Contact Metric Geometry of the Unit Tangent Sphere Bundle

Calvaruso, Giovanni

doi:10.1007/0-8176-4424-5_4

Cited by 12 publications

(7 citation statements)

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“…Geometries determined by the three metrics above are very much similar to one another, and they often showed a quite "rigid" behaviour, in the sense that many curvature properties on T 1 M , equipped with one of these metrics, imply strong restrictions on the base manifold itself. Surveys on the geometry of (T 1 M, g S ) and (T 1 M, η,ḡ) can be found in [6] and [7], respectively. The first author and M. Sarih [5] investigated geometric properties of "g-natural" metrics on the tangent bundle T M .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

$g$-natural metrics of constant curvature on unit tangent sphere bundles

Abbassi¹,

Calvaruso²

2012

Arch. Math. (Brno)

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Section: Introduction and Main Resultsmentioning

confidence: 99%

$g$-natural metrics of constant curvature on unit tangent sphere bundles

Abbassi¹,

Calvaruso²

2012

Arch. Math. (Brno)

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“…Let g round be the metric with constant curvature 1 on S n−1 and g euc be the Euclidean metric on D n (ρ). The metric g = g euc ⊕ g round on D n (ρ) × S n−1 is compatible with the contact form α euc ; see [Cal05].…”

Section: 2mentioning

confidence: 99%

Dynamically exotic contact spheres in dimensions $\geq 7$

Alves

Matthias

2019

Comment. Math. Helv.

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We exhibit the first examples of contact structures on S 2n−1 with n ≥ 4 and on S 3 × S 2 , all equipped with their standard smooth structures, for which every Reeb flow has positive topological entropy. As a new technical tool for the study of the volume growth of Reeb flows we introduce the notion of algebraic growth of wrapped Floer homology. Its power stems from its stability under several geometric operations on Liouville domains. 2010 Mathematics Subject Classification. Primary 37J05, 53D40. Key words and phrases. Topological entropy, contact structure, Reeb dynamics, Floer homology. Marcelo R.R. Alves supported by the Swiss National Foundation. Matthias Meiwes supported by German-Israeli Foundation (GIF).. 1.2. Main results. The main result of this paper is the existence of contact structures with positive entropy on high dimensional manifolds. Theorem 1.1.A) Let S 2n−1 be the (2n − 1) -dimensional sphere with its standard smooth structure. For n ≥ 4 there exists a contact structure on S 2n−1 with positive entropy.B) There exists a contact structure on S 3 × S 2 with positive entropy.Recall that a contact manifold is said to be exactly fillable if it is the boundary of a Liouville domain. From Theorem 1.1 and the methods developed in this paper we obtain the following more general result.Theorem 1.2.♣ If V is a manifold of dimension 2n − 1 ≥ 7 that admits an exactly fillable contact structure, then V admits a contact structure with positive entropy.♦ If V is a 5-manifold that admits an exactly fillable contact structure, then the connected sum V #(S 3 × S 2 ) admits a contact structure with positive entropy.Note that the standard contact structure on spheres as well as the canonical contact structure on S * S 3 ∼ = S 3 × S 2 have a contact form with periodic Reeb flow. In particular these are not

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“…where we have the four 1-dimensional invariants from (14) and three irreducible orthogonal subspaces W i defined by…”

Section: The 3d Differential Systemmentioning

confidence: 99%

Natural SU(2)-structures on tangent sphere bundles

Albuquerque

2016

Preprint

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We define and study natural SU(2)-structures, in the sense of Conti-Salamon, on the total space S of the tangent sphere bundle of any given oriented Riemannian 3-manifold M . We recur to a fundamental exterior differential system of Riemannian geometry. Essentially, two types of structures arise: the contact-hypo and the non-contact and, for each, we study the conditions for being hypo, nearly-hypo or double-hypo. We discover new double-hypo structures on S 3 × S 2 , of which the well-known Sasaki-Einstein are a particular case.Hyperbolic geometry examples also appear. In the search of the associated metrics, we find a Theorem for determining the metric, which applies to all SU(2)-structures in general. Within our application to tangent sphere bundles, we discover a whole new class of metrics specific to 3d-geometry. The evolution equations of Conti-Salamon are considered; leading to a new integrable SU(3)-structure on S × R + associated to any flat M .

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Contact Metric Geometry of the Unit Tangent Sphere Bundle

Cited by 12 publications

References 32 publications

$g$-natural metrics of constant curvature on unit tangent sphere bundles

$g$-natural metrics of constant curvature on unit tangent sphere bundles

Dynamically exotic contact spheres in dimensions $\geq 7$

Natural SU(2)-structures on tangent sphere bundles

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