Horizontal Holonomy for Affine Manifolds

Boutheina Hafassa (1, 2) , Amina Mortada (1, 3) , Yacine Chitour (1) , Petri Kokkonen

1 L2S - Laboratoire des signaux et systèmes
2 Département de mathématiques [Tunis]
3 LU / ULB - الجامعة اللبنانية [بيروت] = Lebanese University [Beirut] = Université libanaise [Beyrouth]

In this paper, we consider a smooth connected finite-dimensional manifold M, an affine connection a double dagger with holonomy group H (a double dagger) and Delta a smooth completely non integrable distribution. We define the Delta-horizontal holonomy group as the subgroup of H (a double dagger) obtained by a double dagger-parallel transporting frames only along loops tangent to Delta. We first set elementary properties of and show how to study it using the rolling formalism Chitour and Kokkonen (2011). In particular, it is shown that is a Lie group. Moreover, we study an explicit example where M is a free step-two homogeneous Carnot group with m >= 2 generators, and a double dagger is the Levi-Civita connection associated to a Riemannian metric on M, and show in this particular case that is compact and strictly included in H (a double dagger) as soon as m >= 3.

Mots clés: Holonomy group; Affine holonomy group; Rolling (Development of) manifolds; (Sub)Riemannian Geometry; Theory of Lie groups

Domaines: Automatique / Robotique

Boutheina Hafassa, Amina Mortada, Yacine Chitour, Petri Kokkonen. Horizontal Holonomy for Affine Manifolds. Journal of Dynamical and Control Systems, 2016, 22 (3), pp.413-440. ⟨10.1007/s10883-015-9274-7⟩. ⟨hal-02307515⟩