Horizontal Holonomy for Affine Manifolds

TL;DR

This paper introduces the concept of horizontal holonomy groups for affine manifolds, exploring their properties and providing explicit examples, notably in Carnot groups, revealing new insights into their structure and relation to the full holonomy group.

Contribution

It defines and analyzes the $ abla$-horizontal holonomy group $H^{ abla}_ riangle$, establishing its Lie group structure and illustrating its properties through examples involving Carnot groups.

Findings

$H^{ abla}_ riangle$ is a Lie group.

In Carnot groups, $H^{ abla}_ riangle$'s identity component is compact.

$H^{ abla}_ riangle$ is a subgroup of $H^{ abla}$, often strictly smaller.

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 $H^{\nabla}$.