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). 2010 Mathematics Subject Classification. 53C29,53C03,53C12.

Asymptotic expansion of holonomy

“…This result was found using ideas from [4,13]. By applying the new theory of horizontal holonomy found in [3], we are able to show the following (for more details, see Section 5). Theorem 1.3.…”

“…It was shown in [3] that any equiregular subbundle D has at least one selector. Actually, from the construction in [3, Lemma 2.7], we know that D has a selector such that χ(D j ) ⊆ j−1 i=1 D i ⊕ D j−i for any j = 2, .…”

A de Rham decomposition type theorem for contact sub-Riemannian manifolds

“…In this paper we deal with holonomy determined by a class of connections introduced in [13] for contact sub-Riemannian manifolds, and prove a theorem that can be considered as a version of de Rham decomposition theorem for Riemannian manifolds. Different approaches to sub-Riemannian holonomy and some other problems involving it are treated, e.g., in [7,9].…”

On $\mathrm{G}_2$ and Sub-Riemannian Model Spaces of Step and Rank Three

“…Hence we have that Isom(N ) has the maximal dimension 13 = dim(N ) + (rank(E))(rank(E) − 1)/2. However, orientation reversing isometries can not be integrated because it follows from [8,Example 4.1] that the free nilpotent Lie group F [4,2] is the only model space with step two and rank four that is a Carnot group. We will define the free nilpotent Lie group F[n, r] of step n and rank r in Example 3.6.…”

“…Similarly as for regular affine connections, we can define a holonomy group Hol ∇ (x) at a point x ∈ M by considering parallel transport along loops based at x and horizontal to H. If H is bracket-generating then Hol ∇ (x) is a Lie group and it will be connected whenever M is simply connected. More details on holonomy of partial connections can be found in [4].…”