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. 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).