On 𝐾-contact manifolds with minimal number of closed characteristics

Rukimbira, Philippe

doi:10.1090/s0002-9939-99-05217-x

Cited by 12 publications

(9 citation statements)

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“…We have to show that the Hessian of µ A is non-degenerate in the normal directions. The computation is structurally similar to the K-contact case, see [40].…”

Section: The Canonical Torus Action On An Irregular K-cosymplectic Ma...mentioning

confidence: 83%

“…In this section we derive a relation between the topology of the union C of closed Reeb orbits on a compact K-cosymplectic manifold M and the (basic) cohomology of M , similar to the K-contact case treated in [19]. As in [19], the proof follows from considering the equivariant cohomology of the torus action defined by the closure of the Reeb field, using the statement that C arises as the critical locus of a generic component of the momentum map, which was proven in the K-contact case by Rukimbira [40]. The analogous statement for fixed points of Hamiltonian group actions on symplectic manifolds is well-known, see e.g.…”

Section: Closed Reeb Orbits and Basic Cohomologymentioning

confidence: 88%

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K-Cosymplectic manifolds

Bazzoni

Goertsches

2014

Ann Glob Anal Geom

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In this paper we study K-cosymplectic manifolds, i.e., smooth cosymplectic manifolds for which the Reeb field is Killing with respect to some Riemannian metric. These structures generalize coKähler structures, in the same way as K-contact structures generalize Sasakian structures. In analogy to the contact case, we distinguish between (quasi-)regular and irregular structures; in the regular case, the K-cosymplectic manifold turns out to be a flat circle bundle over an almost Kähler manifold. We investigate de Rham and basic cohomology of K-cosymplectic manifolds, as well as cosymplectic and Hamiltonian vector fields and group actions on such manifolds. The deformations of type I and II in the contact setting have natural analogues for cosymplectic manifolds; those of type I can be used to show that compact K-cosymplectic manifolds always carry quasi-regular structures. We consider Hamiltonian group actions and use the momentum map to study the equivariant cohomology of the canonical torus action on a compact K-cosymplectic manifold, resulting in relations between the basic cohomology of the characteristic foliation and the number of closed Reeb orbits on an irregular K-cosymplectic manifold.

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“…We have to show that the Hessian of µ A is non-degenerate in the normal directions. The computation is structurally similar to the K-contact case, see [40].…”

Section: The Canonical Torus Action On An Irregular K-cosymplectic Ma...mentioning

confidence: 83%

Section: Closed Reeb Orbits and Basic Cohomologymentioning

confidence: 88%

K-Cosymplectic manifolds

Bazzoni

Goertsches

2014

Ann Glob Anal Geom

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“…This proposition is well-known, see for example Section 2 of [30]. For the convenience of the reader we include a proof that the critical set is exactly C. For the nondegeneracy of the Hessian in normal directions, see Lemma 1.…”

Section: The Contact Moment Map and Equivariant Cohomologymentioning

confidence: 94%

Equivariant cohomology of K-contact manifolds

2012

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We investigate the equivariant cohomology of the natural torus action on a K-contact manifold and its relation to the topology of the Reeb flow. Using the contact moment map, we show that the equivariant cohomology of this action is Cohen-Macaulay, the natural substitute of equivariant formality for torus actions without fixed points. As a consequence, generic components of the contact moment map are perfect Morse-Bott functions for the basic cohomology of the orbit foliation F of the Reeb flow. Assuming that the closed Reeb orbits are isolated, we show that the basic cohomology of F vanishes in odd degrees, and that its dimension equals the number of closed Reeb orbits. We characterize K-contact manifolds with minimal number of closed Reeb orbits as real cohomology spheres. We also prove a GKM-type theorem for K-contact manifolds which allows to calculate the equivariant cohomology algebra under the nonisolated GKM condition.2010 Mathematics Subject Classification. 53D35, 53D20, 55N25.

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“…Rukimbira [43] has shown that on a (2n−1)-dimensional closed K-contact manifold (a metric contact manifold whose Reeb vector field is Killing), there are at least n periodic Reeb orbits. If the manifold is simply connected and there are precisely n periodic orbits, the manifold is homeomorphic to a sphere [44].…”

Section: Traps and Plugs In Symplectic Dynamicsmentioning

confidence: 99%

Lectures on controlled Reeb dynamics

Geiges

2018

Preprint

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These are notes based on a mini-course at the conference RIEMain in Contact, held in Cagliari, Sardinia, in June 2018. The main theme is the connection between Reeb dynamics and topology. Topics discussed include traps for Reeb flows, plugs for Hamiltonian flows, the Weinstein conjecture, Reeb flows with finite numbers of periodic orbits, and global surfaces of section for Reeb flows. The emphasis is on methods of construction, e.g. contact cuts and lifting group actions in Boothby-Wang bundles, that might be useful for other applications in contact topology.

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On 𝐾-contact manifolds with minimal number of closed characteristics

Abstract: Abstract. We prove that closed simply connected K-contact manifolds with minimal number of closed characteristics are homeomorphic to odd-dimensional spheres.

Cited by 12 publications

References 11 publications

K-Cosymplectic manifolds

K-Cosymplectic manifolds

Equivariant cohomology of K-contact manifolds

Lectures on controlled Reeb dynamics

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