under transverse holomorphic deformations of the characteristic foliation by the Reeb vector field. In particular, we prove that when the first Betti number $b_1(M)$ and the basic Hodge number $h^{0,2}_B({\oldmathcal S})$ vanish, then ${\oldmathcal S}$ is stable under deformations of the transverse Kahler flow. In addition we show that an irreducible transverse hyperkahler Sasakian structure is ${\oldmathcal S}$-unstable, whereas, an irreducible transverse Calabi-Yau Sasakian structure is ${\oldmathcal S}$-stable when $\dim M\geq 7$. Finally, we prove that the standard Sasaki join operation (transverse holonomy $U(n_1)\times U(n_2)$) as well as the fiber join operation preserve ${\oldmathcal S}$-stability.