# On Holonomy in (regular) Riemannian Foliations

Right now, I am trying to understanding the role of holonomy fields on Riemannian foliations, which lead me to the following (probably topological) groupoid: Let $\mathcal{F}\subset M$ be a Rieamannian foliation on a complete Riemannian manifold. We consider the groupoid of infinitesimal holonomy transformations $\mathcal{E}\rightrightarrows M$ as the collection of linear isomorphisms of the vertical spaces induced by holonomy transformations. These isomorphisms can be thought in two ways:

1) Given a horizontal curve $c:[0,1]\to M$, it induces a diffeomorphism between a neighbourhood of the leaf through $c(0)$ and a neighborhood of the leaf through $c(1)$. The isomorphism is just the differential of this diffeomorphism.
2) Let $c$ be a horizontal curve as above, then an $h$ is such an isomorphism between the verticals at $c(0)$, $\mathcal V_{c(0)}$, and the verticals at $c(1)$, if it sends $\xi\in \mathcal{V}_{c(0)}$ to $\bar\xi(1)$, where $\bar \xi(t)$ is the holonomy field along $c$ with $\bar \xi(0)=\xi(0)$.

The groupoid structure of $\mathcal{E}$ is the natural on, given by the source and target points of the isomorphism. I would like to know when $\mathcal E$ admits a smooth structure that makes it a Lie groupoid.

Please observe that it can't be the general case, since not all pair of points can be reached through a horizontal curve. Therefore, I suggest to assume this hypothesis, together with whatever other hypothesis you feel comfortable (such as bounded holonomy fields, compactness of M - the case of submersions with compact holonomy seems quite simple, so please do not assume it).

Based on Robert Bryant's answer here, an optimal solution would be some kind of regularity between horizontal curves and the leaves: for example (here we assume that every two points can be connected through a horizontal curve), for each $p\in M$, there is a neighborhood $U$ of the leaf around $p$ and a smooth map $\Phi:U\times [0,1]\to M$ such that, for each $x$, $\Phi(x,t)$ is a horizontal curve starting at $p$ and $\Phi|_{U\times \{1\}}$ is a diffeomorphism of $U$.

I appreciate comments and suggestions.

• I'm not sure what you mean by 'horizontal curve'. I would guess that you mean 'curve whose tangent vectors are orthogonal to the tangent spaces to the leaves', but I'm not sure. Could you clarify? Oct 12, 2015 at 21:22
• Sorry for that. Yes, by a horizontal curve I mean a curve whose velocity is always orthogonal to the tangent spaces of the leaves. Analogously, I call 'verticals' the vectors tangent to the leaves. Thank you for the comment! Oct 12, 2015 at 21:31
• What will be the Lie algebroid of this groupoid if the distribution normal to the foliation is not integrable ? Oct 13, 2015 at 21:25
• I do have a positive answer to this question (sort of). Should post it soon. Oct 12, 2016 at 17:31