$\pi\circ\widetilde{\gamma}|_{k\in\nu_{\tau}(A),\quad
t\in\nu_{\tau}(t_{0}),\quad s\in[1/2,1]}$

using the formula appearing in the last item of the statement. For other
values of $k$ and $t$, we set
$\pi\circ\widetilde{\gamma}(k,d,s)=\pi\circ\gamma^{\prime}(k)$.

The family $\widetilde{\gamma}(k,d,s)|_{s\in[1/2,1]}$ itself is given from its
projection by lifting horizontally. This is done with Lemmas 5.3 and 5.4. Here
is where the smallness of $r$ enters. It must be sufficiently small to control
the error in the estimated-displacement of ${\mathcal{C}}$ (which amounts to
the error in the estimated-displacement of its tangles; Lemma 6.29). How small
$r$ must be depends only on the graphical model $(V,{\mathcal{D}})$. It
follows that $r$ being small enough implies that a lift exists for all times
and, due to the properties of tangles, it yields embedded curves. ∎

####  7.3.1. Controllability

We now address how the insertion of a controller allows us to control the
endpoint of the corresponding horizontal curves.

######  Lemma 7.7.

Consider the setup and conclusions of Proposition 7.6. Consider the endpoint
map

${\mathfrak{ep}}:K\times[-h,h]^{n-q}\quad\longrightarrow\quad{\mathbb{R}}^{n-q}$

defined by ${\mathfrak{ep}}(k,d):=\pi_{|}\circ\widetilde{\gamma}(k,d,1)(1)$.
Then the following estimate holds:

${\mathfrak{ep}}(k,d)=\gamma(k)+(0,d)(1+O(r)+O(\delta))$

for all $k\in A$.

###### Proof.

This is immediate from the analogous statement about tangles, namely Lemma
6.29. ∎

As in Lemma 4.5 we can be more precise and say that there is a constant $C$,
depending only on the graphical model, such that the error is bounded above by
$C.(r+\delta)$. It follows that imposing $\delta,r<<1/C$ implies that
${\mathfrak{ep}}(k,-)$ is an equidimensional embedding whose image contains a
ball of radius $h/2$, centered at $\gamma(k)$.

####  7.3.2. Insertion in the $\varepsilon$-horizontal case

For our purposes, we will need the following variation of Proposition 7.6:

######  Lemma 7.8.

Let $r$, $h$, $A$, $t_{0}$, $\eta$, $R$, $S$, and $\delta$ be as in
Proposition 7.6. Given
$\gamma:K\rightarrow{\mathfrak{Emb}}^{\varepsilon}([0,1];D_{r},{\mathcal{D}})$,
there is a family
$\widetilde{\gamma}:K\times[-h,h]^{n-q}\times[0,1]\rightarrow{\mathfrak{Emb}}([0,1];V)$
such that:

  * •

$\widetilde{\gamma}(k,d,0)=\gamma(k)$.

  * •

$\widetilde{\gamma}(k,d,s)=\gamma(k)$ outside of
$\nu_{2\tau}(A)\times\nu_{2\tau}(t_{0})$.

  * •

The length of $\widetilde{\gamma}(k,d,s)$ in the region
$\nu_{2\tau}(t_{0})\setminus\nu_{\tau}(t_{0})$ is bounded above by $\eta$.

  * •

For all $k\in\nu_{\tau}(A)$, all $t\in\nu_{\tau}(t_{0})$, and all
$s\in[1/2,1]$, it holds that

$\pi\circ\widetilde{\gamma}(k,d,s)(t)={\mathcal{C}}(d,(2s-1)\theta(k))\left(\dfrac{t-t_{0}-\tau}{2\tau}\right)+\pi\circ\gamma(k)(t_{0}).$

###### Proof.

We use Lemmas 5.3 and 5.4 to horizontalise $\gamma$, yielding some family
$\gamma^{\prime}$. We apply to it Proposition 7.6, yielding a family
$\widetilde{\gamma}^{\prime}$ that is also horizontal. We then adjust its
vertical component to yield the claimed $\widetilde{\gamma}$. Due to the
displacement introduced by the controller, it may be the case that
$\widetilde{\gamma}$ is not $\varepsilon$-horizontal. However, it is still
graphical over ${\mathbb{R}}^{q}$. ∎

##  8\. h-Principles for horizontal curves

In this section we prove our main Theorem 1.4, the classification of regular
horizontal embeddings. This uses all the tools that we have presented in
previous sections. The corresponding statement for immersions, Theorem 1.2,
will follow from simplified versions of the same arguments. In Subsection 8.5
we explain how this is done.

###  8.1. Relative version of Theorem 1.4

We explained in Section 3 that our $h$-principle arguments are relative in
nature. This allows us to state an analogue of Theorem 1.4 that deals with
embedded regular horizontal paths and is relative in parameter and domain.

######  Proposition 8.1.

Let $K$ be a compact manifold. Let $I=[0,1]$. Let $(M,{\mathcal{D}})$ be a
manifold of dimension $\dim(M)>3$, endowed with a bracket–generating
distribution. Suppose we are given a map
$\gamma:K\to{\mathfrak{Emb}}^{\varepsilon}(I;M,{\mathcal{D}})$ satisfying:

  * •

$\gamma(k)\in{\mathfrak{Emb}}^{\operatorname{r}}(I;M,{\mathcal{D}})$ for
$k\in{\mathcal{O}p}(\partial K)$.

  * •

$\gamma(k)(t)$ is horizontal if $t\in{\mathcal{O}p}(\partial I)$.

Then, there exists a homotopy
$\widetilde{\gamma}:K\times[0,1]\to{\mathfrak{Emb}}^{\varepsilon}(I;M,{\mathcal{D}})$
satisfying:

  * •

$\widetilde{\gamma}(k,0)=\gamma(k)$.

  * •

$\widetilde{\gamma}(k,1)$ takes values in
${\mathfrak{Emb}}^{\operatorname{r}}(I;M,{\mathcal{D}})$.

  * •

this homotopy is relative to $k\in{\mathcal{O}p}(\partial K)$ and to
$t\in{\mathcal{O}p}(\partial I)$.

  * •

$\widetilde{\gamma}(k,s)$ is $C^{0}$-close to $\gamma(k)$ for all $s\in[0,1]$.

Do note that the analogous statement where we consider formal horizontal
embeddings instead of $\varepsilon$-horizontal ones follows from this one,
thanks to the results in Subsection 3.2.

Now, as usual, the absolute statement follows from the relative one:

###### Proof of Theorem 1.4 from Proposition 8.1.

The statement follows, according to Subsection 3.2, from the vanishing of the
relative homotopy groups of the pair

$({\mathfrak{Emb}}^{\varepsilon}(M,{\mathcal{D}}),\,{\mathfrak{Emb}}^{\operatorname{r}}(M,{\mathcal{D}})).$

Consider a family
$\gamma:{\mathbb{D}}^{a}\to{\mathfrak{Emb}}^{\varepsilon}(M,{\mathcal{D}})$
that takes values in ${\mathfrak{Emb}}^{\operatorname{r}}(M,{\mathcal{D}})$
along ${\mathbb{S}}^{a-1}$. This represents a class in the $a$th relative
homotopy group. We must deform this family to lie entirely in
${\mathfrak{Emb}}^{\operatorname{r}}(M,{\mathcal{D}})$.

We may assume, by suitable reparametrisation in the parameter, that in a
collar of ${\mathbb{S}}^{a-1}$, the family $\gamma$ is radially constant. This
provides for us an open along the boundary of ${\mathbb{D}}^{a}$ in which all
curves are regular horizontal. We then consider the product space
${\mathbb{D}}^{a}\times{\mathbb{S}}^{1}$ and we make the curves $\gamma$
horizontal in a neighbourhood of the slice ${\mathbb{D}}^{a}\times\\{1\\}$,
using Lemma 5.8. This is a $C^{0}$-small process.

Regarding ${\mathbb{D}}^{a}\times{\mathbb{S}}^{1}$ as ${\mathbb{D}}^{a}\times
I$ yields a family of $\varepsilon$-horizontal paths, horizontal at the
endpoints, with all curves regular close to $\partial{\mathbb{D}}^{a}$. We
apply Proposition 8.1 to it, relatively to
${\mathcal{O}p}({\mathbb{S}}^{a-1})$ in the parameter, and relatively to
${\mathcal{O}p}(\partial I)$ in the domain. The claim follows. ∎

The remainder of this section is mostly dedicated to the proof of Proposition
8.1.

Figure 20. Choosing the time slice ${\mathbb{D}}^{a}\times\\{1\\}$ identifies
the product ${\mathbb{D}}^{a}\times{\mathbb{S}}^{1}$ with
${\mathbb{D}}^{a}\times[0,1]$, allowing us to apply Proposition 8.1.

###  8.2. Setup for the proof of Proposition 8.1

Given the family of curves $\gamma$, we will produce a series of homotopies in
order to obtain $\widetilde{\gamma}(-,1)$. These elementary homotopies are
relative to the boundary and through $\varepsilon$-horizontal curves. Their
concatenation will be the homotopy $\widetilde{\gamma}$.

We denote $\dim(M)=n$, ${{\operatorname{rank}}}({\mathcal{D}})=q$, and
$\dim(K)=K$. We write $\pi_{K}$ and $\pi_{I}$ for the projections of $K\times
I$ to its factors. We fix a metric on $K$, endow $I$ with the euclidean
metric, and endow the product $K\times I$ with the product metric. We will
write $d(-,-)$ to denote distance between subsets; the metric used should be
clear from context.

####  8.2.1. A fine cover of $K\times I$

Our first goal is to subdivide $K\times I$ in a manner that is nicely adapted
to $(M,{\mathcal{D}})$ and $\gamma$. The aim with this is to reduce our
subsequent arguments to constructions happening in very small balls in which
errors are controlled. No homotopy of $\gamma$ is produced in this subsection,
we are just doing some preliminary work.

Introduce a size parameter $N\in{\mathbb{N}}$, to be fixed as we go along in
the proof. We divide $I$ into intervals $I_{j}$ of length $1/N$. Furthermore,
we fix a finite cover of $K$ by charts parametrised by the unit cube. Such
cubes can themselves be divided into cubes of side $3/N$, spaced along the
coordinate axes as $1/N$. We write

$\\{\phi_{i}:[0,3/N]^{k}\,\longrightarrow\,U_{i}\subset K\\}$

for the resulting collection of cubical charts.

####  8.2.2. Bounding the number of intersections between charts

By construction, there is a constant $C_{1}$ such that any intersection
$U_{i_{1}}\cap\cdots\cap U_{i_{C_{1}}}$, involving distinct charts, is empty.
This follows from the properties of cubical subdivision; a detailed argument
can be found in [30, p. 25].

####  8.2.3. Covering the image by graphical models

Given the origin $k\in K$ of the chart $U_{i}$ and the initial time
$t_{j}:=j/N\in I_{j}$, we fix an adapted chart $(V_{i,j},\Psi_{i,j})$ centered
at $\gamma(k)(t_{j})$. These adapted charts are given by Lemma 4.5, meaning
that they all have the same radius $r_{0}>0$ and the difference between their
framing and the coordinate axes is controlled by some constant $C_{2}>0$.

####  8.2.4. Discussion about parameters

The constants $C_{1}$, $C_{2}$, and $r_{0}$ are given to us and depend on
$(M,{\mathcal{D}})$ and the family $\gamma$. For convenience, we introduce new
parameters $0<r_{1}<r_{0}$ and $0<l$, to be fixed later in the proof. We
impose $1/N<<r_{1},l$ in order to ensure that:

(2) $\gamma(k)(t)\in\Psi_{i,j}({\mathbb{D}}_{r_{1}})\text{ for all }(k,t)\in
U_{i}\times I_{j}$ (3) $\Psi_{i,j}^{-1}\circ\gamma(k)|_{I_{j}}\text{ has
euclidean length bounded above by $l$}$

I.e. the curves in each cube $U_{i}\times I_{j}$ are very short and are
located very close to the origin of the corresponding graphical model.

###  8.3. Introducing controllers

Our next goal is to add controllers to $\gamma$ over each cube $U_{i}\times
I_{j}$. We continue using the notation introduced in the previous subsection.
We write $\nu_{r}(A)$ for the $r$-neighbourhood of a subset $A$.

####  8.3.1. Boundary neighbourhoods

We fix a small enough constant $\tau>0$ so that all curves $\gamma(k)$ with
$k\in\nu_{\tau}(\partial K)$ are horizontal and regular. We then consider a
constant $\tau/2<\tau^{\prime}<\tau$ and a subset ${\mathcal{U}}$ of the cover
$\\{U_{j}\times I_{i}\\}$ so that:

  * •

the elements of ${\mathcal{U}}$, together with
$\nu_{\tau^{\prime}}(\partial(K\times I))$, cover $K\times I$,

  * •

all elements in ${\mathcal{U}}$ are disjoint from
$\nu_{\tau/2}(\partial(K\times I))$.

Figure 21. The cover ${\mathcal{U}}$, together with the boundary
neighbourhoods of radii $\tau$ and $\tau/2$. A controller is shown in one of
the $U_{i}\times I_{j}$.

####  8.3.2. Introducing controllers

Given $U_{i}\times I_{j}\in{\mathcal{U}}$, we choose a time $t_{i,j}$ in the
interior of $I_{j}$. We require that these are all distinct. We then introduce
a controller ${\mathcal{C}}_{i,j}$ (Lemma 7.8) along
$U_{i}\times\\{t_{i,j}\\}$. We write:

  * •

$S$ for the size-at-rest of all the controllers.

  * •

$\eta>0$ for the size of the neighbourhood of $U_{i}\times\\{t_{i,j}\\}$ in
which the controllers are contained. If $\eta$ is sufficiently small, the
controllers do not interact with one another.

Later on in the proof we will use the estimated-displacement of the
${\mathcal{C}}_{i,j}$, one pair $(i,j)$ at a time. For now we write
$\gamma^{\prime}$ for the $K$-family of $\varepsilon$-horizontal curves that
has the estimated-displacement of each controller at $0$. Do note that $S$
must be small enough to guarantee $\varepsilon$-horizontality. Furthermore,
$\gamma^{\prime}$ is homotopic to $\gamma$ through $\varepsilon$-horizontal
curves, relative to the complement of all
$\nu_{\tau}(U_{i}\times\\{t_{i,j}\\})$.

There is now a subtlety that we need to take care of: some of the
${\mathcal{C}}_{i,j}$ enter the region $\nu_{\tau}(\partial(K\times I))$,
destroying the horizontality condition there. This cannot be addressed using
the controllers themselves, since the collection $\\{U_{i}\\}$ does not cover
$\nu_{\tau}(\partial K)$ completely. We address it instead using regularity.
Note that this is not a technical point: due to the phenomenon of rigidity
(Subsection 2.2), the usage of regularity at this stage cannot be avoided.

####  8.3.3. Horizontalisation

The family $\gamma^{\prime}$ is horizontal over ${\mathcal{O}p}(\partial K)$,
but not necessarily over the whole band $\nu_{\tau}(\partial K)$, due to our
insertion of controllers. To address this, we reintroduce horizontality, at
the cost of losing control of the endpoints. Namely, given $\mu>0$, any
sufficiently small size-at-rest $S$ will guarantee that there is a family of
horizontal curves $(\alpha(k))_{k\in\nu_{\tau}(\partial K)}$ that satisfies:

  * i.

$\left|\alpha-\gamma|_{\nu_{\tau}(\partial K)}\right|_{C^{0}}<\mu$.

  * i’.

${{\operatorname{len}}}(\alpha(k))<{{\operatorname{len}}}(\gamma(k))+\mu$.

  * ii.

$\alpha$ is homotopic to $\gamma|_{\nu_{\tau}(\partial K)}$ through horizontal
curves.

  * iii.

This homotopy is relative in the parameter to $\nu_{\tau/2}(\partial K)$.

  * iii’.

The homotopy is relative to $\\{t=0\\}$ in the domain.

The family $\alpha$ is constructed inductively, one chart $U_{i}\times I_{j}$
at a time, increasingly in $j$, and arbitrarily in $i$. The inductive step
consists of using each adapted chart $(V_{i,j},\Psi_{i,j})$ to see
$\gamma^{\prime}|_{U_{i}\times I_{j}}$ as a family of curves in the graphical
model $V_{i,j}$. We can then apply the lifting Lemma 5.3 to the projection
$\pi\circ\gamma^{\prime}|_{U_{i}\times I_{j}}$. The lifting process can be
completed over $[0,1]$ because a small $S$ means that
$\pi\circ\gamma^{\prime}|_{U_{i}\times I_{j}}$ is close to
$\pi\circ\gamma|_{U_{i}\times I_{j}}$. This also justifies Conditions (i) and
(i’). The families are homotopic two one another by the birth of the
controller, proving Condition (ii). Lastly, projecting and lifting leaves
horizontal curves invariant, proving Conditions (iii) and (iii’).

We now choose $\mu$ small enough so that the bounds provided by Conditions (i)
and (i’) allows us to invoke the interpolation Lemma 5.5. This allows us to
interpolate through $\varepsilon$-horizontal curves between $\alpha$ and
$\gamma^{\prime}$ in the region $\\{\tau^{\prime}<d(k,\partial K)<\tau\\}$.
The resulting family of curves $\alpha^{\prime}$:

  * •

agrees with $\gamma^{\prime}$ in the complement of $\nu_{\tau}(\partial K)$,

  * •

agrees with $\gamma$ in $\nu_{\tau/2}(\partial K)$,

  * •

is horizontal in $\nu_{\tau^{\prime}}(\partial K)$,

  * •

contains controllers along $U_{i}\times\\{t_{i,j}\\}$.

The issue is that $\alpha(k)(1)$ may be different from
$\gamma^{\prime}(k)(1)=\gamma(k)(1)$ in the region $\\{\tau/2<d(k,\partial
K)<\tau^{\prime}\\}$. This is a feature of the lifting process. Nonetheless,
according to Lemma 5.3, the endpoints are $\mu$-close.

####  8.3.4. Controllability

By hypothesis, the curves $\gamma|_{\nu_{\tau}(\partial K)}$ are horizontal
and regular. It follows that $\gamma|_{\nu_{\tau}(\partial K)}$ is a
controllable family444Do note that $\nu_{\tau}(\partial K)$ is not compact
but, since $\tau$ is arbitrary, we can take a slightly smaller compact
neighbourhood of $\partial K$ and carry the argument there., according to
Lemma 7.4. We deduce that there are constants $c,\delta>0$ such that any
$\delta$-displacement of their endpoints can be followed by a homotopy of the
curves themselves, through horizontal curves, that is $c\delta$-small.

We claim that the family $\alpha^{\prime}|_{\nu_{\tau^{\prime}}(\partial K)}$
is also controllable, with constants $2c$ and $\delta/2$, as long as $S$ is
sufficiently small. Indeed, consider the variations $F$ of $\gamma$ that yield
controllability. Then, the homotopy lifting property, applied to $F$ and the
homotopy of horizontal curves connecting
$\gamma|_{\nu_{\tau^{\prime}}(\partial K)}$ with
$\alpha^{\prime}|_{\nu_{\tau^{\prime}}(\partial K)}$, yields corresponding
variations for small values of the homotopy parameter. They exist for the
whole homotopy if $\mu$ is assumed to be sufficiently small.

Then, assuming that $S$ is sufficiently small, we have that $\mu<\delta/2$ and
we can use the controllability of
$\alpha^{\prime}|_{\nu_{\tau^{\prime}}(\partial K)}$ to yield a family
$\gamma^{\prime\prime}:K\to{\mathfrak{Emb}}^{\varepsilon}(I;M,{\mathcal{D}})$
that:

  * •

agrees with $\gamma$ in ${\mathcal{O}p}(\partial(K\times I))$.

  * •

is horizontal and regular if $k\in\nu_{\tau^{\prime}}(\partial K)$.

  * •

has a family of controllers, still denoted by $\\{{\mathcal{C}}_{i,j}\\}$,
along $U_{i}\times\\{t_{i,j}\\}$.

The situation is depicted in Figure 22.

Figure 22. For each element in the cover ${\mathcal{U}}$, a controller has
been introduced. These are shown as green thin rectangles. Close to the
boundary, some controllers (in red) enter its $\tau$-neighbourhood.
Horizontality is reestablished using the variations given by local
controllability.

####  8.3.5. Discussion about parameters

In this subsection we added controllers $\\{{\mathcal{C}}_{i,j}\\}$ to the
family $\gamma$. Each curve $\gamma(k)$ crosses at most $N.C_{1}$ controllers;
here $C_{1}$ is the upper bound for the intersections between elements in
$\\{U_{i}\\}$. Inserting the controllers produces a deformation $\alpha$ of
$\gamma|_{\nu_{\tau^{\prime}}(\partial K)}$ through horizontal curves. This
deformation displaces the endpoints an amount $\mu$, which we can estimate.
For each controller inserted, the endpoints move a magnitude $O(S)$, the size-
at-rest. This implies that $\mu$ is bounded above by $O(S).N.C_{1}$.

Furthermore, in Subsection 8.2 we showed that $\gamma$ satisfies the size
estimates given in Equations 2 and 3, involving $r_{1}$ and $l$. We want
$\alpha$ and thus $\gamma^{\prime}$ to satisfy these as well. To this end, the
$C^{0}$-distance between $\gamma$, $\alpha$, and $\gamma^{\prime}$ must be
much smaller than $r_{1}$ and $l$.

These considerations force the choice $S<<1/N<<l,r_{1}$.

###  8.4. Using the controllers

In this subsection we complete the proof of Proposition 8.1. The idea is to
use projection and lifting to replace $\gamma^{\prime\prime}$ by a family of
horizontal curves $\beta$, whose endpoints are incorrect. The endpoints will
then be adjusted thanks to the presence of controllers.

####  8.4.1. Horizontalisation

Much like in the proof of Theorem 1.4, we first apply Lemma 5.8 to
$\gamma^{\prime\prime}$ at each time $t_{j}=j/N\in I$. This can be done in a
$C^{0}$-small way, through $\varepsilon$-horizontal curves, by making the
newly created horizontal region sufficiently small. This is relative to
$\nu_{\tau^{\prime}}(\partial K)$ in the parameter. The resulting family is
denoted by $\gamma_{0}$. The proof now focuses on a concrete interval $I_{j}$;
the argument is identical for all of them.

####  8.4.2. Horizontalisation again

We have a family $\gamma_{0}$ with values in
${\mathfrak{Emb}}^{\varepsilon}(I_{j};M,{\mathcal{D}})$ that along the
boundary of $K\times I_{j}$ is horizontal. Suppose $1/N$ is sufficiently
small. We can argue as in Subsection 8.3.3 to construct a $K$-family $\beta$
with values in ${\mathfrak{Emb}}(I_{j};M,{\mathcal{D}})$ such that:

  * •

$\beta(k)=\gamma_{0}(k)$ if $d(k,\partial K)\leq\tau^{\prime}$.

  * •

$\beta(k)(t)=\gamma_{0}(k)(t)$ if $t\in{\mathcal{O}p}(\\{t_{j}\\})$.

  * •

Write $\pi$ for the projection to the base given by each graphical model
$V_{i,j}$. Then, the $C^{\infty}$-closeness of $\pi\circ\beta$ and
$\pi\circ\gamma_{0}$ is controlled by $S$.

  * •

In particular, the two families are homotopic through $\varepsilon$-horizontal
curves.

In particular, the controllers of the family $\gamma_{0}$ define controllers
for $\beta$. We still denote them by $\\{{\mathcal{C}}_{i,j}\\}_{i}$. The goal
now is to use these to produce a homotopy of horizontal curves between $\beta$
and the claimed $\widetilde{\gamma}|_{K\times\\{1\\}}$.

####  8.4.3. Adjusting the endpoint over one chart

The difference $e=|\gamma_{0}(k)(t_{j+1})-\beta(k)(t_{j+1})|$ is certainly
bounded above by $r_{1}$. However, as explained in Subsection 4.2.1, it can
also be bounded above by $C_{2}.r_{1}.l$. It follows that we should impose
$l<<1/(C_{2}.r_{1})$ to make $e$ much smaller than $r_{1}$. By making this
choice, Equations 2 and 3 hold for $\beta$.

We now perform induction on $i$ to correct this difference. We start with the
base case $i=1$, so we work over the chart $U_{1}$. Adjusting the estimated-
displacement of the controller ${\mathcal{C}}_{1,j}$ yields a homotopy of
horizontal curves

$\widetilde{\beta}:U_{1}\times
A\subset{\mathbb{R}}^{n-q}\quad\longrightarrow\quad{\mathfrak{Emb}}(I_{j};M,{\mathcal{D}})$

with $\widetilde{\beta}(k,0)=\beta(k)$. The variable $a\in A$ measures the
endpoint displacement introduced by the controller vertically. We package this
as an endpoint map

$\widetilde{\beta}(-,-)(t_{j+1}):U_{1}\times
A\longrightarrow{\mathbb{R}}^{n-q}$

which satisfies the following error estimate:

$\widetilde{\beta}(k,a)(t_{j+1})=\beta(k)(t_{j+1})+a.(1+C_{3}.(r_{1}+|a|+\delta)).$

According to Lemma 4.5, the constant $C_{3}$ appearing as the coefficient of
the error term $(r_{1}+|a|)$ is independent of the adapted chart and thus
independent of the controller. This forces us to choose
$\delta,r_{1}<<1/C_{3}$. The quantity $|a|$ will be of magnitude $e$ and thus
smaller than $r_{1}$.

This choice tells us that
$\widetilde{\beta}(k,-)(t_{j+1})|_{{\mathbb{D}}_{2r_{1}}}$ is an embedding
whose image contains ${\mathbb{D}}_{r_{1}}$, for all $k\in V_{1}$. In
particular, it contains the desired endpoint $\gamma_{0}(k)(t_{j+1})$. The
inverse function theorem (Lemma 7.3) defines for us a unique function
$a:V_{1}\rightarrow{\mathbb{D}}_{2r_{1}}$ so that
$\widetilde{\beta}(k,a(k))(t_{j+1})=\gamma_{0}(k)(t_{j+1})$.

We now cut-off the function $a$, in order to make the construction relative to
the boundary of $U_{1}\times I_{j}$. Fix a constant $\rho>0$ and write
$W_{i}\subset U_{i}$ for a domain covering $U_{i}$ up to a
$\rho$-neighbourhood of its boundary. We require that the family $\\{W_{i}\\}$
is an open cover of $K\setminus\nu_{\tau^{\prime}}(\partial K)$. This imposes
$\rho<<1/N,\tau$. This allows us to introduce a cut-off function
$\chi_{1}:K\rightarrow[0,1]$ that is one in $W_{1}$ and zero along $\partial
U_{1}$. We set $\beta_{1}(k):=\widetilde{\beta}(k,\chi(k).a(k))$. This is a
family of horizontal curves such that:

  * •

$\beta_{1}(k)$ is horizontal.

  * •

$\beta_{1}$ is homotopic to $\beta$ as maps into
${\mathfrak{Emb}}(I_{j};M,{\mathcal{D}})$.

  * •

The base projections of $\beta_{1}(k)$ and $\gamma_{0}(k)$ agree for all
$k\in{\mathcal{O}p}(\partial U_{1})$.

  * •

$\beta_{1}(k)$ and $\gamma_{0}(k)$ agree over $t\in{\mathcal{O}p}(\partial
I_{j})$, for every $k\in W_{1}$.

The third property allows us to homotope $\beta_{1}$ to a family
$\gamma_{1}:U_{1}\to{\mathfrak{Emb}}^{\varepsilon}(I_{j};M,{\mathcal{D}})$
that is horizontal over $W_{1}$ and agrees with $\gamma_{0}$ in
${\mathcal{O}p}(\partial U_{1})$. We can then use $\gamma_{0}$ to extend
$\gamma_{1}$ to a family
$K\to{\mathfrak{Emb}}^{\varepsilon}(I_{j};M,{\mathcal{D}})$. The two are
homotopic, relative to endpoints and to the complement of $U_{1}$, thanks to
the second property and the fact that $\beta$ was the horizontal lift of
$\gamma_{0}$.

####  8.4.4. The inductive argument

The $i_{0}$th inductive step follows the exact same argument. It produces a
family
$\gamma_{i_{0}}:K\to{\mathfrak{Emb}}^{\varepsilon}(I_{j};M,{\mathcal{D}})$
that is horizontal over the union $\cup_{i\leq i_{0}}W_{i}$. The observation
to be made is the following. Suppose $U_{i_{0}}$ intersects non-trivially some
previous $U_{i}$. Then, in the overlap $U_{i_{0}}\cap W_{i}$, we have that the
family $\gamma_{i_{0}-1}$ is already horizontal over $W_{i}$. It follows that
the associated horizontal family $\beta$ agrees with $\gamma_{i_{0}-1}$ over
$W_{i}$, due to the uniqueness of horizontal lifts. In particular, when we use
the controller ${\mathcal{C}}_{i_{0},j}$ to produce a fully controllable
family over $W_{i_{0}}$, we see that no adjustments must be made over $W_{i}$,
since the endpoint is already correct there. This is immediate from the
uniqueness provided by the inverse function theorem.

Since the $W_{i}$ cover $K\setminus\nu_{\tau}(\partial K)$, the inductive
argument produces the required homotopy $\widetilde{\gamma}$. The proof of
Proposition 8.1 is complete.

####  8.4.5. Final discussion about constants

Two quantities had to be controlled in this section. The first was the error
suffered by the controllers. This was proportional (Lemma 7.7) to the radius
of the graphical models (and could therefore be controlled by $r_{1}$) and by
the smoothing parameter $\delta$. The other quantity was the $C^{0}$-distance
$e$ between $\beta$ and $\gamma_{i-1}$ (particularly at their endpoints). This
was controlled by setting $l<<r_{1}$.

We note that the embedding condition enters the discussion only in the choice
of $r_{1}$. Namely: once the error of the controllers has been bounded, it
follows that the horizontal curves produced by the controller are indeed
embedded. See Lemma 7.7.

The summary is that we require the chain of inequalities
$\delta<<S<<1/N<<l<<r_{1}$. $\Box$

Figure 23. The inductive process. In the $i$th step of the induction we
introduce horizontality over the region $W_{i}$, in dark blue. Appropriate
cut-off functions have been introduced in $V_{i}\setminus W_{i}$ (light red)
to make this homotopy relative to the boundary of the model. At a later stage,
we consider some $U_{i_{0}}$ overlapping with $W_{i}$. In the overlap, the
step $i_{0}$ homotopy is constant, thanks to horizontality.

######  Remark 8.2.

The reader experienced in $h$-principles may wonder why we do not use a
triangulation of $K\times I$, in general position with respect to $\pi_{K}$,
to argue. Indeed, this would have the added advantage of localising our
arguments to balls that do not interact with one another. This was not the
case in the proof we presented.

The issue with the triangulation approach is that we would have to make a
first homotopy that makes our curves horizontal along the codimension-1
skeleton. This is certainly possible, but we have no guarantee that the
produced curves are themselves regular. This is absolutely necessary, since we
need to be able to introduce controllers at the bottom of each top-cell.

In fact, this can be made to work. The local integrability of micro-regular
curves (i.e. curves that are in particular regular over any interval) was
proven in [35, 8], which would allow us to produce regular curves along the
skeleton. However, it seemed preferable to us to keep the proof self-contained
and not invoke additional results. $\Box$

###  8.5. Other $h$-principles for horizontal curves

We now discuss how the proof of Theorem 1.4 adapts to prove Theorems 1.7 and
1.6.

###### Proof of Theorem 1.7.

The absolute statement can be reduced to proving the relative $h$-principle
over the interval (i.e. the analogue of Proposition 8.1). The proof of the
relative statement is identical to the one we presented for embeddings. The
reader can check that the proof goes through line by line. Instead, it is more
interesting to point out how the proof simplifies for immersions.

First note that the construction of tangles (Section 6) and controllers
(Section 7) is less involved if we do not need to take care of self-
intersections. In particular, we do not need to develop all the explicit
models shown in Figures 14 and 15. Similarly, in Subsection 8.4.5, we do not
need to control errors in order to ensure the controller produces embedded
curves.

This crucial difference explains why the statement for immersions goes through
in $\dim(M)=3$. Achieving embeddedness parametrically was not possible in
dimension $3$, but one can certainly produce tangles and controllers that are
immersed. ∎

Similarly:

###### Proof of Theorem 1.6.

The statement reduces once again to the $h$-principle for horizontal paths,
relative both in parameter and domain. We now indicate the differences with
respect to the proof of Proposition 8.1.

First: since there is no first order formal data, we do not need to set up a
convex integration argument to achieve $\varepsilon$-horizontality. Second: we
use the covering arguments as they appear in Subsection 8.2 but we run into
issues in Subsection 8.3, when we try to introduce controllers. In Proposition
7.6 we explained how to introduce them when our curves are embedded/immersed,
which may not be the case here. To address this, we use the “stopping trick”;
see Figure 24. We explain it next.

Given a smooth horizontal arc $\gamma:[a,b]\rightarrow(M,{\mathcal{D}})$, we
can precompose it with a non-decreasing map $\phi:[a,b]\rightarrow[a,b]$ that
is the identity at the endpoints and is constant in
${\mathcal{O}p}(\\{(a+b)/2\\})$. Since $\phi$ is homotopic to the identity rel
boundary, it defines a homotopy between $\gamma$ and a horizontal curve
$\gamma\circ\psi$ whose parametrisation is stationary at the middle point. The
curve $\gamma\circ\psi$ is thus regular. Even more: suppose
$\nu:[0,1]\rightarrow(M,{\mathcal{D}})$ is some other horizontal curve with
$\nu(0)=\gamma\circ\psi((a+b)/2)$. Then $\gamma\circ\psi$ is homotopic to a
smooth horizontal curve $\gamma^{\prime}$ whose image is the concatenation

$(\gamma\circ\psi|_{[(a+b)/2,b]})\bullet\bar{\nu}\bullet\nu\bullet(\gamma\circ\psi|_{[a,(a+b)/2]}).$

Since $\nu$ is arbitrary, it may be chosen to be a curve contained in a
graphical model around $\gamma\circ\psi((a+b)/2)$ and projecting to the base
as whatever direction we require.

The conclusion is that any family $\gamma:K\longrightarrow{\mathcal{L}}(M)$,
horizontal at the boundary, can also be assumed to be regular along $\partial
K$, up to a homotopy through horizontal curves. Furthermore, it can be assumed
to be immersed whenever controllers need to be introduced. This concludes the
proof. ∎

Figure 24. The stopping trick. The space of horizontal loops is extremely
flexible. Any family can be homotoped to a regular family by introducing
stationary points in the parametrisation. This allows us to introduce
controllers.

##  9\. h-Principle for transverse embeddings

In this section we prove Theorem 1.8, the classification of transverse
embeddings.

###  9.1. The relative $h$-principle

The $h$-principle for transverse paths, relative in parameter and domain,
reads:

######  Proposition 9.1.

Let $K$ be a compact manifold. Let $I=[0,1]$. Let $(M,{\mathcal{D}})$ be a
manifold of dimension $\dim(M)>3$, endowed with a bracket–generating
distribution of corank $1$. Suppose that we are given a map
$\gamma:K\to{\mathfrak{Emb_{\mathcal{T}}}}^{f}(I;M,{\mathcal{D}})$ satisfying:

  * •

$\gamma(k)\in{\mathfrak{Emb_{\mathcal{T}}}}(I;M,{\mathcal{D}})$ for
$k\in{\mathcal{O}p}(\partial K)$.

  * •

$\gamma(k)(t)$ is positively transverse if $t\in{\mathcal{O}p}(\partial I)$.

Then, there exists a homotopy
$\widetilde{\gamma}:K\times[0,1]\to{\mathfrak{Emb_{\mathcal{T}}}}^{f}(I;M,{\mathcal{D}})$
satisfying:

  * •

$\widetilde{\gamma}(k,0)=\gamma(k)$.

  * •

$\widetilde{\gamma}(k,1)$ takes values in
${\mathfrak{Emb_{\mathcal{T}}}}(I;M,{\mathcal{D}})$.

  * •

this homotopy is relative to $k\in{\mathcal{O}p}(\partial K)$ and to
$t\in{\mathcal{O}p}(\partial I)$.

  * •

$\widetilde{\gamma}(k,s)$ is $C^{0}$-close to $\gamma(k)$ for all $s\in[0,1]$.

Observe that we do not assume that ${\mathcal{D}}$ is cooriented. We will be
able to pass to the cooriented case (and thus use $\varepsilon$-transverse
embeddings) during the proof.

###### Proof of Theorem 1.8 from Proposition 9.1.

We must prove the vanishing of the relative homotopy groups of the pair

$({\mathfrak{Emb_{\mathcal{T}}}}^{f}(M,{\mathcal{D}}),\,{\mathfrak{Emb_{\mathcal{T}}}}(M,{\mathcal{D}})).$

Given a family $\gamma$ representing a class in the $a$th relative homotopy
group, we must deform it to lie entirely in
${\mathfrak{Emb_{\mathcal{T}}}}(M,{\mathcal{D}})$. Close to
$\partial{\mathbb{D}}^{a}$ the family is transverse because transversality is
an open condition. We then fix a slice
${\mathbb{D}}^{a}\times\\{1\\}\subset{\mathbb{D}}^{a}\times{\mathbb{S}}^{1}$
and apply the transversalisation Lemma 5.13 to $\gamma$ there. This reduces
the argument to a family of paths and thus to Proposition 9.1. ∎

We henceforth focus on the proof of Proposition 9.1

###  9.2. The first triangulation step

We are given a family of formally transverse curves $\gamma$. Our first goal
is to pass to the $\varepsilon$-transverse setting. To do so, we will produce
a very fine subdivision of $K\times I$ so that we can work over balls in
$(M,{\mathcal{D}})$. We can then use the local coorientability of
${\mathcal{D}}$ to introduce $\varepsilon$-transversality.

####  9.2.1. Triangulating

We subdivide $K\times I$ using a triangulation ${\mathcal{T}}$. This should be
compared to the proof of Proposition 8.1, which used a different scheme to
localise the arguments to little balls. The reason was explained in Remark
8.2: Triangulating $K\times I$ would have led to issues due to the phenomenon
of rigidity for horizontal curves. However, there is no rigidity for
transverse curves because they are defined by an open condition.

In order to produce a triangulation, we proceed as follows. We pick a small
constant $\tau>0$ so that $\gamma$ is transverse over
$\nu_{\tau}(\partial(K\times I))$. We choose a closed domain $A\subset K\times
I$ such that $\\{A,\nu_{\tau}(\partial(K\times I))\\}$ covers $K\times I$. We
ask that $\partial A$ is smooth. We then apply the jiggling Corollary 5.2 to
$(A,\ker(d\pi_{K}))$. We do not need to introduce further subdivisions, we
simply choose ${\mathcal{T}}$ fine enough so that each top-cell
$\Delta\in{\mathcal{T}}$ is mapped by $\gamma$ to an adapted chart of
$(M,{\mathcal{D}})$.

According to Lemma 5.1, the following holds: a simplex $\Delta$ is either
transverse to the vertical or is contained in $\partial A$. In the latter
case, $\gamma$ is already transverse in ${\mathcal{O}p}(\Delta)$. It follows
that we can apply Lemma 5.14 to $\gamma$, along the codimension-$1$ skeleton,
in a manner relative to $\partial A$. This yields a homotopy of formal
transverse embeddings, relative to $\partial A$, between $\gamma$ and a family
$\gamma_{1}$. The family $\gamma_{1}$ is transverse on a neighbourhood of the
codimension-$1$ skeleton.

####  9.2.2. $\varepsilon$-transversality

Let $\Delta\in{\mathcal{T}}$ be a top-cell. Since $\gamma_{1}$ maps it to to
an adapted chart $(V,\Phi)$, we have that $\gamma_{1}|_{\Delta}$ takes values
in a manifold endowed with a coorientable distribution. Furthermore, since
$\gamma_{1}|_{\Delta}$ is formally transverse, it defines a preferred
coorientation for ${\mathcal{D}}|_{\Phi(V)}$.

According to Lemma 5.1, the top-cells of ${\mathcal{T}}$ are flowboxes,
meaning that there is a fibre-preserving embedding

$\Psi:{\mathbb{D}}^{k}\times[0,1]\longrightarrow\Delta\subset A\subset K\times
I$

whose image covers most of $\Delta$. In particular, the boundary of this
embedding may be assumed to be contained in the region where $\gamma$ is
transverse.

The conclusion is that $\Phi^{-1}\circ\gamma_{1}\circ\Psi$ is a family of
formally transverse curves that:

  * •

takes values in a cooriented graphical model $(V,{\mathcal{D}}_{V})$,

  * •

is transverse over ${\mathcal{O}p}(\partial({\mathbb{D}}^{k}\times[0,1]))$.

This implies that we can apply the $h$-principle for $\varepsilon$-transverse
curves (Subsection 3.4) to $\Phi^{-1}\circ\gamma_{1}\circ\Psi$, yielding a
family
$\gamma_{2}:{\mathbb{D}}^{k}\longrightarrow{\mathfrak{Emb_{\mathcal{T}}}}^{\varepsilon}([0,1];V,{\mathcal{D}}_{V})$
that is (positively) transverse over
${\mathcal{O}p}(\partial({\mathbb{D}}^{k}\times[0,1]))$.

We have one such family per top-cell $\Delta$. Furthermore, their domains are
disjoint. This implies that it is sufficient for us to work with each
$\gamma_{2}$ individually, homotoping them through $\varepsilon$-transverse
curves (and relative to $\partial({\mathbb{D}}^{k}\times[0,1])$) to a family
of transverse curves.

###  9.3. The second triangulation step

Let us explain how the remainder of the proof goes, morally. Our goal is to
apply the case of horizontal embeddings. The reasoning is that, whenever the
curves $\gamma_{2}(k)$ are graphical over ${\mathcal{D}}_{V}$, we can flatten
them to make them almost horizontal, which will then allow us to manipulate
them through the use of controllers. In order to project towards
${\mathcal{D}}_{V}$, we use the projection
$\pi:{\mathbb{R}}^{n}\rightarrow{\mathbb{R}}^{n-1}$ to the base of the
graphical model.

There are two issues with this idea. The first is that the family $\gamma_{2}$
may have uncontrolled length, since it was produced by the $h$-principle for
$\varepsilon$-transverse curves. To address this, we will triangulate again to
pass to small balls. The other issue will be explained afterwards.

####  9.3.1. Triangulating again…

We proceed as above, applying the jiggling Lemma 5.1 to $\gamma_{2}$ and
$({\mathbb{D}}^{k}\times[0,1],\ker(d\pi_{[0,1]}))$. We obtain a sequence of
triangulations ${\mathcal{T}}_{b}$ inducing a triangulation of the boundary
(do note that it has corners, but this is not an issue). We fix $b$ as in the
horizontal case (Subsection 8.2): each simplex $\Delta$ should have diameter
bounded above by $1/N<<l,r_{1}$. In this manner, $\gamma_{2}(\Delta)$ is
contained in an adapted chart $(U,\Psi)$ of radius $r_{1}$ and each curve in
$\gamma_{2}|_{\Delta}$ has length at most $l$ from the perspective of the
corresponding graphical model $U$. We still refer to these conditions as
Equations 2 and 3.

We now apply Lemma 5.13 to $\gamma_{2}$, along the codimension-1 skeleton of
${\mathcal{T}}_{b}$. This shows that $\gamma_{2}$ is homotopic, relative to
the boundary, to a family $\gamma_{3}$ that is transverse along the
codimension-1 skeleton. This is a $C^{0}$-small process that does not increase
the length of the curves much. It follows that the conditions given by
Equations 2 and 3 apply to $\gamma_{3}$ as well.

As we did earlier, we can embed a copy of ${\mathbb{D}}^{k}\times[0,1]$ into
each top-simplex, in a fibered manner, in such a way that its boundary lies in
the region where $\gamma_{3}$ is transverse. These domains do not interact
with one another, so we argue on each of them separately.

####  9.3.2. … and again

We now encounter the second issue. We cannot flatten $\gamma_{3}$ to make it
almost horizontal along the locus

$\Sigma:=\\{(k,t)\in
D^{k}\times[0,1]\,\mid\,d\pi(\gamma_{3}(k)^{\prime}(t))=0\\}.$

That is, the locus where the velocity vector becomes vertical. Nonetheless,
using Thom transversality [18, p. 17], we can assume that $\Sigma$ is a closed
submanifold of ${\mathbb{D}}^{k}\times[0,1]$.

Figure 25. Schematic representation of ${\mathbb{D}}^{k}\times[0,1]$. The blue
band represents a neighbourhood of the boundary. The thin green curve is the
vertical locus $\Sigma$. Its neighbourhood $W$ is shown in red. The region $B$
is a slight thickening of the complement.

The family $\gamma_{3}$ is vertical along $\Sigma$ and
$\varepsilon$-transverse in general. It follows that $\gamma_{3}$ is
positively transverse on a neighbourhood $W\supset\Sigma$. We can therefore
find a closed subdomain $B\subset{\mathbb{D}}^{k}\times[0,1]$, disjoint from
$W$, whose smooth boundary lies in the region where $\gamma_{3}$ is
transverse. See Figure 25.

We can then proceed as above, applying Lemma 5.1 to $\gamma_{3}$ and
$(B,\ker(d\pi_{[0,1]}))$. We do not need the resulting triangulation
${\mathcal{T}}^{\prime}$ to be thin, since we already achieved quantitative
control in the previous subdivision. Applying Lemma 5.13 shows that
$\gamma_{3}$ is homotopic, relative to $\partial B$, to a family $\gamma_{4}$
that is transverse along the codimension-1 skeleton. This is a $C^{0}$-small
process.

We henceforth argue on each top simplex separately, relative to the boundary.
The punchline is that we have a family

$\gamma_{4}:{\mathbb{D}}^{k}\longrightarrow{\mathfrak{Emb_{\mathcal{T}}}}^{\varepsilon}([0,1];,{\mathcal{D}}_{U}),$

that is transverse along the boundary. Here $(U,{\mathcal{D}}_{U})$ is the
graphical model of radius $r_{1}$ that we fixed earlier (and that we used to
discuss verticality). Each curve of $\gamma_{4}$ has length bounded above by
$l$. Since we avoided $\Sigma$, we can assume that each curve $\gamma_{4}(k)$
is graphical over ${\mathcal{D}}_{U}$.

###  9.4. End of the argument

Choose $\tau>0$ small enough so that $\gamma_{4}$ is transverse in a
$\tau$-neighbourhood of the boundary of ${\mathbb{D}}^{k}\times[0,1]$. We can
apply Corollary 5.9 to $\gamma_{4}$ along the slice
$D={\mathbb{D}}^{k}_{1-\tau/2}\times\\{\tau/2\\}$, in order to yield a family
that is almost transverse in ${\mathbb{D}}^{k}\times[0,\tau]$ and horizontal
in ${\mathcal{O}p}(D)$. This allows us to introduce a controller
${\mathcal{C}}$ along $D$; see Lemma 7.8. The resulting family is called
$\gamma_{5}$. It consists of embedded curves as long as $r_{1}$ was
sufficiently small (Lemma 7.7).

We now adjust the estimated-displacement of ${\mathcal{C}}$ in order to obtain
an almost transverse family $\gamma_{6}$ such that

$e=\left|\gamma_{6}(k)(1-\tau)-\gamma_{5}(k)(1-\tau)\right|$

is small. We require that

$e<<h=\left|\gamma_{5}(k)(1)-\gamma_{5}(k)(1-\tau)\right|.$

If that is the case, we can invoke the fact that $\gamma_{5}(k)$ is transverse
in the interval $[1-\tau,1]$, to extend $\gamma_{6}(k)$ to a curve that is
transverse in $[1-\tau,1]$ and agrees at the end with $\gamma_{5}(k)$. This
preserves embeddedness.

The proof concludes invoking Lemma 5.11, which adds a $C^{1}$-small
perturbation to $\gamma_{6}$ to yield a family that is transverse. $\Box$

###### Proof of Theorem 1.7.

The proof follows from the same arguments. As we already observed for
horizontal embeddings, handling the controller becomes easier in the immersion
case, since self-intersections do not need to be avoided. Because of this
reason, the statement also holds in dimension $3$. ∎

##  10\. Appendix: Technical lemmas on commutators

Given a vector field $Z$ on $M$ we write $\phi^{Z}_{t}$ for its flow at time
$t$. Given a pair of vector fields $X$ and $Y$ on $M$, we want to compare, in
a quantitative manner, the flow of their Lie bracket $\phi^{[X,Y]}_{t}$ with
the commutator of their flows $\phi_{t}^{X}$ and $\phi_{t}^{Y}$. The contents
of this appendix will be an important technical ingredient in the proof of our
main theorems.

Given $1$–parameter families (not necessarily subgroups)
$(\varphi_{t})_{t\in{\mathbb{R}}}$ and $(\psi_{t})_{t\in{\mathbb{R}}}$ in the
diffeomorphism group ${\operatorname{Diff}}(M)$, we can define in a given
local chart the map
$[\psi(t),\varphi(s)]=\varphi_{s}\circ\psi_{t}\circ\varphi_{s}^{-1}\circ\psi_{t}^{-1}(x)$.
Note that if we take $s=t$ then this map is the commutator of the families
taken for each time $t$, and we denote it by
$[\psi_{t},\varphi_{t}]:=[\psi(t),\varphi(s)]$.

######  Lemma 10.1.

Write $X=\frac{\partial}{\partial t}\Big{|}_{t=0}\phi_{t}$ and
$Y=\frac{\partial}{\partial t}\Big{|}_{t=0}\psi_{t}$ and assume
$\phi_{0}=\psi_{0}=Id$. Then the following statements hold:

  * $i)$

There exists a $2-$parametric family of diffeomorphisms
$\varepsilon_{ts}=o(ts)$ such that

$\psi_{t}\circ\phi_{s}(x)=\varepsilon_{ts}\circ\varphi^{X+Y}_{ts}(x)$

  * $ii)$

$\frac{\partial^{k}}{\partial t^{k}}\Big{|}_{t,s=0}[\psi(t),\phi(s)](x)=0$ for
any $k\in\mathbb{N}$,

  * $iii)$

$\frac{\partial^{k}}{\partial s^{k}}\Big{|}_{t,s=0}[\psi(t),\phi(s)](x)=0$ for
any $k\in\mathbb{N}$,

  * $iv)$

$\frac{1}{2}\frac{\partial^{2}}{\partial t\partial
s}\Big{|}_{t=0}[\psi(t),\phi(s)](x)=[X,Y]$.

  * $v)$

There exists a $2-$parametric family of diffeomorphisms
$\varepsilon(ts)=o(ts)$ such that

$[\phi_{t},\psi_{s}]=\varepsilon_{ts}\circ\varphi^{[X,Y]}_{ts}.$

###### Proof.

Part $i)$ follows from Taylor’s Remainder Theorem applied to the composition
map $\psi_{t}\circ\phi_{s}(x)$, $ii)$ and $iii)$ are obvious, $iv)$ follows
from the definition of Lie Bracket and $v)$ follows from an application of
Taylor’s Remainder Theorem together with $ii),iii)$ and $iv)$. ∎

######  Remark 10.2.

A trivial but rather useful observation that we will eventually make use of is
the following one. If $\phi_{t}$, $\psi_{t}$, are two flows such that for some
$1-$parametric family of diffeomorphisms $\varepsilon_{t}$

$\phi_{t}=\varepsilon_{t}\circ\psi_{t}$

then $\varepsilon_{t}$ is also a flow since it is the composition of two flows
$\varepsilon_{t}=\psi^{-1}_{t}\circ\phi_{t}$. Moreover, if $\varepsilon=o(t)$,
then there exists another flow $\tilde{\varepsilon}=o(t)$ such that

$\psi_{t}=\tilde{\varepsilon}_{t}\circ\phi_{t}.$

For this last statement just note that

$\phi_{t}=\varepsilon_{t}\circ\psi_{t}\Longrightarrow\phi_{t}\circ\psi_{t}^{-1}=o(t)\Longrightarrow\psi_{t}\circ\phi_{t}^{-1}=o(t)$

and thus there exists some flow $\tilde{\varepsilon}_{t}=o(t)$ such that
$\psi_{t}=\tilde{\varepsilon}_{t}\circ\phi_{t}$.

The following Lemma formalizes how errors inside a commutator of
$1-$parametric families of diffeomorphisms can be taken out the bracket
expressions when comparing to the flow of the respective brackets.

######  Lemma 10.3.

Consider a flow $\varepsilon(t)=o(t)$ and write $X=\frac{\partial}{\partial
t}\Big{|}_{t=0}\phi_{t}$, $Y=\frac{\partial}{\partial t}\Big{|}_{t=0}\psi_{t}$
and assume $\phi_{0}=\psi_{0}=Id$. Then there exists a $2-$parametric family
of diffeomorphisms $\tilde{\varepsilon}(ts)=o(ts)$ such that

$[\varphi_{s}^{X},\varepsilon_{t}\circ\varphi_{t}^{Y}]=\tilde{\varepsilon}_{ts}\circ\varphi^{[A,B]}_{ts}.$

###### Proof.

The result follows from a direct application of points $i)$ and $v)$ from
Lemma 10.1. ∎

######  Remark 10.4.

In particular, if we take $s=t$ in this previous lemma, we get that
$\tilde{\varepsilon}_{t^{2}}$ is an actual flow, since it is the composition
of two flows as in Remark 10.2,

$\tilde{\varepsilon}_{t^{2}}=\left(\varphi^{[A,B]}\right)^{-1}_{t^{2}}\circ[\varphi_{t}^{X},\varepsilon_{t}\circ\varphi_{t}^{Y}]$

The same remark is true for the family $\varepsilon_{ts}$ in $v)$ from Lemma
10.1.

######  Lemma 10.5.

Consider $X=\frac{\partial}{\partial t}\Big{|}_{t=0}\phi_{t}$,
$Y=\frac{\partial}{\partial t}\Big{|}_{t=0}\psi_{t}$ and assume
$\phi_{0}=\psi_{0}=Id$. Then, if $\phi_{t}=\varepsilon_{t}\circ\psi_{t}$ for
certain $\varepsilon_{t}=o(t)$, then there exists some other
$\tilde{\varepsilon}_{t}=o(t)$ such that

$\phi_{t}=\psi_{t}\circ\tilde{\varepsilon}_{t}$

###### Proof.

By Lemma 10.3 there exists some $h_{t}=o(t)$ such that
$h_{t^{2}}\circ\varepsilon_{t}^{-1}\circ\psi^{-1}_{t}\circ\varepsilon_{t}\circ\psi_{t}=Id$.
So, we have that
$h_{t^{2}}\circ\varepsilon_{t}^{-1}\circ\psi^{-1}_{t}=\phi^{-1}_{t}$. The
result follows from taking inverse flows at both side of the equation. ∎

######  Lemma 10.6.

Write $X=\frac{\partial}{\partial t}\Big{|}_{t=0}\phi_{t}$ and
$Y=\frac{\partial}{\partial t}\Big{|}_{t=0}\psi_{t}$ and assume
$\phi_{0}=\psi_{0}=Id$. Then we have the following estimation:

$\phi_{t}^{-1}\circ\psi_{s}\circ\phi_{t}=\varepsilon_{ts}\circ\varphi^{X+t[X,Y]}_{s}.$

###### Proof.

First, note that

$\displaystyle\phi_{t}^{-1}\circ\psi_{s}\circ\phi_{t}=\psi_{s}\circ\left[\phi_{t},\psi_{s}\right]=\psi_{s}\circ\tilde{\varepsilon}_{ts}\circ\varphi^{[X,Y]}_{ts}=\psi_{s}\circ\tilde{\varepsilon}_{ts}\circ\varphi^{t[X,Y]}_{s},$

where the second equality follows by Lemma 10.1. Nevertheless, this implies
the existence of a $2-$parametric family of diffeomorphisms
$\epsilon_{ts}=o(ts)$ such that
$\psi_{s}\circ\varphi^{t[X,Y]}_{s}\circ\epsilon_{ts}$. But, from point $i)$ in
Lemma 10.1 applied to the composition $\psi_{s}\circ\varphi^{t[X,Y]}_{s}$,
there exist some other $\tilde{\epsilon}_{ts}=o(ts),\varepsilon_{ts}=o(ts)$
such that

$\psi_{s}\circ\varphi^{t[X,Y]}_{s}\circ\epsilon_{ts}=\varphi_{s}^{Y+t[X,Y]}\circ\tilde{\epsilon}_{ts}=\varepsilon_{ts}\circ\varphi_{s}^{Y+t[X,Y]}$

thus yielding the claim.

∎

######  Definition 10.7.

Given two $1-$parametric families of diffeomorphisms $\varphi_{s}$,
$\psi_{t}$, we define their $k-$th iterated commutator
$\left[\psi_{t},\varphi_{s}\right]^{\\#k}$ as follows

$\left[\psi_{t},\varphi_{s}\right]^{\\#k}:=\left(\left(\varphi_{\frac{s}{\sqrt{k}}}^{-1}\right)\circ\left(\psi^{-1}_{\frac{t}{\sqrt{k}}}\right)\circ\left(\varphi_{\frac{s}{\sqrt{k}}}\right)\circ\left(\psi_{\frac{t}{\sqrt{k}}}\right)\right)^{k}.$

The reason for parametrizing the flows by $\frac{s}{\sqrt{k}}$ in the
definition of iterated commutator is justified by the following proposition.

######  Proposition 10.8.

Write $X=\frac{\partial}{\partial t}\Big{|}_{t=0}\phi_{t}$ and
$Y=\frac{\partial}{\partial t}\Big{|}_{t=0}\psi_{t}$ and assume
$\phi_{0}=\psi_{0}=Id$. Then there exists $\varepsilon_{t}=o(t)$ such that

$\left[\psi_{t},\varphi_{s}\right]^{\\#k}=\varepsilon_{ts}\circ\varphi^{[X,Y]}_{ts}$

###### Proof.

By the definition of the $k-$th iterated commutator of flows and by Lemma
10.1, there exist flows
$\varepsilon^{1}_{t}=o(t),\cdots,\varepsilon^{k}_{t}=o(t)$ such that

$\left[\psi_{t},\varphi_{t}\right]^{\\#k}=\left(\varepsilon^{1}_{ts/k}\circ\varphi_{ts/k}^{[X,Y]}\right)\circ\cdots\circ\left(\varepsilon^{k}_{ts/k}\circ\varphi_{ts/k}^{[X,Y]}\right)$

But by Lemma 10.5 there exist flows
$\tilde{\varepsilon}^{1}_{t}=o(t),\cdots,\tilde{\varepsilon}^{k}_{t}=o(t)$
such that

$\left(\varepsilon^{1}_{ts/k}\circ\varphi_{ts/k}^{[X,Y]}\right)\circ\cdots\circ\left(\varepsilon^{k}_{ts/k}\circ\varphi_{ts/k}^{[X,Y]}\right)=\tilde{\varepsilon}^{1}_{ts/k}\circ\cdots\circ\tilde{\varepsilon}^{k}_{ts/k}\circ\left(\varphi_{ts/k}^{[X,Y]}\right)^{k}$

and so the claim follows. ∎

With this battery of technical results at our disposal, we can compare how
taking a given bracket expression behaves with respect to taking flows [31,
Theorem 1]:

######  Proposition 10.9.

Let $X_{1},X_{2},\cdots,X_{\lambda}$ be (possibly repeated) vector fields on a
manifold $M$. Then, for any bracket expression $A(-,\cdots,-)$ of length
$\lambda$ there exists a flow $\varepsilon_{t}=o(t)$ such that

$A\left(\varphi_{t}^{X_{1}},\cdots,\varphi^{X_{\lambda}}_{t}\right)=\varepsilon_{t^{\lambda}}\circ\phi^{A(X_{1},\cdots,X_{\lambda})}_{t^{\lambda}}.$

###### Proof.

We proceed by induction on the length of the formal bracket–expression:

  * •

For $k=2$ the result holds by Proposition 10.8.

  * •

The Induction Hypothesis (IH) says that the statement holds for all
expressions of length $k^{\prime}<k$. By definition, if $A(-,\cdots,-)$ is an
expression of length $k$, there exists $i<k$ and an integer $m$ such that
$A(X_{1},\cdots,X_{k})=[B(X_{1},\cdots,X_{i}),C(X_{i+1},\cdots,X_{k})]^{\\#m}$,
with $B()$ of length $i$ and $C()$ of length $k-i$. Computing we see that
there are flows $f_{t}=o(t)$ and $g_{t}=o(t)$ such that:

(4) $\displaystyle A(\phi^{X_{1}}_{t},\cdots,\phi^{X_{k}}_{t})=$
$\displaystyle\left[B(\phi^{X_{1}}_{t},\cdots,\phi^{X_{i}}_{t}),C(\phi^{X_{i+1}}_{t},\cdots,\phi^{X_{k}}_{t})\right]^{\\#m}$
$\displaystyle\overset{\text{IH}}{=}$
$\displaystyle\left[f_{t^{i}}\circ\phi^{B(X_{1},\cdots,X_{i})}_{t^{i}},g_{t^{k-i}}\circ\phi^{C(X_{i+1},\cdots,X_{k})}_{t^{k-i}}\right]^{\\#m}$

By an application of Proposition 10.8 first, and by Lemma 10.3, there exists a
flow $\varepsilon_{t}=o(t)$ such that

(5)
$\displaystyle\left[f_{t}\circ\phi^{B(X_{1},\cdots,X_{i})}_{t^{i}},g_{t}\circ\phi^{C(X_{i+1},\cdots,X_{k})}_{t^{k-i}}\right]^{\\#m}=\varepsilon_{t^{k}}\circ\left[\phi^{B(X_{1},\cdots,X_{i})}_{t^{i}},\phi^{C(X_{i+1},\cdots,X_{k})}_{t^{k-i}}\right]$

But, since
$\left[\phi^{B(X_{1},\cdots,X_{i})}_{t^{i}},\phi^{C(X_{i+1},\cdots,X_{k})}_{t^{k-i}}\right]=\phi^{A(X_{1},\cdots,X_{k})}_{t^{k}}$
the result follows from combining (4) and (5).

∎

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