On integrability of certain rank 2 sub-Riemannian structures

Kruglikov, Boris; Vollmer, Andreas; Lukes-Gerakopoulos, Georgios

doi:10.1134/s1560354717050033

Cited by 5 publications

(7 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…we have s max (h(0)) = ∞ and therefore we have cuspless trajectory of infinite SR length, recall (25). This fact is also clear from phase portrait of dynamic of vertical part (52), see Fig.…”

Section: Proof Of Theoremmentioning

confidence: 71%

“…In (39), the horizontal part follows from (36) and (31). It is known that the Hamiltonian system (39) is Liouville integrable [27,25], and it was explicitly integrated in [19,22]. In the next subsections, we classify the possible solutions by values of the parameter ξ, and we adapt the explicit solution to our coordinate chart (x, y, θ) ∈ M, where we follow the analogy to the closely related problem in SE (2).…”

Section: Using the Standard Relation Between Poisson And Lie Brackets {Hmentioning

confidence: 99%

“…We are interested in the first cusp time t cusp = min n∈N {t n cusp > 0}, and we call s max the corresponding value of spherical arclength, s.t. t cusp = t(s max ) via (25).…”

Section: Definitionmentioning

confidence: 99%

“…Theorem 5 When moving along a SR geodesic t → γ(t) the first cusp time is computed as t cusp (h(0)) = t(s max (h(0)), recall (25), where…”

Section: Computation Of the First Cusp Timementioning

confidence: 99%

See 3 more Smart Citations

Tracking of Lines in Spherical Images via Sub-Riemannian Geodesics in $${\text {SO(3)}}$$ SO(3)

et al. 2017

View full text Add to dashboard Cite

In order to detect salient lines in spherical images, we consider the problem of minimizing the functional l 0 C(γ (s)) ξ 2 + k 2 g (s) ds for a curve γ on a sphere with fixed boundary points and directions. The total length l is free, s denotes the spherical arclength, and k g denotes the geodesic curvature of γ . Here the smooth external cost C ≥ δ > 0 is obtained from spherical data. We lift this problem to the sub-Riemannian (SR) problem in Lie group SO(3) and show that the spherical projection of certain SR geodesics provides a solution to our curve optimization problem. In fact, this holds only for the geodesics whose spherical projection does not exhibit a cusp. The problem is a spherical extension of a well-known contour perception model, where we extend the model by Boscain and Rossi to the general case ξ > 0, C = 1. For C = 1, we derive SR geodesics and evaluate the first cusp time. We show that these curves have a simpler expression when they are parameterized by spherical arclength rather than by sub-Riemannian arclength. For case C = 1 (data-driven SR geodesics), we solve via a SR Fast Marching method. Finally, we show an experiment of vessel tracking in a spherical image of the retina and study the effect of including the spherical geometry in analysis of vessels curvature.

show abstract

Section: Proof Of Theoremmentioning

confidence: 71%

Section: Using the Standard Relation Between Poisson And Lie Brackets {Hmentioning

confidence: 99%

“…We are interested in the first cusp time t cusp = min n∈N {t n cusp > 0}, and we call s max the corresponding value of spherical arclength, s.t. t cusp = t(s max ) via (25).…”

Section: Definitionmentioning

confidence: 99%

“…Theorem 5 When moving along a SR geodesic t → γ(t) the first cusp time is computed as t cusp (h(0)) = t(s max (h(0)), recall (25), where…”

Section: Computation Of the First Cusp Timementioning

confidence: 99%

See 2 more Smart Citations

Tracking of Lines in Spherical Images via Sub-Riemannian Geodesics in $${\text {SO(3)}}$$ SO(3)

et al. 2017

View full text Add to dashboard Cite

show abstract

“…[20]) are less important. Notice that if coadjoint orbits of G have dimension ě 4, it may well happen that G admits no integrable left-invariant geodesic flows at all [5,16,17,18].…”

Section: Introductionmentioning

confidence: 99%

A Note about Integrable Systems on Low-dimensional Lie Groups and Lie Algebras

Bolsinov

Bao

2019

Regul. Chaot. Dyn.

View full text Add to dashboard Cite

show abstract

Killing tensors in Koutras–Mcintosh spacetimes

Kruglikov¹,

Steneker²

2022

Class. Quantum Grav.

View full text Add to dashboard Cite

The Koutras-McIntosh family of metrics include conformally flat pp-waves and the Wils metric. It appeared in a paper of 1996 by Koutras-McIntosh as an example of a pure radiation spacetime without scalar curvature invariants or infinitesimal symmetries. Here we demonstrate that these metrics have no "hidden symmetries", by which we mean Killing tensors of low degrees. For the particular case of Wils metrics we show the nonexistence of Killing tensors up to degree 6. The technique we use is the geometric theory of overdetermined PDEs and the Cartan prolongation-projection method. Application of those allows to prove the nonexistence of polynomial in momenta integrals for the equation of geodesics in a mathematical rigorous way. Using the same technique we can completely classify all lower degree Killing tensors and, in particular, prove that for generic pp-waves all Killing tensors of degree 3 and 4 are reducible.

show abstract

On integrability of certain rank 2 sub-Riemannian structures

Cited by 5 publications

References 18 publications

Tracking of Lines in Spherical Images via Sub-Riemannian Geodesics in $${\text {SO(3)}}$$ SO(3)

Tracking of Lines in Spherical Images via Sub-Riemannian Geodesics in $${\text {SO(3)}}$$ SO(3)

A Note about Integrable Systems on Low-dimensional Lie Groups and Lie Algebras

Killing tensors in Koutras–Mcintosh spacetimes

Contact Info

Product

Resources

About