# Horizontal holonomy and foliated manifolds

Abstract : 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).
Type de document :
Pré-publication, Document de travail
2017

https://hal-ensta.archives-ouvertes.fr/hal-01268119
Contributeur : Frédéric Jean <>
Soumis le : mercredi 8 mars 2017 - 10:38:07
Dernière modification le : mercredi 15 mars 2017 - 08:22:26

### Fichier

Holonomy_Done.pdf
Fichiers produits par l'(les) auteur(s)

### Identifiants

• HAL Id : hal-01268119, version 1
• ARXIV : 1511.05830

### Citation

Yacine Chitour, Erlend Grong, Frédéric Jean, Petri Kokkonen. Horizontal holonomy and foliated manifolds. 2017. <hal-01268119>

Consultations de
la notice

## 218

Téléchargements du document