Horizontal Holonomy for Affine Manifolds

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 \({H^{\nabla }_{\Delta }}\) as the subgroup of H ^? obtained by ?-parallel transporting frames only along loops tangent to Δ. We first set elementary properties of \({H^{\nabla }_{\Delta }}\) and show how to study it using the rolling formalism Chitour and Kokkonen (2011). In particular, it is shown that \({H^{\nabla }_{\Delta }}\) 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 \({H^{\nabla }_{\Delta }}\) is compact and strictly included in H ^? as soon as m ≥ 3.