Topology and closed characteristics of {$K$}-contact manifolds

Rukimbira, Philippe

doi:10.36045/bbms/1103408725

Cited by 16 publications

(18 citation statements)

References 15 publications

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“…Because the powers of the basic Euler class dα are nonzero elements of H • (M, F ), part (3) of the above theorem gives a new proof of the following result by Rukimbira (Corollary 1 of [29]).…”

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confidence: 90%

“…If the closed Reeb orbits of α are isolated, then we have If not further specified, cohomology is taken with real coefficients. Part (1) of this theorem was originally proven by Rukimbira (Theorem 2 of [29], see also Theorem 7.4.8 of [8]). We remark that contact toric manifolds of Reeb type are examples of K-contact manifolds with isolated closed Reeb orbits (see Proposition 7.3).…”

Section: Introductionmentioning

confidence: 96%

“…In Theorem 2 of[29], Rukimbira showed that H 1 (M ) = 0 if the closed Reeb orbits are isolated and M is a Sasakian manifold. As stated in Theorem 7.4.8 of[8], his proof can be extended to show H 1 (M ) = 0 under the assumption of Theorem 7.4 in the K-contact case.Remark 7.6.…”

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confidence: 99%

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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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confidence: 90%

Section: Introductionmentioning

confidence: 96%

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Equivariant cohomology of K-contact manifolds

2012

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“…For s = 1, i.e., in the K-contact setting, the following theorem was known previously -the statement about the minimal number of closed leaves generalizes [24,Corollary 1], and the equivalence of the four conditions results from [16]. Theorem 6.4.…”

Section: Closed Leaves Of the Characteristic Foliationmentioning

confidence: 93%

On the topology of metric f–K-contact manifolds

Goertsches

Loiudice

2020

Monatsh Math

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We observe that the class of metric f -K-contact manifolds, which naturally contains that of K-contact manifolds, is closed under forming mapping tori of automorphisms of the structure. We show that the de Rham cohomology of compact metric f -K-contact manifolds naturally splits off an exterior algebra, and relate the closed leaves of the characteristic foliation to its basic cohomology.2010 Mathematics Subject Classification. Primary 53C25, 53C15, Secondary 53D10, 32V05.

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“…At the Cagliari conference I learned from David Blair about earlier results in metric contact geometry concerning the minimal number of periodic Reeb orbits, see [4,Section 3.4]. 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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Topology and closed characteristics of {$K$}-contact manifolds

Cited by 16 publications

References 15 publications

Equivariant cohomology of K-contact manifolds

Equivariant cohomology of K-contact manifolds

On the topology of metric f–K-contact manifolds

Lectures on controlled Reeb dynamics

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