Horizontal holonomy and foliated manifolds

TL;DR

This paper introduces horizontal holonomy groups for foliated manifolds, providing methods to compute them and conditions for foliations to be totally geodesic or to admit a principal bundle structure.

Contribution

It defines horizontal holonomy groups, derives analogues of classical theorems for their computation, and establishes criteria for special foliation structures using these groups.

Findings

Explicit computation methods for horizontal holonomy groups.

Necessary and sufficient conditions for totally geodesic foliations.

Criteria for foliations to admit principal bundle structures.

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).