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\title[Asymptotic expansions of holonomy]{Asymptotic expansion of holonomy}
\author[E.~Grong and P.~Pansu]{Erlend Grong and Pierre Pansu}


\address{Universit\'e Paris Sud, Laboratoire des Signaux et Syst\`emes (L2S) Sup\'elec, CNRS, Universit\'e Paris-Saclay, 3 rue Joliot-Curie, 91192
Gif-sur-Yvette, France and University of Bergen, Department of Mathematics, P. O. Box 7803,
5020 Bergen, Norway.}
\email{erlend.grong@gmail.com}

\address{Laboratoire de Math\'ematiques d'Orsay, Universit\'e Paris-Sud, CNRS,
Universit\'e Paris-Saclay, 91405 Orsay, France.}
\email{pierre.pansu@math.u-psud.fr}


\subjclass[2010]{53C29, 41A99, 53C17}

\keywords{asymptotic expansion of holonomy, radial gauge, sub-Riemannian geometry, horizontal holonomy}

\begin{document}
\let\thefootnote\relax\footnotetext{
This paper is supported by the Fonds National de la Recherche Luxembourg (AFR 4736116 and OPEN Project GEOMREV) and by the Research Council of Norway (project number 249980/F20). P.P. acknowledges support from ANR, project SRGI, ANR-15-CE40-0018.}


\begin{abstract}
Given a principal bundle with a connection, we look for an asymptotic expansion of the holonomy of a loop in terms of its length. This length is defined relative to some Riemannian or sub-Riemannian structure. We are able to give an asymptotic formula that is independent of choice of gauge.
\end{abstract}
 

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\input{1_Introduction}
\input{2_Carnot}
\input{3_GaugeFree}
\input{4_General}
\input{5_IncreaseOrder}
\input{6_Examples}
\appendix
\input{A_ODELie}


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