Reference : Horizontal holonomy and foliated manifolds
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Physical, chemical, mathematical & earth Sciences : Mathematics
Horizontal holonomy and foliated manifolds
Chitour, Yacine [Université Paris XI > Laboratoire des Signaux et Systèmes]
Grong, Erlend mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
Jean, Frédérick [Université Paris-Saclay > UMA, ENSTA ParisTech]
Kokkonen, Petri []
Cornell University Library
[en] holonomy ; totally geodesic foliations ; principal connections
[en] We introduce horizontal holonomy groups, which are groups defined using parallel transport only along curves tangent to a given subbundle D of the tangent bundle. We provide explicit means of computing these holonomy groups by deriving analogues of Ambrose-Singer’s and Ozeki’s theorems. We then give necessary and sufficient conditions in terms of the horizontal holonomy groups for existence of solutions of two problems on foliated manifolds: determining when a foliation can be either (a) totally geodesic or (b) endowed with a principal bundle structure. The subbundle D plays the role of an orthogonal complement to the leaves of the foliation in case (a) and of a principal connection in case (b).
This research was partially supported by the iCODE Institute, research project of the IDEX Paris-Saclay, by the Grant ANR-15-CE40-0018 of the ANR, and by the Hadamard Mathematics LabEx (LMH) through the grant number ANR-11-LABX-0056-LMH in the “Programme des In- vestissements d’Avenir”. It is also supported by the Fonds National de la Recherche Luxembourg (AFR 4736116 and OPEN Project GEOMREV).

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