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.

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.

Variations at infinity in contact form geometry

Abstract: Let v be a nonsingular Morse–Smale vector field in the kernel of a contact form α, with Reeb vector field \(\xi\), defined on M3. We establish that the associated variational problem at infinity defined by the action functional on the stratified space \(\bigcup \Gamma_{2k}\) of curves made of \(\xi\)-pieces of orbits alternating with \(\pm v\)-pieces of orbits satisfies the Palais–Smale condition. This result takes a more special form for the standard contact structure of S3.

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.

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.

Killing Vector Fields and Twistor Forms on Generalized Sasakian Space Forms

Abstract: We study generalized Sasakian space form M(f1, f2, f3) when (i) the Reeb vector field of the almost contact metric structure is Killing, (ii) the Ricci tensor satisfies Einstein-like conditions and (iii) the fundamental 2-form of the almost contact metric structure is a twistor form.