Horizontal holonomy and foliated manifolds
Annales de l'Institut Fourier, Volume 69 (2019) no. 3, p. 1047-1086

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).

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).

Received : 2016-06-22
Revised : 2017-09-19
Accepted : 2018-03-20
Published online : 2019-06-03
DOI : https://doi.org/10.5802/aif.3265
Classification:  53C29,  53C03,  53C12
Keywords: holonomy, totally geodesic foliations, principal connections
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     author = {Chitour, Yacine and Grong, Erlend and Jean, Fr\'ed\'eric and Kokkonen, Petri},
     title = {Horizontal holonomy and foliated manifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {69},
     number = {3},
     year = {2019},
     pages = {1047-1086},
     doi = {10.5802/aif.3265},
     zbl = {07067426},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2019__69_3_1047_0}
}
Horizontal holonomy and foliated manifolds. Annales de l'Institut Fourier, Volume 69 (2019) no. 3, pp. 1047-1086. doi : 10.5802/aif.3265. https://aif.centre-mersenne.org/item/AIF_2019__69_3_1047_0/

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