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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 article, the equivalence between embedded contact homology and Seiberg-Witten Floer homology was used to obtain the following improvements on the Weinstein conjecture: if Y is a closed oriented connected 3-manifold with a stable Hamiltonian structure, then R denotes the associated Reeb vector field on Y.
Abstract: We use the equivalence between embedded contact homology and Seiberg–Witten Floer homology to obtain the following improvements on the Weinstein conjecture. Let Y be a closed oriented connected 3–manifold with a stable Hamiltonian structure, and let R denote the associated Reeb vector field on Y . We prove that if Y is not a T 2 –bundle over S , then R has a closed orbit. Along the way we prove that if Y is a closed oriented connected 3–manifold with a contact form such that all Reeb orbits are nondegenerate and elliptic, then Y is a lens space. Related arguments show that if Y is a closed oriented 3–manifold with a contact form such that all Reeb orbits are nondegenerate, and if Y is not a lens space, then there exist at least three distinct embedded Reeb orbits.
85 citations
TL;DR: In this article, the authors give a characterization of real hyper-surface of type B, that is, a tube over a totally geodesic QP n in complex two-plane Grassmannians G2(C m+2 ), where m = 2n, with the Reeb vec- tor belonging to the distribution D, where D denotes a subdistribution in the tangent space such that TxM = D'D? for any point x 2 M and D? = Span{»1,»2,»3 }.
Abstract: In this paper we give a new characterization of real hyper- surfaces of type B, that is, a tube over a totally geodesicQP n in complex two-plane Grassmannians G2(C m+2 ), where m = 2n, with the Reeb vec- tor » belonging to the distribution D, where D denotes a subdistribution in the tangent space TxM such that TxM = D'D? for any point x 2 M and D? = Span{»1,»2,»3 }.
72 citations
TL;DR: In this paper, 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 $\tilde\mu$) is presented.
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.
68 citations
TL;DR: In this article, it was shown that every compact toric contact manifold can be obtained by a contact reduction from an odd dimensional sphere, and that the convexity theorem holds also for compact contact manifolds with an effective action of a torus whose Reeb vector field corresponds to an element of the Lie algebra of the torus.
Abstract: After observing that the well-known convexity theorems of symplectic geometry also hold for compact contact manifolds with an effective action of a torus whose Reeb vector field corresponds to an element of the Lie algebra of the torus, we use this fact together with a recent symplectic orbifold version of Delzant's theorem due to Lerman and Tolman [LT] to show that every such compact toric contact manifold can be obtained by a contact reduction from an odd dimensional sphere.
64 citations
TL;DR: In this paper, 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 ˜ ).
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.
60 citations