Symmetries of the rolling model

Chitour, Yacine; Molina, Mauricio Godoy; Kokkonen, Petri

doi:10.1007/s00209-015-1508-6

Cited by 10 publications

(12 citation statements)

References 33 publications

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“…For a related result in a special case, cf. [14]. Assume that D is bracket generating and equiregular with a selector χ.…”

Section: Two Problems On Foliated Manifoldsmentioning

confidence: 99%

Horizontal holonomy and foliated manifolds

Chitour¹,

Grong²,

Jean³

et al. 2019

Annales De l'Institut Fourier

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We introduce horizontal holonomy groups, which are groups defined using parallel transport only along curves tangent to a given subbundle D of the tangent bundle. We provide explicit means of computing these holonomy groups by deriving analogues of Ambrose-Singer's and Ozeki's theorems. We then give necessary and sufficient conditions in terms of the horizontal holonomy groups for existence of solutions of two problems on foliated manifolds: determining when a foliation can be either (a) totally geodesic or (b) endowed with a principal bundle structure. The subbundle D plays the role of an orthogonal complement to the leaves of the foliation in case (a) and of a principal connection in case (b). 2010 Mathematics Subject Classification. 53C29,53C03,53C12.

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“…For a related result in a special case, cf. [14]. Assume that D is bracket generating and equiregular with a selector χ.…”

Section: Two Problems On Foliated Manifoldsmentioning

confidence: 99%

Horizontal holonomy and foliated manifolds

Chitour¹,

Grong²,

Jean³

et al. 2019

Annales De l'Institut Fourier

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“…Rolling against a space form. In [6,8], the authors study the rolling system when ( M, g) is a space form, that is, a complete and simply connected Riemannian manifold of constant sectional curvature c ∈ R. In this case, the natural projection π Q,M : Q → M is a principal bundle of a special form, which we proceed to explain.…”

Section: 2mentioning

confidence: 99%

“…Theorem 2.1. The projection π Q,M : Q → M admits a G-principal bundle structure with horizontal distribution D R , for some Lie group G, if and only if M is a space form (under some genericity assumptions, see [8,Theorem 4.10]). In this situation, the Lie group G is given by…”

Section: 2mentioning

confidence: 99%

Rolling Against a Sphere: The Non-transitive Case

Chitour

Molina

Kokkonen³

et al. 2015

J Geom Anal

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Abstract. We study the control system of a Riemannian manifold M of dimension n rolling on the sphere S n . The controllability of this system is described in terms of the holonomy of a vector bundle connection which, we prove, is isomorphic to the Riemannian holonomy group of the cone C(M ) of M .Using Berger's list, we reduce the possible holonomies to a few families. In particular, we focus on the cases where the holonomy is the unitary and the symplectic group. In the first case, using the rolling formalism, we construct explicitly a Sasakian structure on M ; and in the second case, we construct a 3-Sasakian structure on M .

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“…In recent years rolling has received a renewed wave of interest -in part because of its importance for robotic manipulation of objects [11]. For example, there has been an interest in understanding rolling from symmetry arguments [12] as well as from purely geometrical considerations [13,14,15]. Further, the interesting shapes known as D-forms are examples of surface structures that are formed by assembling several developable surfaces [16,17,18].…”

Section: Introductionmentioning

confidence: 99%

Cartan ribbonization and a topological inspection

2018

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We develop the concept of Cartan ribbons and a method by which they can be used to ribbonize any given surface in space by intrinsically flat ribbons. The geodesic curvature along the center curve on the surface agrees with the geodesic curvature of the corresponding Cartan development curve, and this makes a rolling strategy successful. Essentially, it follows from the orientational alignment of the two co-moving Darboux frames during the rolling. Using closed center curves we obtain closed approximating Cartan ribbons that contribute zero to the total curvature integral of the ribbonization. This paves the way for a particular simple topological inspection -it is reduced to the question of how the ribbons organize their edges relative to each other. The Gauss-Bonnet theorem leads to this topological inspection of the vertices. Finally, we display two examples of ribbonizations of surfaces, namely of a torus using two ribbons, and of an ellipsoid using its closed curvature lines as center curves for the ribbons. The topological inspection of the torus ribbonization is particularly simple as it has no vertex points, giving directly the Euler characteristic 0. The ellipsoid has 4 vertices -corresponding to the 4 umbilical points -each of degree one and each therefore contributing one-half to the Euler characteristic.

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Symmetries of the rolling model

Cited by 10 publications

References 33 publications

Horizontal holonomy and foliated manifolds

Horizontal holonomy and foliated manifolds

Rolling Against a Sphere: The Non-transitive Case

Cartan ribbonization and a topological inspection

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