# Escape from compact sets of normal curves in Carnot groups

Enrico Le Donne  E. Le Donne; University of Fribourg, Chemin du Musée 23, 1700
Fribourg, Switzerland & Department of Mathematics and Statistics
P.O. Box 35, FI-40014
University of Jyväskylä, Finland
enrico.ledonne<EMAIL_ADDRESS>and  Nicola Paddeu  N. Paddeu;
University of Fribourg, Chemin du Musée 23, 1700 Fribourg, Switzerland
nicola.paddeu<EMAIL_ADDRESS>
(Date: 6 April 2023)

###### Abstract.

In the setting of subFinsler Carnot groups, we consider curves that satisfy
the normal equation coming from the Pontryagin Maximum Principle. We show
that, unless it is constant, each such a curve leaves every compact set,
quantitatively. Namely, the distance between the points at time 0 and time $t$
grows at least of the order of $t^{1/s}$, where $s$ denotes the step of the
Carnot group. In particular, in subFinsler Carnot groups there are no periodic
normal geodesics.

###### Contents

  1. 1 Introduction
    1. 1.1 The basic case of loops in subRiemannian Carnot groups
    2. 1.2 The strategy of the proof
    3. 1.3 Organization of the paper
  2. 2 Preliminaries
    1. 2.1 SubFinsler geometry and Carnot groups
    2. 2.2 Homogeneous spaces and self-similar distances
    3. 2.3 Pontryagin Maximum Principle and normal curves
  3. 3 Proof of main results
    1. 3.1 The differential of the dilations
    2. 3.2 Proof of Theorem 1.1
    3. 3.3 Proof of Corollary 1.2
  4. 4 Some examples
    1. 4.1 Curves with end-point arbitrarely close to the origin
    2. 4.2 Optimality of the exponent in Theorem 1.1

##  1\. Introduction

This paper originates from the following question: can subRiemannian manifolds
have arbitrarily short geodesic loops? Important subRiemannian manifolds are
Carnot groups, which have dilation structures and therefore are self-similar.
Hence, for these spaces, the question rephrases as: do geodesic loops exist in
a Carnot group? We remark that not only geodesic loops do not exist in the
Euclidean spaces, but also do not exist in normed vector spaces with strictly
convex norm. Finite-dimensional normed vector spaces are exactly the Carnot
groups of nilpotency step 1, equipped with geodesic distances. We refer to
[13] and [14] for an introduction to Carnot groups and the fact that when
equipped with their Carnot-Carathéodory distances they are exactly the metric
spaces that are homogeneous, locally compact, geodesic, and admit dilations.
Some difficulties in Carnot groups are that the distance function may not be
convex, that at every scale there exist pairs of points joined by more than
one geodesic, and geodesics may not be globally length-minimizing. However,
the biggest difficulty, as in other subRiemannian problems, is the presence of
geodesics that are singular points of the endpoint map, which we call the
abnormal geodesics. By the Pontryagin Maximum Principle we know that if a
geodesic is not abnormal, then it satisfies a geodesic equation called the
normal equation. The curves satisfying the normal equation, called normal
curves (or _normal trajectories_), are more manageable. In this paper we focus
on normal curves and we prove that, except for the constant ones, they cannot
form loops and actually they leave every compact set in a quantitative way.
The main result of this paper is the following statement.

######  Theorem 1.1 (Normal curves leave compact sets).

Let $G$ be a subFinsler Carnot group of step $s$. Fix a norm $N$ on the space
of right-invariant co-vectors. Then, there exists a constant $\epsilon>0$ such
that for every normal curve $\gamma:\mathbb{R}\rightarrow G$ parametrized by
arc-length there holds

(1)
$d(\gamma(t),\gamma(t^{\prime}))\geq\frac{\epsilon}{N(\lambda)^{\frac{1}{s}}}|t-t^{\prime}|^{\frac{1}{s}}-1,\
\ \forall t,t^{\prime}\in\mathbb{R},$

for every co-vector $\lambda$ associated to $\gamma$, see Definition 33.

As a consequence we rule out the presence of some geodesic loops.

######  Corollary 1.2.

In subFinsler Carnot groups normal loops are constant.

We stress that the above corollary has been know to be true in step 2 Carnot
groups (by the complete integration of geodesics) and in jet spaces by the
work of Bravo-Doddoli [5]. Moreover, a similar question arose in geometric
group theory of the large-scale geometry of nilpotent groups. In fact, because
of the work of Hoda [11], it would be important to prove that Carnot groups
cannot have isometric copies of the standard unit circle. The latter result is
known for finite-dimensional normed spaces and was proven by Creutz, see [8,
Lemma 1.7].

###  1.1. The basic case of loops in subRiemannian Carnot groups

We give here a simpler proof of the non-existence of normal loops in
subRiemannian Carnot groups. The same idea will be pushed to prove Corollary
1.2 and to show the quantitative result of Theorem 1.1.

###### Proof of Corollary 1.2 for subRiemannian Carnot groups..

Let $G$ be a subRiemannian Carnot group with left-invariant scalar product
$\langle\cdot,\cdot\rangle$. Let $\gamma:[0,1]\rightarrow G$ be a normal
geodesic that makes a loop. Without loss of generality we assume
$\gamma(0)=\gamma(1)=1_{G}$.

Since $\gamma$ is normal, by definition its control $u$ satisfies the normal
equation: for some $\lambda\in(T_{\gamma(1)}G)^{*}$ and all controls $v$ we
have

(2) $\lambda(\mathrm{d}\operatorname{End}_{u}v)=\langle u,v\rangle,$

where $\operatorname{End}$ is the end-point map from $1_{G}$, see for example
[1, Corollary 8.8]. When $v=u$ we get

(2bis) $\lambda(\mathrm{d}\operatorname{End}_{u}u)=||u||_{L^{2}}^{2}.$

Now the idea is to consider the curve dilated by the Carnot dilations: for
$\tau>0$ let $\delta_{\tau}:G\to G$ be the dilation of factor $\tau$. On the
one hand, we obviously have $(\delta_{\tau}\circ\gamma)(1)=1_{G}$. On the
other hand, the control of the curve $\delta_{\tau}\circ\gamma$ is $\tau u$.
Hence we have that

(3) $\operatorname{End}(\tau u)=\delta_{\tau}(\operatorname{End}(u))=1_{G},\ \
\forall\tau>0.$

Equation (3) implies that

(4)
$\mathrm{d}\operatorname{End}_{u}u=\frac{\mathrm{d}}{\mathrm{d}\epsilon}\operatorname{End}((1+\epsilon)u)\bigg{|}_{\epsilon=0}=0.$

Consequently, equations (4) and (2bis) imply that $||u||_{L^{2}}=0$ and
therefore $\gamma$ is the constant curve. ∎

###  1.2. The strategy of the proof

We present here the key ideas that we shall use in the proof of Theorem 1.1.
We work in the context of subFinsler spaces, smooth manifolds equipped with a
distribution, in which the distance between two points is given by the infimum
of the lengths of paths tangent to the distribution joining the two points.
The length is measured with respect to a continuously varying norm.
We start by writing the normal equation coming from the Pontryagin Maximum
Principle in terms of the sub-differentials of the energy, see Proposition
2.13. We use this formulation of the normal equation to get the analog of
(2bis), which expresses the energy of a normal control $u$ in terms of some
co-vector $\lambda$ applied to $\mathrm{d}\operatorname{End}_{u}u$. We then
use the properties of the one parameter subgroup of dilations $\delta$ to
define the vector field

$\vec{\delta}(g):=\left.\frac{\mathrm{d}}{\mathrm{d}t}\delta_{t}(g)\right|_{t=1},\forall
g\in G.$

The first key point is that we have
$\vec{\delta}(\operatorname{End}(u))=\mathrm{d}\operatorname{End}_{u}u$, see
(44). In particular we shall get

(5) $||u||^{2}_{L^{2}}=\lambda(\vec{\delta}(\operatorname{End}(u)).$

We show that $g\mapsto\lambda(\vec{\delta}(g))$ is a finite sum of homogeneous
functions of degree of homogeneity smaller than the step $s$, and therefore it
can be bounded by $Cd(1,g)^{s}$ when $d(1,g)>1$, for some constant $C>0$. This
bound together with (5) will give (1).

###  1.3. Organization of the paper

The paper is organized as follows: we dedicate Section 2 to a brief
presentation of the notions of subFinsler geometry that we need in the paper.
We recall the definitions of subFinsler manifolds, Carnot groups, and self-
similar spaces; we define normal curves and we characterize them in terms of
sub-differentials of the energy. In Section 3, we prove the main results of
the paper. We also prove a slight generalization of Corollary 1.2 to self-
similar spaces. Section 4 contains some examples: we show that length-$1$
normal geodesics can stay arbitrarily close to the starting point and that the
exponent $\frac{1}{s}$ in Theorem 1.1 is the biggest possible.

Acknowledgments. The authors were partially supported by the Swiss National
Science Foundation (grant 200021-204501 ‘ _Regularity of sub-Riemannian
geodesics and applications_ ’) and by the European Research Council (ERC
Starting Grant 713998 GeoMeG ‘ _Geometry of Metric Groups_ ’). E.L.D was also
partially supported by the Academy of Finland (grant 322898 ‘ _Sub-Riemannian
Geometry via Metric-geometry and Lie-group Theory_ ’).

##  2\. Preliminaries

###  2.1. SubFinsler geometry and Carnot groups

We start recalling some basic definitions and facts from subFinsler geometry
following [12] and [3].

######  Definition 2.1.

Let $M$ be a smooth manifold. A _distribution_ on $M$ is a sub-bundle of the
tangent bundle. We call a distribution _bracket-generating_ if, in some
neighborhood of every point, the Lie algebra generated by vector fields
tangent to the distribution contains a frame for the tangent bundle on that
neighborhood. A _subFinsler manifold_ is a smooth manifold $M$, with a
bracket-generating distrubution $\Delta$ and a continuously varying norm
$||\cdot||$, equipped with the Carnot-Carathéodory distance

(6)
$\begin{split}d_{cc}(x,y):=\inf\\{\int_{0}^{1}||\dot{\gamma}(t)||\mathrm{d}t\
\big{|}\gamma:[0,1]\rightarrow M\text{ absolutely continuous; }\\\
\dot{\gamma}(t)\in\Delta_{\gamma(t)},\text{ for a.e.
}t\in[0,1];\gamma(0)=x;\gamma(1)=y\\}.\end{split}$

The fact that $d_{cc}$ is a distance is guaranteed by Chow’s Theorem (see for
example [9, Section 1]). In this paper we focus mainly on Carnot groups:
particular Lie groups with left-invariant distributions and norms.

######  Definition 2.2.

A _stratification_ of a Lie algebra $\mathfrak{g}$ is a decomposition of
$\mathfrak{g}$ as a direct sum $\mathfrak{g}=V_{1}\oplus...\oplus V_{s}$ with

(7) $[V_{1},V_{j}]=V_{j+1},\ \ \forall j\in\\{1,...,s\\},$

where $V_{s+1}:=\\{0\\}$. The sub-spaces $V_{i}$ are called _strata_ of the
stratification. A Lie algebra equipped with a stratification is called
_stratified_. A _Carnot group_ is a simply connected Lie group with stratified
Lie algebra, with the left-invariant extension of the first stratum as
distribution and a left-invariant norm, equipped with the Carnot-Carathéodory
distance.

Carnot groups are self-similar, in the sense that there are natural
automorphisms that act as homotheties.

######  Definition 2.3.

Let $G$ be Carnot group with stratified Lie algebra
$\mathfrak{g}=V_{1}\oplus...\oplus V_{s}$. The _dilation_
$\delta_{\lambda}:G\rightarrow G$ of factor $\lambda$, with
$\lambda\in\mathbb{R}$, is the Lie group automorphism defined setting

(8) $(\delta_{\lambda})_{*}(v)=\lambda^{i}v,\ \ ,\forall i\in\\{1,...,s\\},\
\forall v\in V_{i}.$

The presence of dilations will be crucial in our proof of the non-existence of
periodic normal geodesics, see (61).

###  2.2. Homogeneous spaces and self-similar distances

In differential geometry, the term homogeneous space is referred to the
quotient space of a Lie group modulo a closed subgroup, in order to still have
a transitive action of the Lie group. However, in metric geometry and, more
generally in analysis on metric spaces, the term homogeneous is referred to
functions that gets multiplied by some dilations of the space. Every metric
space admitting a dilation, also called homothety, is said to be self-similar.

In subFinsler geometry, self-similar spaces are well characterized. As
differentiable manifold they have a homogeneous structure of a quotient space
of a Carnot group modulo the action of a dilation-invariant subgroup via left-
multiplication. However, the well-defined action on the right is not by
isometries. Hence they are not isometrically homogeneous spaces. They are
still called homogeneous because they admit dilations. To avoid this double
use of the word homogeneity, we shall call them self-similar (subFinsler)
spaces. By the work of Bellaïche ([4], [3]) we know that the metric tangents
of (constant-rank) subFinsler manifolds are (constant-rank) self-similar
spaces.

######  Definition 2.4.

A _self-similar subFinsler space_ is a subFinsler manifold obtained as the
quotient space of a (left-invariant) subFinsler Carnot group with respect to
the left-action of a dilation-invariant subgroup, and it is equipped with the
quotient distribution. Namely, assume $G$ is a Carnot group with distribution
$\Delta$ and left-invariant norm $||\cdot||$ and $H<G$ is a closed dilation-
invariant subgroup, for which $T_{1}H\cap\Delta_{1}=\\{0\\}$. On the quotient
manifold
${\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}:=\\{Hg:g\in
G\\}$ we define a subFinsler structure that makes the projection
$\pi:G\rightarrow{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}$
a submetry: we take
$\Delta_{{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}}:=\pi_{*}\Delta$
as (constant-rank) distribution and we define the continuously varing norm on
${\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}$
setting for all
$p\in{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}$
and for all $v\in(\pi_{*}\Delta)_{p}\subseteq
T_{p}({\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}})$

(9)
$(||v||_{{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}})_{p}:=\inf\\{||w||_{q}:q\in\pi^{-1}(p),w\in
T_{q}\Delta,\mathrm{d}\pi_{q}(w)=v\\}.$

Being $H$ dilation invariant, the dilations of $G$ pass to the quotient and
define a dilation on
${\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}$.
Thus, on a self-similar space
${\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}$
we call $H$ the origin, since it is the only point fixed by dilations. We
remand to [4, Section 7] for a presentation of self-similar spaces.

###  2.3. Pontryagin Maximum Principle and normal curves

In the remaining part of this section we define the end-point map and we state
the Pontryagin Maximum Principle. Afterwards, we present normal curves and we
characterize them using sub-differentials of the energy.

######  Definition 2.5.

_(Controls)_ Let $M$ be a subFinsler manifold with distribution $\Delta$,
choose $X_{1},...,X_{n}\in\operatorname{Vec}(M)$ such that
$\operatorname{Span}(\\{X_{1}(p),...,X_{n}(p)\\})=\Delta_{p}$ for every $p\in
M$. The elements of $\Omega:=L^{2}([0,1],\mathbb{R}^{n})$ are called controls.

Consider a subFinsler manifold $M$ and fix
$X_{1},...,X_{n}\in\operatorname{Vec}(M)$ such that the distribution at every
$p\in M$ coincides with $\operatorname{Span}(\\{X_{1}(p),...,X_{n}(p)\\})$.
Let $\Omega$ be $L^{2}([0,1],\mathbb{R}^{n})$. To each control $u\in\Omega$
and each point $p\in M$ we associate the unique curve
$\gamma_{u}:[0,1]\rightarrow M$ solving

(10)
$\begin{cases}\gamma_{u}^{\prime}(t)=\sum_{i=1}^{n}u_{i}(t)X_{i}(\gamma_{u}(t)),\text{
for a.e. }t\in[0,1];\\\ \gamma_{u}(0)=p.\end{cases}$

The existence and uniqueness of the solution of (10) is guaranteed by
Carathéodory Existence and Uniqueness Theorem ([15, Theorem 3.4], [7, Theorem
1.3]). We shall say that $\gamma_{u}$ is a/the _curve with control $u$_, or
that $u$ is a _control_ of $\gamma_{u}$. Two controls could give the same
curve, because the vector fields $X_{j}$’s might not be linearly independent
at some point. Moreover, every curve of finite length admits a
reparametrization with control in $\Omega$. We can use Equation (10) to define
the flow along a control. This will be useful for the computation of the
differential of the end-point map.

######  Definition 2.6.

_(Flow along a control)_ Let $M$ be a subFinsler manifold with distribution
$\Delta$, let $X_{1},...,X_{n}\in\operatorname{Vec}(M)$ be such that
$\operatorname{Span}(\\{X_{1}(p),...,X_{n}(p)\\})=\Delta_{p}$ for every $p\in
M$ and define $\Omega:=L^{2}([0,1],\mathbb{R}^{n})$. The _flow along a
control_ $u\in\Omega$ is the map $\phi:[0,1]\times M\rightarrow M$ defined as

(11) $\phi^{t}(p):=\gamma_{u}(t),\ \forall t\in[0,1],$

where $\gamma_{u}$ is the curve solving (10). For fixed $s,t\in[0,1]$, with
$s<t$, we will write $\phi_{s}^{t}:M\rightarrow M$ for the flow from $s$ to
$t$:

(12) $\phi_{s}^{t}:=\phi^{t-s}\circ(\phi^{s})^{-1}.$

######  Example 2.7.

Let $G$ be a subFinsler Carnot group with first stratum $V_{1}$. Let
$X_{1},...,X_{n}$ be a left-invariant frame of the first stratum. Using the
frame $X_{1},...,X_{n}$ identify $L^{2}([0,1],\mathbb{R}^{n})$ with
$L^{2}([0,1],V_{1})$. For all $p\in G$ and $u\in L^{2}([0,1],V_{1})$, equation
(10) rewrites as

(13)
$\begin{cases}\gamma_{u}^{\prime}(t)=\mathrm{d}L_{\gamma_{u}(t)}u(t),\text{
for a.e. }t\in[0,1];\\\ \gamma_{u}(0)=p.\end{cases}$

In particular, if $\phi:[0,1]\times G\rightarrow G$ is the flow along a
control $u\in L^{2}([0,1],V_{1})$, see (11), then it is trivial to check that
for all $p\in G$ the curve $t\mapsto L_{p}(\phi^{t}(1_{G}))$ solves (13).
Consequently,

(14) $\phi^{t}(p)=R_{\phi^{t}(1_{G})}(p),\ \forall p\in G,\forall t\in[0,1].$

######  Definition 2.8.

_(End-point map)_ Let $M$ be a subFinsler manifold with distribution $\Delta$.
Choose $X_{1},....,X_{n}\in\operatorname{Vec}(M)$ such that
$\operatorname{Span}(\\{X_{1}(p),...,X_{n}(p)\\})=\Delta_{p}$. Denote
$\Omega:=L^{2}([0,1],\mathbb{R}^{n})$ and fix a point $p\in M$. The _end-point
map_ $\operatorname{End}:\Omega\rightarrow M$ associated to $X_{1},...,X_{n}$
and $p$ is defined as

(15) $\operatorname{End}(u):=\gamma_{u}(1),$

where $\gamma_{u}$ solves (10). In a Carnot group we always assume the point
$p$ to be the identity element and $X_{1},...,X_{n}$ to be a left-invariant
frame of the first stratum.

######  Proposition 2.9.

_(Differential of the end-point map,[1, Proposition 8.5])._ Let $M$ be a
subFinsler manifold with distribution $\Delta$, let
$X_{1},...,X_{m}\in\operatorname{Vec}(M)$ such that
$\Delta_{p}=\operatorname{Span}(\\{X_{1}(p),...,X_{m}(p)\\})$ for every $p\in
M$. Fix $q\in M$. Set $\Omega:=L^{2}([0,1],\mathbb{R}^{m})$ and denote with
$\operatorname{End}:\Omega\rightarrow M$ the end-point map associated to
$X_{1},...,X_{m}$ and $q$. Then $\operatorname{End}$ is smooth on $\Omega$ and
for every $u,v\in\Omega$ we have

(16)
$\mathrm{d}\operatorname{End}_{u}v=\int_{0}^{1}(\phi_{t}^{1})_{*}\sum_{i=1}^{m}v_{i}(t)X_{i}(\gamma(t))\mathrm{d}t;$

where $\phi^{t}$ is the flow associated to the control $u$ and
$\gamma(t):=\phi^{t}(q)$, see (11).

######  Definition 2.10.

_(Energy)_ Let $M$ be a subFinsler manifold with continuously varying norm
$||\cdot||:\Delta\rightarrow\mathbb{R}$. The _energy_ at a point $p\in M$ is
the function $\operatorname{E}_{p}:\Delta_{p}\rightarrow\mathbb{R}$ defined as

$\operatorname{E}_{p}:=\frac{1}{2}||\cdot||_{p}^{2}.$

For an absolutely continuous curve $\gamma:[0,1]\rightarrow M$ with
$\dot{\gamma}\in\Delta$ we define its _energy_ as
$\frac{1}{2}\int_{0}^{1}||\dot{\gamma}(t)||^{2}\mathrm{d}t$.

The curves realizing the infimum in (6) are called _length-minimizing_. It is
well-known that the same infimum is realized by minimizers of the energy,
considering length-minimizing curves re-parametrized by constant speed.
We are now ready to state the well known Pontryagin Maximum Principle, which
gives first-order necessary conditions for curves to be energy-minimizing.

######  Theorem 2.11.

_(PMP,[2, Theorem 12.10])._ Let $M$ be a subFinsler manifold with distribution
$\Delta$, let $X_{1},...,X_{m}\in\operatorname{Vec}(M)$ such that
$\Delta_{p}=\operatorname{Span}\\{X_{1}(p),...,X_{m}(p)\\}$ for every $p\in
M$. Let $\Omega$ be $L^{2}([0,1],\mathbb{R}^{m})$.
For every $\nu\in\mathbb{R}$ and $v=(v_{1},...,v_{m})\in\mathbb{R}^{m}$;
define $h_{v,\nu}:T^{*}M\rightarrow\mathbb{R}$ as

(17)
$h_{v,\nu}(\eta):=\left\langle\eta,\sum_{i=1}^{m}v_{i}X_{i}(\pi(\eta))\right\rangle+\nu\operatorname{E}_{\pi(\eta)}\left(\sum_{i=1}^{m}v_{i}X_{i}(\pi(\eta))\right),\
\ \forall\eta\in T^{*}M,$

where $\pi:T^{*}M\rightarrow M$ is the canonical projection. If a curve
$\gamma:[0,1]\rightarrow M$ with control $u$ is energy-minimizing, then there
exists $\nu\in\\{-1,0\\}$ and an absolutely continuous curve
$\eta:[0,1]\rightarrow T^{*}M$ such that $\eta(0)\in T^{*}_{\gamma(0)}M$ and

(18) $\dot{\eta}(t)=\vec{h}_{u(t),\nu}(\eta(t)),\text{ for a.e. }t\in[0,1];$
(19) $h_{u(t),\nu}(\eta(t))\geq h_{v,\nu}(\eta(t)),\ \ \forall
v\in\mathbb{R}^{m},\text{ for a.e. }t\in[0,1],$

where $\vec{h}_{u(t),\nu}(\eta(t))$ is the Hamiltonian vector field associated
to $h_{u(t),\nu}$, see [2, Section 11.5.2].

In this paper we will use a reformulation of Proposition 2.11. We first need
to recall the definition of sub-differential.

######  Definition 2.12.

_(Sub-differentials)_ Let $V$ be a vector space and
$f:V\rightarrow\mathbb{R}$. A _sub-differential_ of $f$ at $v\in V$ is a
linear function $a:V\rightarrow\mathbb{R}$ such that

(20) $a(u-v)\leq f(u)-f(v),\quad\forall u\in V.$

We will use $\partial_{v}f$ to denote the _set of sub-differentials_ of $f$ at
$v$.

It is an easy exercise to check that the set of sub-differentials of a convex
function is always a non-empty closed convex set.

######  Proposition 2.13.

_(subFinsler PMP revised)_ Let $M$ be a subFinsler manifold with distribution
$\Delta$, let $X_{1},...,X_{m}\in\operatorname{Vec}(M)$ such that
$\Delta_{p}=\operatorname{Span}(\\{X_{1}(p),...,X_{m}(p)\\})$ for every $p\in
M$. If a curve $\gamma:[0,1]\rightarrow M$ with control $u$ is energy-
minimizing then there exists $\lambda\in T^{*}_{\gamma(1)}M$ such that either

(21)
$\left(\phi_{t}^{1}\right)^{*}\lambda\in\partial_{\dot{\gamma}(t)}\operatorname{E}_{\gamma(t)},\
\ \text{ for a.e. }t\in[0,1]$

or

(22) $\lambda\left(\left(\phi_{t}^{1}\right)_{*}X\right)=0,\ \ \forall
t\in[0,1],\ \forall X\in\Delta_{\gamma(t)},$

the function $\phi^{t}$ denoting the flow along the control $u$, see (11).

###### Proof.

By Proposition 2.11 if a curve $\gamma:[0,1]\rightarrow M$ with control $u$ is
energy-minimizing then there exist $\nu\in\\{-1,0\\}$ and an absolutely
continuous curve $\eta:[0,1]\rightarrow T^{*}M$ such that $\eta(0)\in
T^{*}_{\gamma(0)}M$ and equations (18) and (19) hold. It is well know (see for
example [2, Section 12.2]) that Equation (18) rewrites for all $t\in[0,1]$ as

(23) $\begin{cases}\pi(\eta(t))=\gamma(t);\\\
\eta(t)=\left(\phi_{t}^{1}\right)^{*}\eta(1).\end{cases}$

Setting $\lambda:=\eta(1)$, by (23) and the definition of $h_{v,\nu}$ in (17)
we have that

(24)
$h_{v,\nu}(\eta(t))=\left\langle\left(\phi_{t}^{1}\right)^{*}\lambda,\sum_{i=1}^{m}v_{i}X_{i}(\gamma(t))\right\rangle+\nu\operatorname{E}_{\gamma(t)}\left(\sum_{i=1}^{m}v_{i}X_{i}(\gamma(t))\right).$

When $\nu=-1$, Equation (19) is equivalent to, for all $v\in\mathbb{R}^{m}$,

$\left\langle\left(\phi_{t}^{1}\right)^{*}\lambda,\sum_{i=1}^{m}v_{i}X_{i}(\gamma(t))-\dot{\gamma}(t)\right\rangle\leq\operatorname{E}_{\gamma(t)}\left(\sum_{i=1}^{m}v_{i}X_{i}(\gamma(t))\right)-\operatorname{E}_{\gamma(t)}\left(\dot{\gamma}(t)\right).$

When $v$ varies, the vector $\sum_{i=1}^{m}v_{i}X_{i}(\gamma(t))$ gives an
arbitrary element of the domain of $\operatorname{E}_{\gamma(t)}$. Hence, we
have that Equation (19) is equivalent to (21). When $\nu=0$, from (24)
Equation (19) is

$\left\langle\left(\phi_{t}^{1}\right)^{*}\lambda,\sum_{i=1}^{m}v_{i}X_{i}(\gamma(t))-\dot{\gamma}(t)\right\rangle\leq
0.$

Noticing that we are applying $\left(\phi_{t}^{1}\right)^{*}\lambda$ to an
arbitrary element of $\Delta_{\gamma(t)}$, we finally deduce that in this
second case Equation (19) is equivalent to (22). ∎

######  Definition 2.14.

_(Normal and abnormal curves)_ In the setting of Proposition 2.13, we say that
a curve $\gamma:[0,1]\rightarrow M$ is a _normal curve_ if there exists a co-
vector $\lambda\in T^{*}_{\gamma(1)}M$ such that (21) holds. We say that a
curve is an _abnormal curve_ if there exist a co-vector $\lambda\in
T^{*}_{\gamma(1)}M$ such that (22) holds.

We can define normal curves also with domain different than $[0,1]$.

######  Definition 2.15.

Let $M$ be a subFinsler manifold with distribution $\Delta$ and let $I$ be an
interval. A curve $\gamma:I\rightarrow M$ is a _normal curve_ if the
restriction of $\gamma$ to every compact sub-interval of $I$, when affinely
re-parametrized in $[0,1]$, is a normal curve in the sense of Definition 2.14.

The following remark ensures that the restriction of a normal curve to every
sub-interval of its domain is a normal curve and therefore that Definition
2.15 is well posed.

######  Remark 2.16.

Let $M$ be a subFinsler manifold with distribution $\Delta$. Let
$\gamma:[0,1]\rightarrow M$ be a normal curve and $\lambda\in
T^{*}_{\gamma(1)}M$ be a co-vector such that (21) holds. For every
$\alpha\in\mathbb{R}_{>0}$, define the curve
$\gamma_{\alpha}:\left[0,\frac{1}{\alpha}\right]\rightarrow M$ as

(25) $\gamma_{\alpha}(t):=\gamma(\alpha t),\ \ \forall
t\in\left[0,\frac{1}{\alpha}\right].$

Then

(26)
$(\phi^{\frac{1}{\alpha}}_{t;\alpha})^{*}\lambda_{\alpha}\in\partial_{\dot{\gamma}_{\alpha}(t)}\operatorname{E}_{\gamma_{\alpha}(t)},\
\ \forall t\in\left[0,\frac{1}{\alpha}\right],$

where $\lambda_{\alpha}\in T^{*}_{\gamma(1)}M$is defined setting
$\lambda_{\alpha}:=\alpha\lambda$ and $(\phi^{\frac{1}{\alpha}}_{t;\alpha})$
is the flow along the control $\alpha u(\alpha
t)\mathbbm{1}_{\left[0,\frac{1}{\alpha}\right]}$ from $t$ to
$\frac{1}{\alpha}$.

###### Proof.

By definition of sub-differential and of $\operatorname{E}$, if for some $p\in
M$, $v\in T_{p}M$ and $a\in T_{p}^{*}M$ we have
$a\in\partial_{v}\operatorname{E}_{p}$, then $\tau a\in\partial_{\tau
v}\operatorname{E}_{p}$ for every $\tau>0$. Moreover, we have for all
$t\in\left[0,\frac{1}{\alpha}\right]$ that
$\gamma_{\alpha}^{\prime}(t)=\alpha\gamma^{\prime}(\alpha t)$ and that
$\phi^{\frac{1}{\alpha}}_{t;\alpha}$ is equal to the flow $\phi_{\alpha
t}^{1}$ along $u$ from $\alpha t$ to $1$. Consequently, since $(\phi_{\alpha
t}^{1})^{*}\lambda\in\partial_{\dot{\gamma}(\alpha
t)}\operatorname{E}_{\gamma(\alpha t)}$ for all
$t\in\left[0,\frac{1}{\alpha}\right]$, we have that
$(\phi^{\frac{1}{\alpha}}_{t;\alpha})^{*}(\alpha\lambda)\in\partial_{\dot{\gamma}_{\alpha}(t)}\operatorname{E}_{\gamma_{\alpha}(t)}$
for all $t\in\left[0,\frac{1}{\alpha}\right]$. ∎

From Proposition 2.13 and the formula of the differential of
$\operatorname{End}$ in Proposition 2.9 we can get the equivalent of (2bis)
for subFinsler manifolds.

######  Corollary 2.17.

_(Normal equation and end-point map)_ Let $M$ be a subFinsler manifold with
distribution $\Delta$. Choose $X_{1},...,X_{m}\in\operatorname{Vec}(M)$ such
that $\operatorname{Span}(\\{X_{1}(p),...,X_{m}(p)\\})=\Delta_{p}$ for every
$p\in M$ and fix $q\in M$. Set $\Omega:=L^{2}([0,1],\mathbb{R}^{m})$ and
denote with $\operatorname{End}:\Omega\rightarrow M$ the end-point map
associated to $X_{1},...,X_{m}$ and $q$. Let $\gamma:[0,1]\rightarrow M$ be a
normal curve with control $u$ such that $\gamma(0)=q$. Let $\lambda\in
T^{*}_{\gamma(1)}M$ be a co-vector for which (21) holds. Then

(27)
$\lambda(\mathrm{d}\operatorname{End}_{u}u)=\int_{0}^{1}||\dot{\gamma}(t)||_{\gamma(t)}^{2}\mathrm{d}t,$

###### Proof.

By Proposition 2.9 we have

(28)
$\lambda(\mathrm{d}\operatorname{End}_{u}v)=\int_{0}^{1}\lambda\left(\left(\phi_{t}^{1}\right)_{*}\sum_{i=1}^{m}v_{i}(t)X_{i}(\gamma(t))\right)\mathrm{d}t,\
\ \forall v\in\Omega.$

By (21) we have

(29)
$\left(\left(\phi_{t}^{1}\right)^{*}\lambda\right)(\dot{\gamma}(t))=||\dot{\gamma}(t)||^{2},\text{
for a.e. }t\in[0,1],$

see for example [10, Lemma 2.19]. Equations (28) and (29) imply (27). ∎

In the case of Carnot groups we always choose a left-invariant frame as the
set of vector fields spanning the distribution $\Delta$ at every point.
Therefore, we can interpret controls as functions $u\in
L^{2}([0,1],\Delta_{1})$ and associate to each absolutely continuous curve a
unique control: the left translation at the origin of the derivative of the
curve. The flow along a control coincides with the right-translation along the
associated curve, hence we can rewrite Proposition 2.13 and Corollary 2.17 as
follows:

######  Proposition 2.18.

Let $G$ be a subFinsler Carnot group equipped with first stratum $V_{1}$ and a
left-invariant norm, let $\\{X_{1},...,X_{m}\\}$ be a left-invariant frame of
the first stratum. Let $\operatorname{End}$ be the end-point map associated to
$X_{1},...,X_{m}$ and $1_{G}$. Let $\gamma:[0,1]\rightarrow G$ be a normal
curve with control $u\in L^{2}([0,1],V_{1})$ and such that $\gamma(0)=1_{G}$.
Then, there exist a right-invariant co-vector $\lambda$ such that

(30)
$\lambda\circ\operatorname{Ad}_{\gamma(t)}\in\partial_{u(t)}\operatorname{E}_{1},\text{
for a.e. }t\in[0,1],$

and

(31) $\lambda(\mathrm{d}\operatorname{End}_{u}u)=||u||_{L^{2}}^{2}.$

###### Proof.

Let $\bar{\lambda}\in T^{*}_{\gamma(1)}G$ be a co-vector for which (21) holds.
Define $\lambda$ to be the right-invariant extension of $\bar{\lambda}$. Since
the flow along $u$ is $\phi^{t}=R_{\gamma(t)}$, see (14), there holds

$\left(\phi^{1}_{t}\right)^{*}\bar{\lambda}=\bar{\lambda}\circ\mathrm{d}R_{\gamma(1)}\circ\mathrm{d}R_{\gamma(t)}^{-1},\forall
t\in[0,1].$

Consequently, equation (21) rewrites as

(32)
$\bar{\lambda}\circ\mathrm{d}R_{\gamma(1)}\circ\mathrm{d}R_{\gamma(t)}^{-1}\in\partial_{\dot{\gamma}(t)}\operatorname{E}_{\gamma(t)},\text{
for a.e. }t\in[0,1].$

Being $\dot{\gamma}=\mathrm{d}L_{\gamma}u$ (see (13)) and being
$\operatorname{E}$ left-invariant, equation (32) is equivalent to

$\bar{\lambda}\circ\mathrm{d}R_{\gamma(1)}\circ\mathrm{d}R_{\gamma(t)}^{-1}\circ\mathrm{d}L_{\gamma(t)}\in\partial_{u(t)}\operatorname{E}_{1},\text{
for a.e. }t\in[0,1],$

which is exactly (30) being $\lambda$ the right-invariant extension of
$\bar{\lambda}$. Equation (31) follows from (27), being
$\bar{\lambda}=\lambda_{\gamma(1)}$ and
$||u||_{L^{2}}=\int_{0}^{1}||\dot{\gamma}(t)||_{\gamma(t)}^{2}\mathrm{d}t$. ∎

From Proposition 31 and Remark 2.16 we have that if $G$ is a Carnot group,
$I\subseteq\mathbb{R}$ is an interval, $\gamma:I\rightarrow G$ is a normal
geodesic and $u:=\mathrm{d}L_{\gamma}^{-1}\dot{\gamma}$, then there exists a
right-invariant co-vector $\lambda$ such that
$\lambda\circ\operatorname{Ad}_{\gamma(t)}\in\partial_{u(t)}\operatorname{E}_{1}$
for every $t\in I$.

######  Definition 2.19.

Let $G$ be a Carnot group, $I$ be an interval and $\gamma:I\rightarrow G$ be a
normal curve. Set $u:=\mathrm{d}L_{\gamma}^{-1}\dot{\gamma}$. We say that
$\lambda\in T^{*}G$ is a _co-vector associated to_ $\gamma$ if it is right-
invariant and

(33)
$\lambda\circ\operatorname{Ad}_{\gamma(t)}\in\partial_{u(t)}\operatorname{E}_{1},\text{
for a.e. }t\in I.$

##  3\. Proof of main results

###  3.1. The differential of the dilations

This sub-section is devoted to a computation of the differential of the one-
parameter family $\delta$ of dilations in Carnot groups. The aim is to write
the vector field

(34)
$\vec{\delta}(g):=\frac{\mathrm{d}}{\mathrm{d}\tau}\delta_{\tau}(g)\bigg{|}_{\tau=1}$

as a linear combination of right-invariant vector-fields. Then, using the
particular form of the coefficients of this combination, we will get an
estimate of $\lambda(\vec{\delta}(g))$ in terms of the distance $d(1,g)$ for
every right-invariant co-vector $\lambda$.

######  Definition 3.1.

Let $G$ be a Carnot group with stratified Lie algebra
$\mathfrak{g}=V_{1}\oplus...\oplus V_{s}$. An ordered basis $X_{1},...,X_{n}$
of $\mathfrak{g}$ is _adapted to the stratification_ if

$X_{i}\in V_{j},\ \ \forall j\in{1,...,s},\forall i\in\\{1,...,n\\}\text{ s.t.
}\dim(V_{j-1})<i\leq\dim(V_{j}),$

where we are setting $V_{0}:=\\{0\\}$.
We say that a vector $X\in\mathfrak{g}$ has _degree of homogeneity_
$j\in\\{1,...,s\\}$ if $X\in V_{j}$. We say that a function
$f:G\rightarrow\mathbb{R}$ is _homogeneous_ of degree $\alpha\in\mathbb{R}$ if
$f\circ\delta_{\tau}=\tau^{\alpha}f$ for all $\tau\geq 0$.

######  Lemma 3.2.

Let $G$ be a Carnot group, fix a basis $X_{1},...,X_{m}$ adapted to the
stratification and let $\vec{\delta}$ be as in (34). Then

(35) $\vec{\delta}=\sum_{i=1}^{m}P_{i}X_{i}^{\dagger},$

where, for each $i\in\\{1,...,m\\}$, we denote by $X_{i}^{\dagger}$ the right-
invariant extension of $X_{i}$ and $P_{i}:G\rightarrow\mathbb{R}$ are
homogeneous functions of degree $d_{i}$, the integer $d_{i}\in\mathbb{N}$
being the degree of homogeneity of $X_{i}$.

###### Proof.

Fix $\tau>0$. For all $g\in G$ we have

(36)
$\left(\left(\delta_{\tau}\right)_{*}\vec{\delta}\right)(g)=\frac{\mathrm{d}}{\mathrm{d}\epsilon}\delta_{\tau}\circ\delta_{1+\epsilon}(\delta_{\tau}^{-1}(g))\bigg{|}_{\epsilon=0}=\frac{\mathrm{d}}{\mathrm{d}\epsilon}\delta_{1+\epsilon}(g)\bigg{|}_{\epsilon=0}=\vec{\delta}(g).$

Moreover, for all $i\in\\{1,...,m\\}$ there holds

(37)
$\left(\delta_{\tau}\right)_{*}X_{i}^{\dagger}=\tau^{d_{i}}X_{i}^{\dagger},$

since
$\left(\delta_{\tau}\right)_{*}X_{i}^{\dagger}(1_{G})=\tau^{d_{i}}X_{i}^{\dagger}(1_{G})$
and $\left(\delta_{\tau}\right)_{*}X_{i}^{\dagger}$ is a right-invariant
vector field, being $\delta_{\tau}$ a group homomorphism and $X_{i}^{\dagger}$
right-invariant.
Since $\vec{\delta}$ is a smooth vector field there exist smooth functions
$P_{1},...,P_{m}:G\rightarrow\mathbb{R}$ such that (35) holds. To conclude the
proof of the lemma we have to show that

(38) $P_{i}\circ\delta_{\tau}=\tau^{d_{i}}P_{i},\ \ \forall
i\in\\{1,...,m\\}.$

For all $i\in\\{1,...,m\\}$ we have

$\displaystyle\sum_{i=1}^{m}P_{i}X_{i}^{\dagger}$
$\displaystyle\stackrel{{\scriptstyle\eqref{e differenzialedelta}}}{{=}}$
$\displaystyle\vec{\delta}$ $\displaystyle\stackrel{{\scriptstyle\eqref{e
pushforwarddelta}}}{{=}}$
$\displaystyle\left(\delta_{\tau}\right)_{*}\vec{\delta}$
$\displaystyle\stackrel{{\scriptstyle\eqref{e differenzialedelta}}}{{=}}$
$\displaystyle\left(\delta_{\tau}\right)_{*}\left(\sum_{i=1}^{m}P_{i}X_{i}^{\dagger}\right)$
$\displaystyle\stackrel{{\scriptstyle\eqref{e pushforwardXi}}}{{=}}$
$\displaystyle\sum_{i=1}^{m}\tau^{d_{i}}(P_{i}\circ\delta_{\tau}^{-1})X_{i}^{\dagger},$

and consequently (38) holds. ∎

The identity found in Lemma 3.2 allows us to give an estimate of the value
that we get applying a right-invariant co-vector to the vector
$\vec{\delta}(g)$ defined in (34). Up to constants, the bound we find depends
only on the co-vector and on the distance of $g$ from the origin. To get a
quantitative estimate we fix a norm on the Lie algebra.

######  Definition 3.3.

Let $G$ be a Lie group and $\lambda$ a right-invariant co-vector. Choose a
basis $\\{X_{1},...,X_{n}\\}$ of $\,T_{1}G$ and define

(39) $N(\lambda):=\sum_{i=1}^{n}|\lambda(X_{i})|.$

Up to a multiplicative constant the norm $N$ is equivalent to every other norm
on the space of right-invariant co-vectors.

######  Lemma 3.4.

Let $G$ be a Carnot group. Define $\vec{\delta}$ as in (34). There exists a
constant $C>0$ such that for every right-invariant co-vector $\lambda$ we have

(40) $\lambda(\vec{\delta}(g))\leq
CN(\lambda)\max(d(1,g),d(1,g)^{s}),\qquad\forall g\in G,$

where $N(\lambda)$ is defined as in Definition 3.3.

###### Proof.

Choose a basis $X_{1},...,X_{n}$ adapted to the stratification; call
$\\{X_{i}^{\dagger}\\}_{i\in\\{1,...,n\\}}$ the right-invariant extension of
this basis. Let $d_{i}\in\mathbb{N}$ be the degree of homogeneity of $X_{i}$.
From Lemma 3.2 we have that (35) holds. For all $i\in\\{1,...,n\\}$, being the
function $P_{i}$ homogeneous of degree $d_{i}$, there exist a constant
$C_{i}>0$ such that

(41) $P_{i}(g)\leq C_{i}d(1,g)^{d_{i}},\ \ \forall g\in G.$

Setting $C=\max_{1,...,n}C_{i}$, for every right-invariant co-vector
$\lambda$, by definition of $N(\lambda)$, we have

(42) $\lambda(\vec{\delta}(g))\stackrel{{\scriptstyle\eqref{e
differenzialedelta}}}{{=}}\sum_{i=1}^{n}P_{i}(g)\lambda(X_{i})\stackrel{{\scriptstyle\eqref{e
normalambda},\eqref{e
inequalitihomogeneity}}}{{\leq}}CN(\lambda)\max(d(1,g),d(1,g)^{s}),\ \ \forall
g\in G.$

This concludes the proof of the lemma. ∎

###  3.2. Proof of Theorem 1.1

We are now ready to prove the main result of this paper.

###### Proof of Theorem 1.1.

Let $\gamma:\mathbb{R}\rightarrow G$ be a normal curve parametrized by arc-
length. Without loss of generality we can assume $t^{\prime}=0$, $t>0$ and
$\gamma(0)=1_{G}$. Since all norms on the space of right-invariant co-vectors
are equivalent, we don’t lose generality assuming that $N$ is the norm defined
by (39). Choose a co-vector $\lambda$ associated to $\gamma$ (see Definition
33). Denote with $V_{1}$ the first stratum of the stratification of
$\operatorname{Lie}(G)$ and for $t>0$ denote with $u_{t}\in
L^{2}([0,1],V_{1})$ the control of the curve $\gamma_{t}:[0,1]\rightarrow G$,
$\gamma_{t}(\tau):=\gamma|_{[0,t]}(t\tau)$ for all $\tau\in[0,1]$. For every
$t>0$, we have

(43)
$\mathrm{d}\operatorname{End}_{u_{t}}u_{t}=\frac{\mathrm{d}}{\mathrm{d}\tau}\operatorname{End}(\tau
u_{t})\bigg{|}_{\tau=1}.$

Being $\tau u_{t}$ the control of $\delta_{\tau}\gamma_{t}$, we can rewrite
(43) as

(44)
$\mathrm{d}\operatorname{End}_{u_{t}}u_{t}=\frac{\mathrm{d}}{\mathrm{d}\tau}\delta_{\tau}(\operatorname{End}(u_{t}))\bigg{|}_{\tau=1}=\vec{\delta}(\operatorname{End}(u_{t})),\
\ \forall t>0,$

where $\vec{\delta}$ is defined by (34). Applying Lemma 3.4 we have that there
exists a constant $C>0$ such that

(45) $\lambda(\vec{\delta}(\operatorname{End}(u_{t})))\leq
CN(\lambda)\max(d(1,\operatorname{End}(u_{t})),d(1,\operatorname{End}(u_{t}))^{s}),\
\ \forall t>0.$

By (44) and (45), and being $\operatorname{End}(u_{t})=\gamma(t)$ for all
$t>0$, we have

(46) $\lambda(\mathrm{d}\operatorname{End}_{u_{t}}u_{t})\leq
CN(\lambda)\max(d(1,\gamma(t)),d(1,\gamma(t))^{s}),\ \ \forall t>0.$

Moreover, we have from Remark 2.16 and Proposition 31 that

(47)
$t\lambda(\mathrm{d}\operatorname{End}_{u_{t}}u_{t})=\int_{0}^{1}||t\dot{\gamma}(tx)||^{2}\mathrm{d}x=t^{2},\
\ \forall t>0.$

From equations (46) and (47) we get for every $t>0$ that

(48) $t\leq CN(\lambda)\max(d(1,\gamma(t)),d(1,\gamma(t))^{s}).$

In particular, for $t>CN(\lambda)$ we must have

$d(1,\gamma(t))>1,$

and therefore

$\max(d(1,\gamma(t))),d(1,\gamma(t))^{s})=d(1,\gamma(t))^{s}.$

Thus, for all $t>CN(\lambda)$, Equation (48) becomes

(49) $t\leq CN(\lambda)d(1,\gamma(t))^{s}.$

Set $\epsilon:=C^{-\frac{1}{s}}$. If $t>CN(\lambda)$ equation (1) is a
conseqeuence of (49). If $t\leq CN(\lambda)$ equation (1) is trivially true
being the right-hand side less or equal than $0$. We showed that (1) holds for
every $t>0$, thus we concluded the proof of the theorem. ∎

###  3.3. Proof of Corollary 1.2

This section is devoted to the proof of Corollary 1.2. We will actually prove
a stronger statement:

######  Theorem 3.5.

In self-similar spaces, normal loops starting from the origin are constant.

We start showing that normal curves in self-similar spaces can be lifted to
normal curves in Carnot groups.

######  Lemma 3.6.

Let $G$ be a Carnot group with stratified Lie algebra
$\mathfrak{g}=V_{1}\oplus...\oplus V_{s}$ and let $H<G$ be a dilation-
invariant subgroup, with $\operatorname{Lie}(H)\cap V_{1}=\\{0\\}$. Denote
with
$\pi:G\rightarrow{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}$
the projection. Let
$\gamma:[0,1]\rightarrow{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}$
be a normal curve. For every $g\in\pi^{-1}(\gamma(0))$ there exists a normal
curve $\tilde{\gamma}:[0,1]\rightarrow G$ such that $\tilde{\gamma}(0)=g$ and
$\pi\circ\tilde{\gamma}=\gamma$. Moreover, we can choose a co-vector
$\tilde{\lambda}\in(T_{\tilde{\gamma}(1)}G)^{*}$ associated to
$\tilde{\gamma}$ such that

(50) $\tilde{\lambda}(\ker(\mathrm{d}\pi_{\tilde{\gamma}(1)}))=\\{0\\}.$

###### Proof.

Denote with $\Delta$ the distribution of $G$ and with $\bar{\Delta}$ the one
of
${\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}$,
see Definition 2.4 . Fix a left-invariant frame $\\{X_{1},...,X_{m}\\}$ such
that $\operatorname{Span}\\{X_{1}(g),...,X_{m}(g)\\}=\Delta_{g}$ for all $g\in
G$. We have
$\operatorname{Span}\\{\mathrm{d}\pi_{g}X_{1}(g),...,\mathrm{d}\pi_{g}X_{m}(g)\\}=\mathrm{d}\pi_{g}\Delta_{g}=\bar{\Delta}_{\pi(g)}$
for all $g\in G$, thus we can see $\Omega:=L^{2}([0,1],\mathbb{R}^{m})$ both
as the set of controls of curves in $G$ associated to $\\{X_{1},...,X_{m}\\}$
and as the set of controls of curves in
${\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}$
associated to $\\{\pi_{*}X_{1},...,\pi_{*}X_{m}\\}$. We remark that by
definition of
$||\cdot||_{{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}}$
in (9), and being the norm on $\Delta$ left-invariant, we have for every
$p\in{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}$,
every $q\in\pi^{-1}(p)$ and every $v\in\mathbb{R}^{m}$ that

$||\sum_{i=1}^{m}v_{i}\left(\pi_{*}X_{i}\right)(p)||_{{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}}=||\sum_{i=1}^{m}v_{i}X_{i}(q)||_{G}$

Thus for the control $u\in\Omega$ of $\gamma$ we have

(51)
$||\dot{\gamma}(t)||_{{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}}=||\sum_{i=1}^{m}u_{i}(t)X_{i}(q)||_{G},\text{
for a.e. }t\in[0,1],\forall q\in\pi^{-1}(\gamma(t)).$

To simplify the notation we write $u(t)\cdot X(q)$ for
$\sum_{i=1}^{m}u_{i}(t)X_{i}(q)$. We claim that for a.e. $t\in[0,1]$, for all
$q\in\pi^{-1}(\gamma(t))$ and for all linear functions
$a:T_{\gamma(t)}({\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}})\rightarrow\mathbb{R}$
there holds

(52)
$a\in\partial_{\dot{\gamma}(t)}\left(\frac{\left(||\cdot||_{{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}}\right)_{\gamma(t)}^{2}}{2}\right)\implies
a\circ\mathrm{d}\pi_{q}\in\partial_{u(t)\cdot
X(q)}\left(\frac{\left(||\cdot||_{G}\right)_{q}^{2}}{2}\right).$

Indeed, by definition of sub-differential if
$a\in\partial_{\dot{\gamma}(t)}\left(\frac{\left(||\cdot||_{{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}}\right)_{\gamma(t)}^{2}}{2}\right)$
and $q\in\pi^{-1}(\gamma(t))$, then

(53)
$\begin{split}\frac{||\dot{\gamma}(t)||_{{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}}^{2}}{2}\leq\frac{||\mathrm{d}\pi_{q}(w)||^{2}_{{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}}}{2}-a(\mathrm{d}\pi_{q}(w)-\dot{\gamma}(t)),\forall
w\in T_{q}G.\end{split}$

Consequently, by (51) we have

(54) $\frac{||u(t)\cdot
X(q)||_{G}^{2}}{2}\leq\frac{||\mathrm{d}\pi_{q}(w)||^{2}_{{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}}}{2}-a(\mathrm{d}\pi_{q}(w)-\dot{\gamma}(t)),\forall
w\in T_{q}G.$

By definition of
$||\cdot||_{{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}}$
we have
$||\mathrm{d}\pi_{q}(w)||^{2}_{{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}}\leq||w||_{G}^{2}$
and therefore Equation (54) implies

(55) $\frac{||u(t)\cdot
X(q)||_{G}^{2}}{2}\leq\frac{||w||_{G}^{2}}{2}-(a\circ\mathrm{d}\pi_{q})(w-u(t)\cdot
X(q)),\forall w\in T_{q}G,$

which proves the claim (52).
If we call $\tilde{\phi}^{t}$ the flow along $u$ in $G$ and $\phi^{t}$ the
flow along $u$ in
${\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}$,
we have that

(56) $\pi\circ\tilde{\phi}^{t}=\phi^{t}\circ\pi,\ \ \forall t\in[0,1],$

since for every $q\in G$ both $\pi\circ\tilde{\phi}^{t}(q)$ and
$\phi^{t}\circ\pi(q)$ solve the differential equation

$\begin{cases}\sigma^{\prime}(t)=\sum_{i=1}^{m}u_{i}(t)\pi_{*}X_{i}(\sigma(t));\\\
\sigma(0)=\pi(q).\end{cases}$

In particular, for all $g\in\pi^{-1}(\gamma(0))$, the curve
$\tilde{\gamma}:[0,1]\rightarrow G$, defined as
$\tilde{\gamma}(t):=\tilde{\phi}^{t}(g)$ for all $t\in[0,1]$, is such that
$\pi\circ\tilde{\gamma}=\gamma$ and $\tilde{\gamma}(0)=g$.
Being $\gamma$ normal there exists $\lambda\in
T^{*}_{\gamma(1)}({\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}})$
such that

(57)
$\left(\phi_{t}^{1}\right)^{*}\lambda\in\partial_{\dot{\gamma}(t)}\left(\frac{\left(||\cdot||_{{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}}\right)_{\gamma(t)}^{2}}{2}\right),\text{
for a.e. }t\in[0,1].$

We prove that $\tilde{\gamma}$ is normal showing that for
$\tilde{\lambda}:=\lambda\circ\mathrm{d}\pi_{\tilde{\gamma}(1)}$ there holds

(58)
$\left(\tilde{\phi}_{t}^{1}\right)^{*}\tilde{\lambda}\in\partial_{\dot{\tilde{\gamma}}(t)}\left(\frac{\left(||\cdot||_{G}\right)_{\tilde{\gamma}(t)}^{2}}{2}\right),\text{
for a.e. }t\in[0,1].$

Indeed, we know from (52) and (57) that

(59)
$\left(\left(\phi_{t}^{1}\right)^{*}\lambda\right)\circ\pi_{*}\in\partial_{\dot{\tilde{\gamma}}(t)}\left(\frac{\left(||\cdot||_{G}\right)_{\tilde{\gamma}(t)}^{2}}{2}\right),\text{
for a.e. }t\in[0,1],$

and by (56) we have

(60)
$\left(\tilde{\phi}_{t}^{1}\right)^{*}\tilde{\lambda}=\left(\left(\phi_{t}^{1}\right)^{*}\lambda\right)\circ\pi_{*},\
\ \forall t\in[0,1].$

Finally, Equation (50) come trivially from the definition of
$\tilde{\lambda}$. ∎

We are now ready to prove that in self-similar spaces normal loops starting
from the origin are constant.

###### Proof of Theorem 3.5.

Let $G$ be a Carnot group and $H<G$ be a subgroup invariant under the action
of dilations, with $\operatorname{Lie}(H)\cap V_{1}=\\{0\\}$. Denote with
$\pi:G\rightarrow{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}$
the canonical projection. Let
$\gamma:[0,1]\rightarrow{\mathchoice{\raisebox{-3.41666pt}{$\displaystyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\displaystyle{G}$}}{\raisebox{-3.41666pt}{$\textstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{3.41666pt}{$\textstyle{G}$}}{\raisebox{-2.39166pt}{$\scriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{2.39166pt}{$\scriptstyle{G}$}}{\raisebox{-1.70833pt}{$\scriptscriptstyle{H}$}\mkern-4.0mu\diagdown\mkern-5.0mu\raisebox{1.70833pt}{$\scriptscriptstyle{G}$}}}$
be a normal curve such that $\gamma(0)=\gamma(1)=\pi(1_{G})$. From Lemma 50 we
know there exists exists a normal curve $\tilde{\gamma}:[0,1]\rightarrow G$
with control $u$ such that $\pi\circ\tilde{\gamma}=\gamma$ and
$\tilde{\gamma}(0)=1_{G}$. In particular $\tilde{\gamma}(1)\in H$ and thus

(61)
$\mathrm{d}\operatorname{End}_{u}u=\frac{\mathrm{d}}{\mathrm{d}\tau}\delta_{\tau}(\tilde{\gamma}(1))\bigg{|}_{\tau=1}\in
T_{\tilde{\gamma}(1)}H,$

being $H$ dilation invariant. On the other hand, by (50) we know that
$\tilde{\gamma}$ solves Equation (31) with respect to a covector
$\tilde{\lambda}\in T^{*}_{\tilde{\gamma}(1)}G$ such that
$\tilde{\lambda}(T_{\tilde{\gamma}(1)}H)=\\{0\\}$, thus

(62) $||u||_{L^{2}}=\tilde{\lambda}(\mathrm{d}\operatorname{End}_{u}u)=0.$

We proved that $\tilde{\gamma}$ is constant and thus $\gamma$ is constant.
Since $\gamma$ was an arbitrary normal loop starting from the origin, we
concluded the proof of the theorem. ∎

##  4\. Some examples

###  4.1. Curves with end-point arbitrarely close to the origin

Even if by Theorem 3.5 we know that in Carnot groups non-constant normal loops
don’t exists, we can find normal curves of length $1$ with end-point
arbitrarily close to the origin. We present a simple example in the Heisemberg
group.

######  Definition 4.1.

The _3-dimensional Heisemberg group_ is the Carnot group
$(\mathbb{R}^{3},\cdot)$ with product law

$(x,y,z)\cdot(x^{\prime},y^{\prime},z^{\prime}):=(x+x^{\prime},y+y^{\prime},z+z^{\prime}-\frac{1}{2}(x^{\prime}y-xy^{\prime})),\forall
x,y,z,x^{\prime},y^{\prime},z^{\prime}\in\mathbb{R}.$

In the Heisemberg group we choose as orthonormal left-invariant frame of the
first stratum the two vector fields $\\{X,Y\\}$ defined for all
$x,y,z\in\mathbb{R}$ by

$\displaystyle X(x,y,z):=\partial_{x}-\frac{y}{2}\partial_{z};$ $\displaystyle
Y(x,y,z):=\partial_{y}+\frac{x}{2}\partial_{z}.$

######  Proposition 4.2.

Let $G:=(\mathbb{R}^{3},\cdot)$ be the subRiemannian 3-dimensional Heisemberg
group. For every $\epsilon>0$ there exists a normal curve
$\gamma:[0,1]\rightarrow\mathbb{R}^{3}$, parametrized by arclength, such that
$\gamma(0)=1_{G}$ and $d(\gamma(1),1_{G})<\epsilon$.

###### Proof.

For every $N\in\mathbb{N}$, in the coordinates of Definition 4.1, consider the
curve $\gamma_{N}:[0,1]\rightarrow G$ defined as
$\gamma_{N}(t):=(x_{N}(t),y_{N}(t),z_{N}(t))$ with

$x_{N}(t):=\frac{\cos(2\pi Nt)-1}{2\pi N};$ $y_{N}(t):=\frac{\sin(2\pi
Nt)}{2\pi N};$ $z_{N}(t):=\frac{2\pi Nt-\sin(2\pi Nt)}{8(\pi N)^{2}}.$

The curve $\gamma_{N}$ is the horizontal lift of a circle of radius
$\frac{1}{2\pi N}$ travelled $N$ times and is a normal geodesic parametrized
by arc-length (see for example [12, Section 1.4.1] for a characterization of
geodesics in the Heisemberg group). We have $\gamma_{N}(0)=1_{G}$ and
$\gamma_{N}(1)=(0,0,\frac{1}{4\pi N})$. Therefore, for every $\epsilon>0$
there exists $N>0$ such that $\gamma_{N}(1)\in B(1_{G},\epsilon)$. We showed
that for every $\epsilon>0$ there exists a normal curve of length $1$ having
end-point in $B(1_{G},\epsilon)$, thus the proof of the proposition is
concluded. ∎

###  4.2. Optimality of the exponent in Theorem 1.1

In an arbitrary Carnot group we cannot obtain an estimate as the one in
Theorem 1.1 with an exponent smaller than the reciprocal of the step. Indeed,
for every $s\in\mathbb{N}$, we provide an example of a normal curve $\gamma$,
in a filiform group of step $s$, for which the distance of $\gamma(t)$ from
the origin is bounded above by a constant times $t^{\frac{1}{s}}$.

######  Definition 4.3.

The _subRiemannian filiform group of first type_ is the Carnot group with
stratified Lie algebra $\mathfrak{g}=V_{1}\oplus...\oplus V_{s}$ with basis
$X_{1},Y_{1},...,Y_{s}$, $\\{X_{1},Y_{1}\\}$ being an orthonormal basis of
$V_{1}$, $V_{j}=\operatorname{Span}(Y_{j})$ for $j=2,...,s$ and only non-
trivial bracket relations $[X_{1},Y_{i}]=Y_{i+1}$, for every $i=1,...,s-1$.

We recall that in the subRiemannian filiform groups of first type all length-
minimizing curves are normal (see for example [6, Proposition 4.1]).

######  Proposition 4.4.

Let $G$ be the sub-Riemmanian filiform group of first type of step $s$. Let
$Y_{n}$ be a vector spanning the $s$-th stratum and
$\gamma:\mathbb{R}\rightarrow G$ be a normal curve with $\gamma(0)=1_{G}$,
$\gamma(1)=\exp(Y_{n})$ and $\gamma|_{[0,1]}$ length-minimizing. Then there
exists $C>0$ such that for every $t>1$ there holds

(63) $d(1,\gamma(t))<Ct^{\frac{1}{s}}.$

###### Proof.

Let $\\{X_{1},X_{2}\\}$ be an orthonormal frame of the first stratum. Let
$\lambda$ be a co-vector associated to $\gamma$, see Definition 33. Since the
energy at the origin is differentiable with differential

$\mathrm{d}(\operatorname{E}_{1})_{w}v=\langle w,v\rangle,\ \ \forall v,w\in
T_{1}G,$

equation (33) rewrites as

(64) $\lambda(\operatorname{Ad}_{\gamma(t)}X_{i})=\langle u(t),X_{i}\rangle,\
\ \forall t\in\mathbb{R},\forall i\in\\{1,2\\}.$

We claim that

(65) $\gamma(m+s)=\exp(mY_{n})\gamma(s),\ \ \forall m\in\mathbb{N},\forall
s\in\mathbb{R}.$

Indeed, the curve $\exp(mY_{n})\gamma(s)$ solves the normal Equation (64) with
respect to the covector $\lambda$:

(66)
$\begin{split}(\exp(mY_{n})\gamma(s))^{\prime}=\mathrm{d}L_{\exp(mY_{n})}\gamma(s)^{\prime}=\\\
=\mathrm{d}L_{\exp(mY_{n})}\sum_{i=1}^{2}\lambda(\operatorname{Ad}_{\gamma(s)}X_{i})X_{i}(\gamma(s))=\\\
=\sum_{i=1}^{2}\lambda(\operatorname{Ad}_{\exp(mY_{n})\gamma(s)}X_{i})X_{i}(\exp(mY_{n})\gamma(s)),\end{split}$

where in the second equality we used that $\gamma$ solves (64) and the last
equality comes from the fact that $\exp(mY_{n})$ is in the center.
We have

(67)
$d(1,\exp(mY_{n}))=d(1,\delta_{m^{\frac{1}{s}}}(\exp(Y_{n})))=m^{\frac{1}{s}}d(1,\exp(Y_{n})),\
\ \forall m\in\mathbb{N}.$

Equation (67) together with (65) and the triangle inequality give us

(68) $d(1,\gamma(m+s))\leq m^{\frac{1}{s}}d(1,\exp(Y_{n}))+d(1,\gamma(s)),\ \
\forall m\in\mathbb{N},\forall s\in\mathbb{R}.$

In particular, being $\gamma|_{[0,1]}$ length minimizing and
$\gamma(1)=\exp(Y_{n})$, we have

(69) $\begin{split}d(1,\gamma(m+s))\leq
m^{\frac{1}{s}}d(1,\exp(Y_{n}))+d(1,\exp(Y_{n}))\leq\\\ \leq
d(1,\exp(Y_{n}))(m^{\frac{1}{s}}+1),\ \ \forall m\in\mathbb{N},\forall
s\in[0,1].\end{split}$

Consequently, setting $C=2d(1,\exp(Y_{n}))$, we have for $m>1$ that

(70) $d(1,\gamma(m+s))<Cm^{\frac{1}{s}}<C(m+s)^{\frac{1}{s}},\ \ \forall
m\in\mathbb{N},\forall s\in[0,1].$

This concludes the proof of the proposition. ∎

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