# Horizontal Holonomy for Affine Manifolds

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## 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 \({H^{\;\nabla }_{\Delta }}\) 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.

## Keywords

Holonomy group Affine holonomy group Rolling (Development of) manifolds (Sub)Riemannian Geometry Theory of Lie groups.## Mathematics Subject Classification (2010)

53B05 53C05 53B21 53C17## Notes

### Acknowledgments

The authors thank F. Jean and M. Sigalotti for fruitful discussions and insights regarding the proof of Proposition 5.7.

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