On a quasihyperbolic plane

Gribanova, Irina

doi:10.1007/s11202-999-0005-8

Cited by 13 publications

(13 citation statements)

References 4 publications

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“…Therefore the sequence converges to a C 0 -Finsler metric. But in this case it is straightforward to see thatF in (9) is infinite (just analyze small neighborhoods of the origin), what contradicts Theorem 5.3. Therefore the sequence of induced Hausdorff metrics doesn't converge to a metric on R. Example 6.3.…”

Section: Further Examplesmentioning

confidence: 93%

Sequence of Induced Hausdorff Metrics on Lie Groups

Neyra

Fukuoka

Neves

2019

Bull Braz Math Soc, New Series

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Let ϕ : G × (M, d) → (M, d) be a left action of a Lie group on a differentiable manifold endowed with a metric d, which is compatible with its topology. Let X be a compact subset of M . Then the isotropy subgroup of X is defined as H X := {g ∈ G; gX = X} and it is closed in G. The induced Hausdorff metric is a metric on the left coset manifold G/H X definedSuppose that ϕ is transitive and that there exist p ∈ M such that H X = Hp.Then gH X → gp is a diffeomorphism that identifies G/H X and M . In this work we define a discrete dynamical system of metrics on M . Let d 1 =d X , whered X stands for the intrinsic metric induced by d X . We can iterate the process on ϕ : G×(M ≡ G/H X , d 1 ) → (M ≡ G/H X , d 1 ), in order to get d 2 , d 3 and so on. We study the particular case where M = G, ϕ : G × (G, d) → (G, d) is the usual product, d is bounded above by a right invariant intrinsic metric on G and X is a finite subset of G containing the identity element. We prove that d i converges pointwise to a metric d ∞ . In addition, if d is complete and the semigroup generated by X is dense in G, then d ∞ is the distance function of a right invariant C 0 -Carnot-Carathéodory-Finsler metric. The case where d ∞ is C 0 -Finsler is studied in detail.

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Section: Further Examplesmentioning

confidence: 93%

Sequence of Induced Hausdorff Metrics on Lie Groups

Neyra

Fukuoka

Neves

2019

Bull Braz Math Soc, New Series

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“…In 1999 Gribanova [25] found all inner metrics on the upper half plane which are invariant under the action of the group Γ : x ′ = αx + β, y ′ = αy, α > 0, −∞ < β < +∞, as well as their geodesics. It follows from [7] that every such metric must be Finslerian.…”

Section: Examplementioning

confidence: 99%

“…Note that nowadays many authors apply the term Busemann space to a geodesic space with a local or global condition of nonpositive curvature in the Busemann sense [35]. However, there exist metrically homogeneous Finsler 2-manifolds among the so-called quasihyperbolic planes, which are Busemann G-spaces with no geodesically convex balls of positive radius (the assertion was stated in [16] and proved in [25]). Weaker assertions have been proved in [19].…”

Section: Introductionmentioning

confidence: 99%

Locally G-homogeneous Busemann G-spaces

Berestovskiî

Halverson²,

Repovš³

2011

Differential Geometry and its Applications

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We present short proofs of all known topological properties of general Busemann G-spaces (at present no other property is known for dimensions more than four). We prove that all small metric spheres in locally G-homogeneous Busemann G-spaces are homeomorphic and strongly topologically homogeneous. This is a key result in the context of the classical Busemann conjecture concerning the characterization of topological manifolds, which asserts that every n-dimensional Busemann G-space is a topological nmanifold. We also prove that every Busemann G-space which is uniformly locally G-homogeneous on an orbal subset must be finite-dimensional.

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“…For centrally symmetrical sets Ω, this problem was investigated in [27]. In the present paper, we find all geodesics for all possible sets Ω.…”

Section: Introductionmentioning

confidence: 99%

Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometry

2021

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We consider a series of optimal control problems with 2-dimensional control lying in an arbitrary convex compact set Ω. The considered problems are well studied for the case when Ω is a unit disc, but barely studied for arbitrary Ω. We derive extremals to these problems in general case by using machinery of convex trigonometry, which allows us to do this identically and independently on the shape of Ω. The paper describes geodesics in (i) the Finsler problem on the Lobachevsky hyperbolic plane; (ii) left-invariant sub-Finsler problems on all unimodular 3D Lie groups (SU(2), SL(2), SE(2), SH(2)); (iii) the problem of rolling ball on a plane with distance function given by Ω; (iv) a series of "yacht problems" generalizing Euler's elastic problem, Markov-Dubins problem, Reeds-Shepp problem and a new sub-Riemannian problem on SE(2); and (v) the plane dynamic motion problem.

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On a quasihyperbolic plane

Cited by 13 publications

References 4 publications

Sequence of Induced Hausdorff Metrics on Lie Groups

Sequence of Induced Hausdorff Metrics on Lie Groups

Locally G-homogeneous Busemann G-spaces

Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometry

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