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 |
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Format: | Artikel |
Sprache: | eng |
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Zusammenfassung: | 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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ISSN: | 1777-5310 0373-0956 1777-5310 |
DOI: | 10.5802/aif.3265 |