Horizontal holonomy and foliated manifolds

Horizontal holonomy and foliated manifolds

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....

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Veröffentlicht in:Annales de l'Institut Fourier 2019-01, Vol.69 (3), p.1047-1086
Hauptverfasser: Chitour, Yacine, Grong, Erlend, Jean, Frédéric, Kokkonen, Petri
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Mathematics
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container_title Annales de l'Institut Fourier
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creator Chitour, Yacine
Grong, Erlend
Jean, Frédéric
Kokkonen, Petri
description 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).
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Mathematics
title Horizontal holonomy and foliated manifolds
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