Horizontal Holonomy for Affine Manifolds

Abstract

In this paper, we consider a smooth connected finite-dimensional manifold M, an affine connection ? with holonomy group H? and Δ a smooth completely non integrable distribution. We define the Δ-horizontal holonomy group HΔ?${H^{\;\nabla }_{\Delta }}$ as the subgroup of H? obtained by ?-parallel transporting frames only along loops tangent to Δ. We first set elementary properties of HΔ?${H^{\;\nabla }_{\Delta }}$ and show how to study it using the rolling formalism Chitour and Kokkonen (2011). In particular, it is shown that HΔ?${H^{\;\nabla }_{\Delta }}$ 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 ? is the Levi-Civita connection associated to a Riemannian metric on M, and show in this particular case that HΔ?${H^{\;\nabla }_{\Delta }}$ is compact and strictly included in H? as soon as m ? 3.