Open problems and questions about geodesics

Burns, Keith; Matveev, Vladimir S.

doi:10.1017/etds.2019.73

Cited by 27 publications

(23 citation statements)

References 161 publications

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“…Similarly, because the geodesics are diverging in the case of the hyperbolic iso-disparity family, the curvature of visual space is negative. The numerical value cannot be readily obtained in both cases as the problem of getting the metric from unparametrized geodesics is still open in Riemannian geometry [5].…”

Section: Numerical Results and A Comparison With Other Studiesmentioning

confidence: 99%

Riemannian Geometries of Visual Space: Variable Curvature and Horizon

Turski

2021

Preprint

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This is a study of the phenomenal geometries constructed in the Riemannian geometry framework from simulated iso-disparity conics in the horizontal visual plane of the binocular system with the asymmetric eyes (AEs). The iso-disparity conic's arcs in the Cyclopean direction are the frontal visual geodesics. For the eyes' resting vergence posture, which depends on parameters of the AE, the iso-disparity conics are frontal straight lines in physical space. For all other fixations, the iso-disparity conics consist of families of the ellipses or hyperbolas depending on both the AE's parameters and the bifoveal fixation. An assumption underlying the relevant architecture of the human visual system is combined with results from simulated iso-disparity straight lines, giving the relative depth as a function of the distance. This establishes the metric tensor in binocular space of fixations for the eyes' resting vergence posture. The resulting geodesics in the gaze direction, give the distance to the horizon and zero curvature. For all other fixations, only the sign of the curvature can be inferred from the global behavior of the simulated iso-disparity conics.

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Section: Numerical Results and A Comparison With Other Studiesmentioning

confidence: 99%

Riemannian Geometries of Visual Space: Variable Curvature and Horizon

Turski

2021

Preprint

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“…We also would like to draw attention to two recent papers containing open problems on integrable natural Hamiltonian systems on two-dimensional manifolds: Burns & Matveev [59] (see §10.2 there), and Butler [60] (see § §3.3-3.5 there).…”

Section: Conjecture 32 ([42]mentioning

confidence: 99%

Open problems, questions and challenges in finite- dimensional integrable systems

Bolsinov

Matveev

Miranda

et al. 2018

Phil. Trans. R. Soc. A.

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The paper surveys open problems and questions related to different aspects of integrable systems with finitely many degrees of freedom. Many of the open problems were suggested by the participants of the conference 'Finite-dimensional Integrable Systems, FDIS 2017' held at CRM, Barcelona in July 2017.This article is part of the theme issue 'Finite dimensional integrable systems: new trends and methods'.

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“…Such geodesics have been ruled out for a large class of rank 1 surfaces [Wu15], but the question of whether such examples can exist in general remains open, and is related to the (difficult) question of ergodicity of the geodesic flow. See [BM13] for an interesting discussion of these issues.…”

Section: Introductionmentioning

confidence: 99%

Fat Flats in Rank One Manifolds

Constantine¹,

Lafont²,

McReynolds³

et al. 2019

Michigan Math. J.

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We study closed nonpositively curved Riemannian manifolds M which admit 'fat k-flats': that is, the universal coverM contains a positive radius neighborhood of a k-flat on which the sectional curvatures are identically zero. We investigate how the fat k-flats affect the cardinality of the collection of closed geodesics. Our first main result is to construct rank 1 nonpositively curved manifolds with a fat 1-flat which corresponds to a twisted cylindrical neighborhood of a geodesic on M . As a result, M contains an embedded closed geodesic with a flat neighborhood, but M nevertheless has only countably many closed geodesics. Such metrics can be constructed on finite covers of arbitrary odd-dimensional finite-volume hyperbolic manifolds. Our second main result is to prove a closing theorem for fat flats, which implies that a manifold M with a fat k-flat contains an immersed, totally geodesic k-dimensional flat closed submanifold. This guarantees the existence of uncountably many closed geodesics when k ≥ 2. Finally, we collect results on thermodynamic formalism for the class of manifolds considered in this paper.

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Open problems and questions about geodesics

Cited by 27 publications

References 161 publications

Riemannian Geometries of Visual Space: Variable Curvature and Horizon

Riemannian Geometries of Visual Space: Variable Curvature and Horizon

Open problems, questions and challenges in finite- dimensional integrable systems

Fat Flats in Rank One Manifolds

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