Horizontal Holonomy for Affine Manifolds

Abstract

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