Calibrated geodesic foliations of hyperbolic space

Godoy, Yamile; Salvai, Marcos

doi:10.1090/proc/12834

Cited by 5 publications

(6 citation statements)

References 17 publications

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“…On the other hand, if N is the unit normal vector field on M pointing outwards, then, after the identification with Jacobi fields, N ([γ u ]) = I ∈ J γu satisfying I(0) = −n(p) and I (0) = 0. Indeed, by (11),…”

Section: Lemma 14mentioning

confidence: 98%

“…We also include the expressions of the associated fundamental forms (see [12,17,18,6,1,9] (Theorem 1 in [18] is valid also for κ = 0, 1). The geometry of these spaces of geodesics has been studied for instance in [10,11,19]. See for instance in [8] or [14] the proof that C is diffeomorphic to S 2 × S 2 .…”

Section: Kähler Structures On the Manifolds Of Oriented Geodesicsmentioning

confidence: 99%

“…The group G κ acts smoothly and transitively on G κ as follows: g The Killing form of Lie (G κ ) provides G κ with a bi-invariant metric and thus there exists a unique pseudo-Riemannian metric gK on G κ /H κ such that the canonical projection π : G κ → G κ /H κ is a pseudo-Riemannian submersion. The diffeomorphism φ turns out to be an isometry onto G κ endowed with a constant multiple of the metric g K defined in (11).…”

Section: Lemma 14mentioning

confidence: 99%

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Outer billiards on the manifolds of oriented geodesics of the three dimensional space forms

Godoy¹,

Harrison²,

Salvai³

2021

Preprint

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For κ = 0, 1, −1, let M κ be the three dimensional space form of curvature κ, that is, R 3 , S 3 and hyperbolic 3-space H 3 . Let G κ be the manifold of all oriented (unparametrized) complete geodesics of M κ , i.e., G 0 and G −1 consist of oriented lines and G 1 of oriented great circles.Given a strictly convex surface S of M κ , we define an outer billiard map B κ on G κ . The billiard table is the set of all oriented geodesics not intersecting S, whose boundary can be naturally identified with the unit tangent bundle of S. We show that B κ is a diffeomorphism under the stronger condition that S is quadratically convex.We prove that B 1 and B −1 arise in the same manner as Tabachnikov's original construction of the higher dimensional outer billiard on standard symplectic space R 2n , ω . For that, of the two canonical Kähler structures that each of the manifolds G 1 and G −1 admits, we consider the one induced by the Killing form of Iso (M κ ). We prove that B 1 and B −1 are symplectomorphisms with respect to the corresponding fundamental symplectic forms. Also, we discuss a notion of holonomy for periodic points of B −1 .

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Section: Lemma 14mentioning

confidence: 98%

Section: Kähler Structures On the Manifolds Of Oriented Geodesicsmentioning

confidence: 99%

Section: Lemma 14mentioning

confidence: 99%

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Outer billiards on the manifolds of oriented geodesics of the three dimensional space forms

Godoy¹,

Harrison²,

Salvai³

2021

Preprint

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“…One can see E as the set of curves in the plane congruent to the circle via the group G. Let K be a Lie group acting on a manifold N. Canonical K-invariant pseudo Riemannian metrics on spaces of K-congruent curves in N have proved to be useful in the study of foliations of N by such curves (see for instance [9,3,4]).…”

Section: The Canonical Lorentz Metric On the Space Of Centered Ellipsesmentioning

confidence: 99%

Centro-affine invariants and the canonical Lorentz metric on the space of centered ellipses

Salvai¹

2017

Kodai Math. J.

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We consider smooth plane curves which are convex with respect to the origin. We describe centro-affine invariants (that is, GL + (2, R)-invariants), such as centro-affine curvature and arc length, in terms of the canonical Lorentz structure on the three dimensional space of all the ellipses centered at zero, by means of null curves of osculating ellipses. This is the centro-affine analogue of the approach to conformal invariants of curves in the sphere introduced by Langevin and O'Hara, using the canonical pseudo Riemannian metric on the space of circles.

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“…Warren applied them to the volume maximization problem of the special Lagrangian submanifolds of a pseudo-Euclidean space (see also [10]). For curved manifolds, they were also useful for instance in relation to optimal transport [14] or foliations [5].…”

Section: Introductionmentioning

confidence: 99%

A split special Lagrangian calibration associated with helicity

Salvai¹

2022

Preprint

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Let M be an oriented three dimensional Riemannian manifold. We define a notion of helicity of local sections of the bundle SO (M ) → M of all its positively oriented orthonormal tangent frames. When M is a space form, we relate the concept to a suitable invariant split pseudo-Riemannian metric on Iso o (M ) ∼ = SO (M ): A local section has positive helicity if and only if it determines a spacelike submanifold. In the Euclidean case we find explicit homologically volume maximizing sections using a split special Lagrangian calibration. We introduce the concept of optimal helicity and give an optimal screwed global section for the three-sphere. We prove that it is also homologically volume maximizing (now using a common one-point split calibration). Besides, we show that no optimal section can exist in the Euclidean and hyperbolic cases.

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Calibrated geodesic foliations of hyperbolic space

Cited by 5 publications

References 17 publications

Outer billiards on the manifolds of oriented geodesics of the three dimensional space forms

Outer billiards on the manifolds of oriented geodesics of the three dimensional space forms

Centro-affine invariants and the canonical Lorentz metric on the space of centered ellipses

A split special Lagrangian calibration associated with helicity

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