no. 3
Horizontal holonomy and foliated manifolds
[Holonomie horizontale et variétés feuilletées]
Chitour, Yacine1 ; Grong, Erlend2, 3 ; Jean, Frédéric4 ; Kokkonen, Petri5
1 Laboratoire des Signaux et Systèmes (L2S) Supélec 3 rue Joliot-Curie Université Paris XI 91192 Gif-sur-Yvette (France)
2 Laboratoire des Signaux et Systèmes (L2S) Supélec Université Paris-Sud CNRS, Université Paris-Saclay 3 rue Joliot-Curie 91192 Gif-sur-Yvette (France)
3 and Department of Mathematics University of Bergen P. O. Box 7803 5020 Bergen, (Norway)
4 UMA, ENSTA ParisTech Université Paris-Saclay 828 bd des Maréchaux 91762 Palaiseau (France)
5 Helsinki (Finland)
Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 1047-1086.

Dans cet article, nous introduisons les groupes d’holonomie horizontale associés à un sous-fibré D du fibré tangent d’une variété différentielle M munie d’une connexion linéaire. Ces groupes sont construits comme l’holonomie par le transport parallèle (pour la connexion) uniquement le long des lacets tangents à D. Nous faisons une étude détaillée de ces groupes et donnons en particulier des analogues des théorèmes d’Ambrose–Singer et Ozeki sous une hypothèse d’équirégularité du sous-fibré D. D’autre part nous appliquons l’holonomie horizontale à l’étude de problèmes de feuilletages et obtenons ainsi des conditions nécessaires et suffisantes pour que les feuilles d’un feuilletage donné soient (a) totalement géodésiques, ou (b) les fibres d’un fibré principal. Le sous-fibré D est choisi comme le complément orthogonal des feuilles dans le cas (a), et comme la connexion principale dans le cas (b).

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).

Reçu le :
Révisé le :
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DOI : 10.5802/aif.3265
Classification : 53C29, 53C03, 53C12
Keywords: holonomy, totally geodesic foliations, principal connections
Mot clés : l’holonomie, des feuilletages totalement géodésiques, des connexions principals
Chitour, Yacine 1 ; Grong, Erlend 2, 3 ; Jean, Frédéric 4 ; Kokkonen, Petri 5

1 Laboratoire des Signaux et Systèmes (L2S) Supélec 3 rue Joliot-Curie Université Paris XI 91192 Gif-sur-Yvette (France)
2 Laboratoire des Signaux et Systèmes (L2S) Supélec Université Paris-Sud CNRS, Université Paris-Saclay 3 rue Joliot-Curie 91192 Gif-sur-Yvette (France)
3 and Department of Mathematics University of Bergen P. O. Box 7803 5020 Bergen, (Norway)
4 UMA, ENSTA ParisTech Université Paris-Saclay 828 bd des Maréchaux 91762 Palaiseau (France)
5 Helsinki (Finland)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Horizontal holonomy and foliated manifolds},
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     pages = {1047--1086},
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Chitour, Yacine; Grong, Erlend; Jean, Frédéric; Kokkonen, Petri. Horizontal holonomy and foliated manifolds. Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 1047-1086. doi : 10.5802/aif.3265. https://aif.centre-mersenne.org/articles/10.5802/aif.3265/

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