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 $\operatorname{Diff}_{0}\left(\mathbb{R}^{q}\right)$.
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 $(M, \mathcal{F})$.
However, I can't see how this happen. So my questions are listed as follows:
- 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?
- If I understand incorrect for the above question, how can the authors used Remark 2.7 (2) to imply the other direction?
- 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?