Relationship between the holonomy pseudogroup and holonomy homomorphism (foliation)

Question

This question is surely a duplication of https://math.stackexchange.com/questions/4343635/relationship-between-the-holonomy-pseudogroup-and-holonomy-homomorphism-foliati , however, I got no replies. It may be quite elementary and I am not expert in this topic.

First, let me state some basics mainly coming from Introduction to foliations and Lie groupoids written by I. Moerdijk and J. Mrcun.

A codimension $q$ foliation $\mathcal{F}$ on a smooth n-manifold $M$ is given by the following data: An open cover $\mathcal{U}:=\left\{U_{i}\right\}_{i \in I}$ of $\mathrm{M}$.

• A $q$-dimensional smooth manifold $T_{0}$.
• For each $U_{i} \in \mathcal{U}$ a submersion $f_{i}: U_{i} \rightarrow T_{0}$ with connected fibers (these fibers are called plaques).
• For all intersections $U_{i} \cap U_{j} \neq \emptyset$ a local diffeomorphism $\gamma_{i j}$ of $T_{0}$ such that $f_{j}=\gamma_{i j} \circ f_{i}.$

We call $T=\coprod_{U_{i} \in \mathcal{U}} f_{i}\left(U_{i}\right)$ the transverse manifold of $\mathcal{F} .$ The local diffeomorphisms $\gamma_{i j}$ generate a pseudogroup $\Gamma$ of transformations on $T$ (called the holonomy pseudogroup). The space of leaves $M / \mathcal{F}$ of the foliation $\mathcal{F}$ can be identified with $T / \Gamma$.

Also, for a transversal section $S$ at $x\in L$ one obtains the map $$ \mathrm{hol}^{S}=\mathrm{hol}^{S, S}: \pi_{1}(L, x) \longrightarrow \operatorname{Diff}_{x}(S) $$ which is a group homomorphism to obtain a homomorphism of groups hol: $\pi_{1}(L, x) \longrightarrow \operatorname{Diff}_{0}\left(\mathbb{R}^{q}\right)$ which is called the holonomy homomorphism of $L$, and is determined uniquely up to a coniugation in $\operatorname{Diff}_{0}\left(\mathbb{R}^{q}\right)$.

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The motivation for me to compare these two concepts coming from the following statement, (the above book page 26, paragraph -2):

For a given foliation $\mathcal{F}$ on $M$, a Riemannian structure on the normal bundle of $\mathcal{F}$ determines a transverse metric (i.e., $\mathcal{F}$ is Riemann) if and only if this structure is holonomy invariant. One half of this is stated in the following proposition, the other half in Remark $2.7$ (2).

And the following proposition should imply the necessariness:

Proposition $2.5$ Let $(\mathcal{F}, g)$ be a Riemannian foliation of $M$. Let $L$ be a leaf of $\mathcal{F}, \alpha$ a path in $L$, and let $T$ and $S$ be transversal sections of $\mathcal{F}$ with $\alpha(0) \in T$ and $\alpha(1) \in S$. Then $$ \mathrm{hol}^{S, T}(\alpha):(T, \alpha(0)) \longrightarrow(S, \alpha(1)) $$

As the authors claimed, the other direction can be proved by Remark 2.7 (2) as follows:

Let $\mathcal{F}$ be a foliation of $M$ given by a Haefliger cocycle $\left(U_{i}, s_{i}, \gamma_{i j}\right)$. If each submersion $s_{i}: U_{i} \rightarrow s_{i}\left(U_{i}\right)$ has connected fibres, then any transverse metric on $(M, \mathcal{F})$ induces a Riemannian metric on $s_{i}\left(U_{i}\right)$, for any $i$, such that the diffeomorphisms $\gamma_{i j}$ are isometries. Conversely, if each $s_{i}\left(U_{i}\right)$ is a Riemannian manifold and if each $\gamma_{i j}$ is an isometry, then the pull-back of the Riemannian structure on $s_{i}\left(U_{i}\right)$ along $s_{i}$ gives a transverse metric on $\left(U_{i},\left.\mathcal{F}\right|_{U_{i}}\right)$, and these transverse metrics amalgamate to a transverse metric on $(M, \mathcal{F})$.

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However, I can't see how this happen. So my questions are listed as follows:

1. I think Remark 2.7 (2) is saying $$\mathcal{F} \text{ is Riemann}\iff \text{ transverse manifold } T \text{ has a } \Gamma\text{-invariant Riemannian metric,}$$ am I right?
2. If I understand incorrect for the above question, how can the authors used Remark 2.7 (2) to imply the other direction?
3. Why we call the name holonomy pseudogroup? What is the relationship between the holonomy pseudogroup and holonomy homomorphism, also, holonomy-invariant and holonomy pseudogroup-invariant?

Answer 1

I will answer even though this question was asked a year ago. Hopefully my answer is of some help.

1. Yeah that's right.
2. I don't quite understand the question, the Remark 2.7 says that any transverse Riemannian metric induces a metric on the fibers and conversely a Riemannian metric on $s_i(U_i)$ amalgamates to a transverse Riemannian metric, so that is both of the directions.
3. The holonomy pseudogroup its a pseudogroup because not all elements can be multipliead i.e. given arbitrary $\gamma_{i_1,j_1}$ and $\gamma_{i_2,j_2}$, the composition $\gamma_{i_1,j_1}\circ\gamma_{i_2,j_2}$ may not be defined (it depends on the intersection of $\{U_{i_1},U_{i_2},U_{j_1},U_{j_2}\}$) and the relationship between the holonomy pseudogroup and the holonomy group can be understood as a way to measure how a string of Haeflinger's cycles changes after walking in a loop. This turns out to be homotopy-invariant and base point invariant modulo conjugation in group of homeomorphisms/difeomorphisms of the transverse space. Also by definition every holonomy pseudogroup invariant is holonomy invariant but I don't know if the other way is true.