Horizontal holonomy and foliated manifolds
Differential Geometry
2015-11-19 v1
Abstract
We introduce horizontal holonomy groups, which are groups defined using parallel transport only along curves tangent to a given subbundle 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 plays the role of an orthogonal complement to the leaves of the foliation in case (a) and of a principal connection in case (b).
Keywords
Cite
@article{arxiv.1511.05830,
title = {Horizontal holonomy and foliated manifolds},
author = {Y. Chitour and E. Grong and F. Jean and P. Kokkonen},
journal= {arXiv preprint arXiv:1511.05830},
year = {2015}
}