Showing papers on "Reeb vector field published in 2011"

Notes on contact Ricci solitons

Abstract: A compact contact Ricci soliton (whose potential vector field is the Reeb vector field) is Sasaki–Einstein. A compact contact homogeneous manifold with a Ricci soliton is Sasaki–Einstein.

Closed trajectories on symmetric convex Hamiltonian energy surfaces

Abstract: In this article, let $\Sigma\subset\mathbf{R}^{2n}$ be a compact convex Hamiltonian energy surface which is symmetric with respect to the origin, where $n\ge 2$. We prove that there exist at least two geometrically distinct symmetric closed trajectories of the Reeb vector field on $\Sigma$.

Contact Homology of Orbit Complements and Implied Existence

Abstract: For Reeb vector fields on closed 3-manifolds, cylindrical contact homology is used to show that the existence of a set of closed Reeb orbit with certain knotting/linking properties implies the existence of other Reeb orbits with other knotting/linking properties relative to the original set. We work out a few examples on the 3-sphere to illustrate the theory, and describe an application to closed geodesics on $S^2$ (a version of a result of Angenent in [1]).

The weinstein conjecture in the presence of submanifolds having a legendrian foliation

Abstract: Helmut Hofer introduced in 1993 a novel technique based on holomorphic curves to prove the Weinstein conjecture. Among the cases where these methods apply are all contact 3-manifolds (M, ξ) with π2(M) ≠ 0. We modify Hofer's argument to prove the Weinstein conjecture for some examples of higher-dimensional contact manifolds. In particular, we are able to show that the connected sum with a real projective space always has a closed contractible Reeb orbit.

Hopf hypersurfaces in complex two-plane grassmannians with lie parallel normal jacobi operator

Abstract: In this paper we give some non-existence theorems for Hopf hypersurfaces in the complex two-plane Grassmannian G2(C) with Lie parallel normal Jacobi operator R̄N and totally geodesic D and D ⊥ components of the Reeb flow. 0. Introduction The Jacobi fields along geodesics of a given Riemannian manifold (M̄, ḡ) play an important role in the study of differential geometry. It satisfies a very well-known differential equation. This classical differential equation naturally inspires the so-called Jacobi operators. That is, if R̄ is the curvature operator of M̄ and X is any vector field tangent to M̄ , the Jacobi operator with respect to X at x ∈ M̄ , R̄X ∈ End(TxM̄), is defined as R̄X(Y )(x) = (R̄(Y,X)X)(x) for all Y ∈ TxM̄ , being a self-adjoint endomorphism of the tangent bundle TM̄ of M̄ . Clearly, each vector field X tangent to M̄ provides a Jacobi operator with respect to X (See [7] and [9]). If the structure vector field ξ = −JN of a real hypersurface M in complex projective space Pn(C) is invariant under the shape operator, ξ is said to be Hopf, where J denotes a Kähler structure of Pn(C), and N is a unit normal vector field of M in Pn(C). In the quaternionic projective space HP Pérez and Suh [10] classified the real hypersurfaces in HP with D⊥-parallel curvature tensor ∇ξνR = 0 for ν = 1, 2, 3, where R denotes the curvature tensor of M in HP and D⊥ is a distribution defined by D⊥ = Span {ξ1, ξ2, ξ3}. In this case they are congruent to a tube of radius π4 over a totally geodesic quaternionic submanifold HP k in HP, 2 ≤ k ≤ m− 2. Received September 22, 2009; Revised July 2, 2010. 2010 Mathematics Subject Classification. Primary 53C40; Secondary 53C15.

The Weinstein conjecture with multiplicities on spherizations

Abstract: Soit M une variete lisse fermee et considerons sont fibre cotangent T*M muni de la structure symplectique usuelle induite par la forme de Liouville. Une hypersurface S de T*M$ est dite etoilee fibre par fibre si pour tout point q de M, l'intersection Sq de S avec la fibre au dessus de q est le bord d'un domaine etoile par rapport a l'origine 0q de la fibre T*qM. Un flot est naturellement associe a S, il s'agit de l'unique flot genere par le champ de Reeb le long de S, le flot de Reeb. L'existence d'une orbite orbite fermee du flot de Reeb sur S fut annoncee par Weinstein dans sa conjecture en 1978. Independamment, Weinstein et Rabinowitz ont montre l'existence d'une orbite fermee sur les hypersurfaces de type etoilees dans l'espace reel de dimension 2n. Sous les hypotheses precedentes, l'existence d'une orbite fermee fut demontree par Hofer et Viterbo. Dans le cas particulier du flot geodesique, l'existence de plusieurs orbites fermees fut notamment etudiee par Gromov, Paternain et Paternain-Petean. Dans cette these, ces resultats sont generalises. Les resultats principaux de cette these montrent que la structure topologique de la variete M implique, pour toute hypersurface etoilee fibre par fibre, l'existence de beaucoup d'orbites fermees du flot de Reeb. Plus precisement, une borne inferieure de la croissance du nombre d'orbites fermees du flot de Reeb en fonction de leur periode est mise en evidence. /Let M be a smooth closed manifold and denote by T*M the cotangent bundle over M endowed with its usual symplectic structure induced by the Liouville form. A hypersurface S of T*M is said to be fiberwise starshaped if for each point q in M the intersection Sq of S with the fiber at q bounds a domain starshaped with respect to the origin 0q in T*qM. There is a flow naturally associated to S, generated by the unique Reeb vector field R along S , the Reeb flow. The existence of one closed orbit was conjectured by Weinstein in 1978 in a more general setting. Independently, Weinstein and Rabinowitz established the existence of a closed orbit on star-like hypersurfaces in the 2n-dimensional real space. In our setting the Weinstein conjecture without the assumption was proved in 1988 by Hofer and Viterbo. The existence of many closed orbits has already been well studied in the special case of the geodesic flow, for example by Gromov, Paternain and Paternain-Petean. In this thesis we will generalize their results.The main result of this thesis is to prove that the topological structure of $M$ forces, for all fiberwise starshaped hypersurfaces S, the existence of many closed orbits of the Reeb flow on S. More precisely, we shall give a lower bound of the growth rate of the number of closed Reeb-orbits in terms of their periods.

A new characterization of Finsler metrics with constant flag curvature 1

Abstract: The purpose of this article is to derive an integral inequality of Ricci curvature with respect to Reeb field in a Finsler space and give a new geometric characterization of Finsler metrics with constant flag curvature 1.

K-contact Lie groups of dimension five or greater

Abstract: We prove that a K-contact Lie group of dimension five or greater is the central extension of a symplectic Lie group by complexifying the Lie algebra and applying a result from complex contact geometry, namely, that, if the adjoint action of the complex Reeb vector field on a complex contact Lie algebra is diagonalizable, then it is trivial.