A unifying approach for rolling symmetric spaces

Krakowski, Krzysztof; Machado, Luís; Leite, F. Silva

doi:10.3934/jgm.2020016

Cited by 3 publications

(6 citation statements)

References 47 publications

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“…Theorem 5.3. For n > 1, the control system (19) (or, equivalently, the kinematic equations (18)) which describe the rolling of H n κ (r) over its affine tangent space at p 0 = [r 0 • • • 0] , is controllable on the Lie group G = SO I κ+1 (n + 1) × IR n .…”

Section: Theorem 52 ([15]mentioning

confidence: 99%

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Pure rolling motion of hyperquadrics in pseudo-Euclidean spaces

Marques

Leite

2022

JGM

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<p style='text-indent:20px;'>This paper is devoted to rolling motions of one manifold over another of equal dimension, subject to the nonholonomic constraints of no-slip and no-twist, assuming that these motions occur inside a pseudo-Euclidean space. We first introduce a definition of rolling map adjusted to this situation, which generalizes the classical definition of Sharpe [<xref ref-type="bibr" rid="b26">26</xref>] for submanifolds of an Euclidean space. We also prove some important properties of these rolling maps. After presenting the general framework, we analyse the particular rolling of hyperquadrics embedded in pseudo-Euclidean spaces. The central topic is the rolling of a pseudo-hyperbolic space over the affine space associated with its tangent space at a point. We derive the kinematic equations, as well as the corresponding explicit solutions for two specific cases, and prove the existence of a rolling map along any curve in that rolling space. Rolling of a pseudo-hyperbolic space on another and rolling of pseudo-spheres are equally treated. Finally, for the central theme, we write the kinematic equations as a control system evolving on a certain Lie group and prove its controllability. The choice of the controls corresponds to the choice of a rolling curve.</p>

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Section: Theorem 52 ([15]mentioning

confidence: 99%

“…After the formal definition of rolling map introduced by Sharpe [26] in 1996 for submanifolds of Euclidean spaces, a number of papers have been devoted to the rolling of certain Riemannian manifolds and the work in this area continues to expand. Examples of this are, for instance, [14], [1], [29], [16], [11], [9], [10], and [18].…”

mentioning

confidence: 99%

Pure rolling motion of hyperquadrics in pseudo-Euclidean spaces

Marques

Leite

2022

JGM

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“…In relation to the last part of the proof of Proposition 7, we point out that in [8, Section 3.3] a complete answer was given to the problem of extending intrinsic rollings to extrinsic ones. We also refer to [18] for non-twist conditions in the case of embedded sub-Euclidean manifolds.…”

Section: 3mentioning

confidence: 99%

“…Sharpe in [25] is the extrinsic rolling, which makes use of the isometric embedding of the manifolds in an ambient (semi)-Euclidean vector space V , so that the rolling is described in terms of the action of the group SE(V ) of oriented isometries of V . More recent works that use the extrinsic rolling are, for instance, [26,12,16,11,6,20,18,22]. As far as we know, only in [8] and [21] both notions of rolling were addressed, the first for the Riemannian case and the second for the semi-Riemannian case.…”

Section: Introductionmentioning

confidence: 99%

Symmetric spaces rolling on flat spaces

Jurdjevic¹,

Markina²,

Leite³

2022

Preprint

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The objective of the current paper is essentially twofold. Firstly, to make clear the difference between two notions of rolling a Riemannian manifold over another, using a language accessible to a wider audience, in particular to readers with interest in applications. Secondly, we concentrate on rolling an important class of Riemannian manifolds. In the first part of the paper, the relation between intrinsic and extrinsic rollings is explained in detail, while in the second part we address rollings of symmetric spaces on flat spaces and complement the theoretical results with illustrative examples.

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“…Another definition of rolling initiated by Nomizu in [23] and presented more formally by Sharpe in [25] is the extrinsic rolling, which makes use of the isometric embedding of the manifolds in an ambient (semi)-Euclidean vector space V , so that the rolling is described in terms of the action of the group SE(V ) of oriented isometries of V . More recent works that use the extrinsic rolling are, for instance, [6,11,12,16,18,20,22,26]. As far as we know, only in [8] and [21] both notions of rolling were addressed, the first for the Riemannian case and the second for the semi-Riemannian case.…”

Section: Introductionmentioning

confidence: 99%