scispace - formally typeset
Search or ask a question
Topic

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.


Papers
More filters
Journal ArticleDOI
TL;DR: In this paper, it was shown that every Reeb vector field on Riemann surfaces has a periodic orbit which is unknotted and has self-linking number equal to 1.
Abstract: It is well known that a Reeb vector field on $S^3$ has a periodic solution. Sharpening this result we shall show in this note that every Reeb vector field $X$ on $S^3$ has a periodic orbit which is unknotted and has self-linking number equal to $-1$. If the contact form $\lambda$ is non-degenerate, then there is even a periodic orbit $P$ which, in addition, has an index $\mu (P) \in \{2,3\}$, and which spans an embedded disc whose interior is transversal to $X$. The proofs are based on a theory for partial differential equations of Cauchy-Riemann type for maps from punctured Riemann surfaces into ${\mathbb R} \times S^3$, equipped with special almost complex structures related to the contact form $\lambda$ on $S^3$.

34 citations

Journal ArticleDOI
TL;DR: In this article, the authors considered a sub-Riemannian Laplacian with an oriented contact distribution and established a quantum ergodicity (QE) theorem for the eigenfunctions of any associated sR LaplACian under the assumption that the Reeb flow is ergodic.
Abstract: This is the first paper of a series in which we plan to study spectral asymptotics for sub-Riemannian Laplacians and to extend results that are classical in the Riemannian case concerning Weyl measures, quantum limits, quantum ergodicity, quasi-modes, trace formulae.Even if hypoelliptic operators have been well studied from the point of view of PDEs, global geometrical and dynamical aspects have not been the subject of much attention. As we will see, already in the simplest case, the statements of the results in the sub-Riemannian setting are quite different from those in the Riemannian one. Let us consider a sub-Riemannian (sR) metric on a closed three-dimensional manifold with an oriented contact distribution. There exists a privileged choice of the contact form, with an associated Reeb vector field and a canonical volume form that coincides with the Popp measure. We establish a Quantum Ergodicity (QE) theorem for the eigenfunctions of any associated sR Laplacian under the assumption that the Reeb flow is ergodic. The limit measure is given by the normalized Popp measure.This is the first time that such a result is established for a hypoelliptic operator, whereas the usual Shnirelman theorem yields QE for the Laplace-Beltrami operator on a closed Riemannian manifold with ergodic geodesic this http URL prove our theorem, we first establish a microlocal Weyl law, which allows us to identify the limit measure and to prove the microlocal concentration of the eigenfunctions on the characteristic manifold of the sR Laplacian. Then, we derive a Birkhoff normal form along this characteristic manifold, thus showing that, in some sense, all 3D contact structures are microlocally equivalent. The quantum version of this normal form provides a useful microlocal factorization of the sR Laplacian. Using the normal form, the factorization and the ergodicity assumption, we finally establish a variance estimate, from which QE follows.We also obtain a second result, which is valid without any ergodicity assumption: every Quantum Limit (QL) can be decomposed in a sum of two mutually singular measures: the first measure is supported on the unit cotangent bundle and is invariant under the sR geodesic flow, and the second measure is supported on the characteristic manifold of the sR Laplacian and is invariant under the lift of the Reeb flow. Moreover, we prove that the first measure is zero for most QLs.

32 citations

Posted Content
TL;DR: In this paper, the authors studied normal CR compact manifolds in dimension 3 and proved that the underlying manifolds are topologically finite quotiens of the 3-sphere or of a circle bundle over a Riemann surface of positive genus.
Abstract: We study normal CR compact manifolds in dimension 3. For a choice of a CR Reeb vector field, we associate a Sasakian metric on them, and we classify those metrics. As a consequence, the underlying manifolds are topologically finite quotiens of the 3-sphere or of a circle bundle over a Riemann surface of positive genus. In the latter case, we prove that their CR automorphisms group is a finite extension of U(1), and we classify the normal CR structures on these manifolds.

32 citations

Journal ArticleDOI
TL;DR: In this paper, the vanishing Lie derivative of the shape operator along the direction of the Reeb vector field was used to characterize real hypersurfaces of type A in a complex two-plane Grassmannian G2(C m+2 ) which are tubes over totally geodesic G 2(Cm+1 ) in G2m+2 ).
Abstract: In this paper we give a characterization of real hypersurfaces of type A in a complex two- plane Grassmannian G2(C m+2 ) which are tubes over totally geodesic G2(C m+1 ) in G2(C m+2 ) in terms of the vanishing Lie derivative of the shape operator A along the direction of the Reeb vector field �.

31 citations

Journal ArticleDOI
TL;DR: In this paper, a general scheme which associates conjugacy classes of tori in the contactomorphism group to transverse almost complex structures on a compact contact manifold is described, and for tori of Reeb type whose Lie algebra contains a Reeb vector field one can associate a Sasaki cone.
Abstract: I describe a general scheme which associates conjugacy classes of tori in the contactomorphism group to transverse almost complex structures on a compact contact manifold. Moreover, to tori of Reeb type whose Lie algebra contains a Reeb vector field one can associate a Sasaki cone. Thus, for contact structures D of K-contact type one obtains a configuration of Sasaki cones called a bouquet such that each Sasaki cone is associated with a conjugacy class of tori of Reeb type.

31 citations

Network Information
Related Topics (5)
Manifold
18.7K papers, 362.8K citations
89% related
Symplectic geometry
18.2K papers, 363K citations
85% related
Scalar curvature
12.7K papers, 296K citations
84% related
Differential geometry
10.9K papers, 305K citations
84% related
Holomorphic function
19.6K papers, 287.8K citations
84% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20221
202126
202028
201918
201813
201721