# Higher order Goh conditions for singular extremals of corank $1$

Francesco Boarotto<EMAIL_ADDRESS>Università di Padova,
Dipartimento di Matematica “Tullio Levi-Civita”, via Trieste 63, 35121 Padova,
Italy  ,  Roberto Monti<EMAIL_ADDRESS>Università di Padova,
Dipartimento di Matematica “Tullio Levi-Civita”, via Trieste 63, 35121 Padova,
Italy  and  Alessandro Socionovo<EMAIL_ADDRESS>Università di Padova,
Dipartimento di Matematica “Tullio Levi-Civita”, via Trieste 63, 35121 Padova,
Italy

###### Abstract.

We prove Goh conditions of order $n$ for strictly singular length minimizing
curves of corank $1$, under the assumption that the lower order intrinsic
differentials of the end-point map vanish. This result relies upon the proof
of an open mapping theorem for maps with non-singular $n$th differential.

##  1\. Introduction

One of the main open problems in sub-Riemannian geometry is the regularity of
length minimizing curves. Its difficulty is due to the singularities of the
end-point map, i.e., to the presence of points where its differential is not
surjective. In this paper, we study the end-point map up to order $n$
obtaining necessary conditions of Goh type for optimal trajectories.

A sub-Riemannian manifold is a triplet $(M,\Delta,g)$, where $M$ is a smooth,
i.e., $C^{\infty}$ manifold, $\Delta\subset TM$ is a distribution of rank
$2\leq d<\dim(M)$, and $g$ is metric on $\Delta$. In a neighborhood $U\subset
M$ of any point $q\in M$, there exist vector-fields
$f_{1},\dots,f_{d}\in\mathrm{Vec}(U)$ such that
$\Delta=\mathrm{span}\\{f_{1},\dots,f_{d}\\}$ on $U$. Since our considerations
are local, we can assume in the sequel that $U=M$. We also assume that
$\Delta$ satisfies Hörmander’s condition

$\mathrm{Lie}\\{f_{1},\dots,f_{d}\\}(p)=T_{p}M,\quad p\in M,$ (1.1)

i.e., that $\Delta$ is completely non-integrable. For an exhaustive
introduction to sub-Riemannian geometry, we refer the reader to [2, 3, 5, 12,
19, 23].

Let $I=[0,1]$ be the unit interval. A curve $\gamma\in AC(I;M)$ is
_horizontal_ if $\dot{\gamma}\in\Delta_{\gamma}$ a.e. on $I$, that is

$\dot{\gamma}(t)=\sum_{i=1}^{d}u_{i}(t)f_{i}(\gamma(t)),\quad\text{for
a.e.\leavevmode\nobreak\ $t\in I$},$ (1.2)

for some unique $u=(u_{1},\dots,u_{d})\in L^{1}(I;\mathbb{R}^{d})$, called
control of $\gamma$. Without loss of generality, we can assume that $g$ makes
$f_{1},\dots,f_{d}$ orthonormal, in which case the length of $\gamma$ is the
$L^{1}$-norm of its control. We can also replace the Banach space
$L^{1}(I;\mathbb{R}^{d})$ with the smaller Hilbert space
$X=L^{2}(I;\mathbb{R}^{d})$.

The end-point map $F_{q}:X\to M$ with base-point $q\in M$ is defined letting
$F_{q}(u)=\gamma_{u}(1)$, where $\gamma_{u}$ is the unique solution to (1.2)
with $\gamma_{u}(0)=q$. The point $\bar{q}=F_{q}(u)$ is the end-point of the
curve $\gamma_{u}$. Since $q\in M$ is fixed, we shall simply write $F=F_{q}$.
Controls $u\in X$ where the differential $d_{u}F$ is not surjective are called
singular and horizontal curves driven by singular controls are called
_singular_ or _abnormal_ extremals.

In his ground-breaking work [18], Montgomery first proved that singular curves
can be as a matter of fact length-minimizing. Also _nice abnormal extremals_ ,
see [17], are locally length minimizing. Examples of purely Lipschitz and
spiral-like abnormal curves in Carnot groups are presented in [14, 15], and an
algorithm for producing many new examples is proposed in [9]. The length-
minimality property of all these examples is not yet well-understood.

A recent approach to the regularity problem of length-minimizing curves is
based on the analysis of specific singularities such as corners, spiral-like
curves or curves with no straight tangent line. This approach does not use
general open mapping theorems but it rather relies on the ad hoc construction
of shorter competitors, see [4, 10, 11, 16, 20, 21, 22].

On the other hand, necessary conditions for the minimality of singular
extremals can be obtained from the differential study of the end-point map.
The theory is well-known till the second order and was initiated by Goh [8]
and developed by Agrachev and Sachkov in [3]. Using second order open mapping
theorems (index theory), for a strictly singular length minimizing curve
$\gamma$ and for any adjoint curve $\lambda$ they prove the validity of the
following Goh conditions:

$\langle\lambda,[f_{i},f_{j}](\gamma)\rangle=0,\quad\textrm{
$i,j=1,\ldots,d$.}$ (1.3)

The first order conditions $\langle\lambda,f_{i}(\gamma)\rangle=0$ are ensured
by Pontryagin Maximum Principle. Partial necessary conditions of the third
order are obtained in [7].

Our goal is to extend the second order theory of [1] to any order $n\geq 3$
and to get necessary conditions as in (1.3) involving brackets of $n$ vector
fields. There is a clear connection between the geometry of $\Delta$ and the
expansion of the end-point map $F$. In particular, the commutators of length
$n$ should appear in the $n$-th order term of the expansion of $F$. In Section
5, we provide a first positive answer to this idea. In the case of a singular
curve $\gamma_{u}$ of corank-one, i.e., such that $\mathrm{Im}(d_{u}F)$ has
codimension $1$ in $T_{\gamma_{u}(1)}M$, we get the following result. For the
definition of adjoint curve, see Section 9.

######  Theorem 1.1.

Let $(M,\Delta,g)$ be a sub-Riemannian manifold, $\gamma=\gamma_{u}\in
AC(I;M)$ be a strictly singular length minimizing curve of corank $1$, and
assume that

$\mathscr{D}^{h}_{u}F=0,\ \ h=2,\dots,n-1.$ (1.4)

Then any adjoint curve $\lambda\in AC(I;T^{*}M)$ satisfies

$\langle\lambda(t),[f_{j_{n}},[\dots[f_{j_{2}},f_{j_{1}}]\dots]](\gamma(t))\rangle=0,$
(1.5)

for all $t\in I$ and for all $j_{1},\dots,j_{n}=1,\dots,d$.

We call the differentials $\mathscr{D}^{h}_{u}F$ appearing in (1.4) _intrinsic
differentials_ of $F$. For $v_{1},\ldots,v_{h}\in X$, at the point $u=0$ we
define

$D_{0}^{h}F(v_{1},\dots,v_{h})=\frac{d^{h}}{dt^{h}}F\Big{(}\sum_{h=1}^{n}\frac{t^{h}v_{h}}{h!}\Big{)}\bigg{|}_{t=0}.$

We first restrict $D_{0}^{h}F$ to a suitable domain
$\mathrm{dom}(\mathscr{D}_{0}^{h}F)\subset X^{h-1}$ that, roughly speaking,
consists of points where the lower order differentials
$d_{0}F,D_{0}^{2}F,\dots,D_{0}^{h-1}F$ vanish. Then we define
$\mathscr{D}_{0}^{h}F=\mathrm{pr}(D_{0}^{h}F)$, with $\mathrm{pr}$ projection
onto $\mathrm{coker}(d_{0}F)$, see Definition 2.2. A motivation for this
definition is the fact that $\mathscr{D}_{0}^{h}F$ behaves covariantly, in the
sense that, for a given diffeomorphism $P\in C^{\infty}(M;M)$,
$\mathscr{D}_{0}^{h}(P\circ F)$ depends only on the first order derivatives of
$P$.

The proof of Theorem 1.1 relies on an open mapping argument applied to the
extended end-point map $F_{J}=(F,J):X\to M\times\mathbb{R}$, where
$J(u)=\frac{1}{2}\|u\|^{2}_{L^{2}(I;\mathbb{R}^{d})}$ is the energy of
$\gamma=\gamma_{u}$. Minimizing the energy is in fact equivalent to minimizing
the length, because for horizontal curves parameterized by arc-length the
$L^{2}$-norm of the control coincides with its $L^{1}$-norm.

Motivated by this application, in Section 3 we develop a theory about open
mapping theorems of order $n$ for functions $F:X\to\mathbb{R}^{m}$ between
Banach spaces. In our opinion, this preliminary study is worth of interest on
its own. It adapts in a geometrical perspective some ideas presented in [24].

######  Theorem 1.2.

Let $X$ be a Banach space and let $F\in C^{\infty}(X;\mathbb{R}^{m})$,
$m\in\mathbb{N}$, be a smooth mapping such that $\mathscr{D}_{0}^{h}F=0$ for
all $2\leq h<n$, for some $n\geq 2$, and with regular $\mathscr{D}_{0}^{n}F$
at the critical point $0\in X$. Then $F$ is open at $0$.

We refer to Section 3 for precise definitions. The notion of “regularity” used
in Theorem 1.2 is delicate because the intrinsic differential
$\mathscr{D}_{0}^{n}F:\mathrm{dom}(\mathscr{D}_{0}^{n}F)\to\mathrm{coker}(d_{0}F)$
is a non-linear mapping defined on a domain without linear structure. The
notion depends on an extension theorem for
$\mathrm{dom}(\mathscr{D}_{0}^{n}F)$ and is fixed in Definition 3.2. When
$0\in X$ is a critical point of corank $1$ it becomes more effective, see
Corollary 3.5.

The rest of the paper is devoted to the study of the open mapping property for
the extended end-point map $F_{J}$ around a singular control $u$. In fact, we
will study the auxiliary map $G$, called _variation map_ , defined by
$G(v)=F(u+v)$, in order to move the base-point from $u$ to $0$.

A crucial ingredient in our analysis is the definition of non-linear sets
$V_{h}\subset X$, $h\in\mathbb{N}$, consisting of controls with vanishing
iterated integrals for any order $h\leq n-1$, see (4.4). Using such controls
we are able to catch the geometric structure of the $n$th differential
$\mathscr{D}_{0}^{n}G$ in terms of Lie brackets. The algebraic properties of
the sets $V_{h}$ are studied in Section 4.

In Section 5, we use the formalism of chronological calculus [3, Chapter 2] to
compute the $n$th differential $\mathscr{D}_{0}^{n}G$ of the variation map and
the final outcome is formula (5.13). This formula contains a localization
parameter $s>0$ that can be used to shrink the support of the control in a
neighborhood of some point $t_{0}\in[0,1)$. Passing to the limit as $s\to
0^{+}$, we obtain a new map $\mathscr{G}^{n}_{t_{0}}:X\to\mathbb{R}$:

$\mathscr{G}^{n}_{t_{0}}(v)=\int_{\Sigma_{n}}\langle\lambda,[g_{v(t_{n})}^{t_{0}},[\dots,[g_{v(t_{2})}^{t_{0}},g_{v(t_{1})}^{t_{0}}]]\ldots](\bar{q})\rangle
dt_{1}\dots dt_{n},$

where $\Sigma_{n}=\\{0\leq t_{n}\leq t_{n-1}\leq\ldots\leq t_{1}\leq 1\\}$ is
the standard simplex, $\lambda\in T^{*}_{\bar{q}}M$ is a fixed covector
orthogonal to $\mathrm{coker}(d_{u}F)$, $\bar{q}=F(q)\in M$ is the end-point,
and $g_{v(t_{i})}^{t_{0}}$ is the pull-back of the time-dependent vector field
$f_{v}=v^{1}f_{1}+\ldots+v^{d}f_{d}$ along the flow of $u$. In the corank $1$
case, we show that if there exists $v\in V_{n-1}$ with
$\mathscr{G}^{n}_{t_{0}}(v)\neq 0$ then the extended map $G_{J}$ is open at
$0$, see our crucial Theorems 6.3 and 6.4. So $\mathscr{G}^{n}_{t_{0}}=0$ on
$V_{n-1}$ becomes a necessary condition for the length-minimality of singular
extremals.

In Sections 7 and 8, we study the geometric implications of equation
$\mathscr{G}^{n}_{t_{0}}=0$. First, we explore the symmetries of
$\mathscr{G}^{n}_{t_{0}}$, showing how the shuffle algebra of iterated
integrals interacts with generalized Jacobi identities of order $n$, see
Theorem 7.1. In spite of the non-linear structure of $V_{n-1}$, we are able to
polarize the equation $\mathscr{G}^{n}_{t_{0}}=0$ on linear subspaces of
$V_{n-1}$ of arbitrarily large dimension, thus de facto bypassing the non-
linearity of the problem.

At this point, we regard the quantities in (1.5) as unknowns of a nonsingular
system of linear equations, thus proving their vanishing. To get this
nonsingularity, we work with families of trigonometric functions having sparse
and high frequences, see Theorems 8.1 and 8.2.

Our argument leading to the final proof of Theorem 1.1 is summarized in
Section 9. In Section 10 we complete the study of a well-known example of
singular curve. In $M=\mathbb{R}^{3}$ we consider the distribution spanned by
the vector-fields

$\displaystyle f_{1}=\frac{\partial}{\partial x_{1}}\quad\textrm{and}\quad
f_{2}=(1-x_{1})\frac{\partial}{\partial
x_{2}}+x_{1}^{n}\frac{\partial}{\partial x_{3}},$ (1.6)

where $n\in\mathbb{N}$ is a parameter. Using the theory developed in Sections
2-6, we show that for odd $n$ the curve $\gamma(t)=(0,t,0)$, $t\in[0,1]$, is
not length minimizing. This is interesting because, for even $n$, this
singular curve is on the contrary locally length minimizing. In fact, for even
$n$ the $n$th differential of the end-point map is not regular according to
Definition 3.2 part i) and our open mapping theorem does not apply. See our
discussion in Remark 10.4.

We conclude this introductory part commenting on the assumptions made in
Theorem 1.1. Hypothesis (1.4) is restrictive. Its geometric meaning is
explained in Remark 9.2: in fact, it implies (but is not equivalent to)
$\langle\lambda,[f_{j_{h}},[\dots[f_{j_{2}},f_{j_{1}}]\dots]](\gamma)\rangle=0$
for $h\leq n-1$. Weakening (1.4) requires a substantial improvement of Theorem
1.2. The corank $1$ assumption on the length minimizing curve is used when
Theorem 1.2 is applied to the end-point map. We think that it should be
possible to drop this assumption, but this certainly requires some new deep
idea.

##  2\. Intrinsic differentials

Let $X$ be a Banach space and $F\in C^{\infty}(X;\mathbb{R}^{m})$,
$m\in\mathbb{N}$, be a smooth map. For any $n\in\mathbb{N}$ we define the
$n$th differential of $F$ at $0\in X$ as the map
$d_{0}^{n}F:X\to\mathbb{R}^{m}$

$d_{0}^{n}F(v)=\frac{d^{n}}{dt^{n}}F(tv)\big{|}_{t=0},\quad v\in X.$

With abuse of notation, the associated $n$-multilinear differential is the map
$d_{0}^{n}F:X^{n}\to\mathbb{R}^{m}$ defined in one of the following two
equivalent ways, for $v_{1},\dots,v_{n}\in X$,

$\begin{split}d_{0}^{n}F(v_{1},\dots,v_{n})=&\frac{\partial^{n}}{\partial
t_{1}\dots\partial
t_{n}}F\Big{(}\sum_{h=1}^{n}t_{h}v_{h}\Big{)}\bigg{|}_{t_{1}=\dots=t_{n}=0}\\\
=&\frac{1}{n!}\frac{\partial^{n}}{\partial t_{1}\dots\partial
t_{n}}d_{0}^{n}F\Big{(}\sum_{h=1}^{n}t_{h}v_{h}\Big{)}\bigg{|}_{t_{1}=\dots=t_{n}=0}.\end{split}$
(2.1)

We have the identity $d_{0}^{n}F(v)=d_{0}^{n}F(v,\dots,v)$. The differential
$d_{0}^{n}F$ is symmetric, in the sense that
$d_{0}^{n}F(v_{1},\dots,v_{n})=d_{0}^{n}F(v_{\sigma_{1}},\dots,v_{\sigma_{n}})$
for any permutation $\sigma\in S_{n}$. Here and hereafter, we use the notation
$\sigma_{i}=\sigma(i)$ for a permutation $\sigma$ and for $i=1,\ldots,n$.

A different $n$-multilinear differential for $F$ at $0$ is the map
$D_{0}^{n}F:X^{n}\to\mathbb{R}^{m}$ defined by the formula

$D_{0}^{n}F(v_{1},\dots,v_{n})=\frac{d^{n}}{dt^{n}}F\Big{(}\sum_{h=1}^{n}\frac{t^{h}v_{h}}{h!}\Big{)}\bigg{|}_{t=0},\quad
v_{1},\dots,v_{n}\in X.$ (2.2)

The multilinear differential $D_{0}^{n}F$ is not symmetric.

The $n$-multilinear differentials $d_{0}^{n}F$ and $D_{0}^{n}F$ are related
via the Faà di Bruno formula [13]. We denote by $\mathscr{I}_{n}$ the set of
$n$-multi-indices, i.e.,

$\mathscr{I}_{n}=\left\\{\alpha\mid\alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{N}^{n}\right\\},$

where $\mathbb{N}=\\{1,2,\dots\\}$ starts from $1$. When the naturals start
from $0$ we use the notation $\mathscr{I}_{n}^{0}$. Also, for $d\in\mathbb{N}$
we use the notation $\mathscr{I}_{n,d}$ for the sets of $n$-multi-indices
$\alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathscr{I}_{n}$ with values in
$\\{1,\dots,d\\}^{n}$. For $\alpha\in\mathscr{I}_{n}$, we use the standard
notation $|\alpha|=\alpha_{1}+\dots+\alpha_{n}$ for the length (or weight) of
$\alpha$ and $\alpha!=\alpha_{1}!\dots\alpha_{n}!$ for its factorial.

######  Proposition 2.1 (Faà di Bruno).

For any $n\in\mathbb{N}$ and $v_{1},\dots,v_{n}\in X$ we have

$D_{0}^{n}F(v_{1},\dots,v_{n})=\sum_{h=1}^{n}\sum_{\alpha\in\mathscr{I}_{h},|\alpha|=n}\frac{n!}{h!\alpha!}d_{0}^{h}F(v_{\alpha}),$
(2.3)

where, for $\alpha\in\mathscr{I}_{h}$, we set
$v_{\alpha}=(v_{\alpha_{1}},\dots,v_{\alpha_{h}})\in X^{h}$.

The $n$-differential $D^{n}_{0}F$, $n\geq 2$, does not transform covariantly,
in the sense that, for a generic diffeomorphism $P\in\
C^{\infty}(\mathbb{R}^{m};\mathbb{R}^{m})$ the differential $D^{n}_{0}(P\circ
F)$ depends also on the derivatives of $P$ of order $2$ and higher. In order
to have an “intrinsic” $n$-differential, we need to restrict $D_{0}^{n}F$ to a
suitable domain and project it onto $\mathrm{coker}(d_{0}F)$. We denote by
$\mathrm{pr}:\mathbb{R}^{m}\to\mathrm{coker}(d_{0}F)$ the standard projection
(quotient map). We can fix coordinates in $X$ and $\mathbb{R}^{m}$ in order to
have the splittings

$X=\ker(d_{0}F)\oplus\mathbb{R}^{m-\ell},\qquad\mathbb{R}^{m}=\mathbb{R}^{\ell}\oplus\mathrm{Im}(d_{0}F),$
(2.4)

where $\ell=\mathrm{dim}(\mathrm{coker}(d_{0}F))$ and
$d_{0}F:\mathbb{R}^{m-\ell}\to\mathrm{Im}(d_{0}F)$ is a linear isomorphism.

######  Definition 2.2 (Intrinsic $n$-differential).

Let $F\in C^{\infty}(X;\mathbb{R}^{m})$. By induction on $n\geq 2$, we define
a domain $\mathrm{dom}(\mathscr{D}_{0}^{n}F)\subset X^{n-1}$ and a map
$\mathscr{D}_{0}^{n}F:\mathrm{dom}(\mathscr{D}_{0}^{n}F)\to\mathrm{coker}(d_{0}F)$,
called _intrinsic $n$-differential_ of $F$ at $0$, in the following way.

When $n=2$ we let $\mathrm{dom}(\mathscr{D}_{0}^{2}F)=\left\\{v\in X\mid
D_{0}F(v)=0\right\\}=\ker(d_{0}F)\subset X$ and we define
$\mathscr{D}_{0}^{2}F:\mathrm{dom}(\mathscr{D}_{0}^{2}F)\to\mathrm{coker}(d_{0}F)$

$\mathscr{D}_{0}^{2}F(v)=\mathrm{pr}(D_{0}^{2}F(v,*)),\quad
v\in\mathrm{dom}(\mathscr{D}_{0}^{2}F).$ (2.5)

Inductively, for $n>2$ we set

$\mathrm{dom}(\mathscr{D}_{0}^{n}F)=\left\\{v\in\mathrm{dom}(\mathscr{D}_{0}^{n-1}F)\times
X\mid D_{0}^{n-1}F(v)=0\right\\}\subset X^{n-1},$

and we define
$\mathscr{D}_{0}^{n}F:\mathrm{dom}(\mathscr{D}_{0}^{n}F)\to\mathrm{coker}(d_{0}F)$
as

$\mathscr{D}_{0}^{n}F(v)=\mathrm{pr}(D_{0}^{n}F(v,*)),\quad
v\in\mathrm{dom}(\mathscr{D}_{0}^{n}F).$ (2.6)

######  Remark 2.3.

The definition of $\mathscr{D}_{0}^{n}F$ in (2.5) and (2.6) does not depend on
the last argument $*\in X$ of $D_{0}^{n}F$. Indeed, by formula (2.3) with
$v=(v_{1},\dots,v_{n})$, so with $*=v_{n}$ in the notation above, we have

$D_{0}^{n}F(v)=d_{0}F(v_{n})+\sum_{h=2}^{n}\sum_{\alpha\in\mathscr{I}_{h},|\alpha|=n}\frac{n!}{h!\alpha!}d_{0}^{h}F(v_{\alpha}),$
(2.7)

where $v_{\alpha}$ does not contain $v_{n}$ when $|\alpha|=n$ and $h\geq 2$,
and $\mathrm{pr}(d_{0}F(v_{n}))=0$.

While $\mathrm{dom}(\mathscr{D}_{0}^{2}F)=\ker(d_{0}F)$ has finite codimension
in $X$, this might not be the case when $n>2$. In order to develop the theory
within a consistent setting we need some additional assumption on $F$.

######  Proposition 2.4.

Let $F\in C^{\infty}(X;\mathbb{R}^{m})$ be a smooth map. If
$\mathscr{D}_{0}^{h}F=0$ for all $2\leq h<n$, with $n\geq 3$, then
$\mathrm{dom}(\mathscr{D}_{0}^{n}F)\subset X^{n-1}$ is a nonempty affine
bundle over $\ker(d_{0}F)$ that is diffeomorphic to $\ker(d_{0}F)^{n-1}$.

###### Proof.

The proof is by induction on $n\geq 3$. When $n=3$ the domain of
$\mathscr{D}_{0}^{3}F$ is

$\mathrm{dom}(\mathscr{D}_{0}^{3}F)=\big{\\{}(v_{1},v_{2})\in\ker(d_{0}F)\times
X\mid D_{0}^{2}F(v_{1},v_{2})=0\big{\\}},$

where, as in (2.7),
$D_{0}^{2}F(v_{1},v_{2})=d_{0}F(v_{2})+d^{2}_{0}F(v_{1},v_{1}).$

We use the splittings (2.4). Since the map
$d_{0}F:\mathbb{R}^{m-\ell}\to\mathrm{Im}(d_{0}F)$ is invertible, we can
define $\varphi\in C^{\infty}(\ker(d_{0}F),\mathbb{R}^{m-\ell})$ letting

$\varphi(v_{1})=-(d_{0}F)^{-1}(d^{2}_{0}F(v_{1},v_{1})).$

This is well-defined because, by assumption, we have $\mathscr{D}_{0}^{2}F=0$
and this implies $\mathrm{pr}(d^{2}_{0}F(v_{1},v_{1}))=0$. Now, letting

$\Phi(v_{1},v_{2})=(v_{1},v_{2}+\varphi(v_{1})),$

we obtain a diffeomorphism
$\Phi:\ker(d_{0}F)^{2}\to\mathrm{dom}(\mathscr{D}_{0}^{3}F)$.

Suppose the theorem is true for $n$ and let us prove it for $n+1$. Our
inductive assumption is the existence of a diffeomorphism $\Phi\in
C^{\infty}(\ker(d_{0}F)^{n-1},\mathrm{dom}(\mathscr{D}_{0}^{n}F))$. Now we
have

$\begin{split}\mathrm{dom}(\mathscr{D}_{0}^{n+1}F)&=\left\\{(v,w)\in\mathrm{dom}(\mathscr{D}_{0}^{n}F)\times
X\mid D_{0}^{n}F(v,w)=0\right\\}\\\ &=\left\\{(\Phi(u),w)\in X^{n}\mid
u\in\ker(d_{0}F)^{n-1},\,D_{0}^{n}F(\Phi(u),w)=0\right\\},\end{split}$

and, by (2.7), equation $D_{0}^{n}F(\Phi(u),w)=0$ reads

$w=\psi(u)=-(d_{0}F)^{-1}\sum_{h=2}^{n}\sum_{\alpha\in\mathscr{I}_{h},|\alpha|=n}\frac{n!}{h!\alpha!}d_{0}^{h}F(\Phi(u)_{\alpha}).$

The function $\psi$ is well-defined because $\mathscr{D}_{0}^{n}F=0$. Now
$\Psi(u,z)=(u,z+\psi(u))$ is a diffeomorphism from $\ker(d_{0}F)^{n}$ to
$\mathrm{dom}(\mathscr{D}_{0}^{n+1}F)$.

∎

The following theorem guaranties that for any finite set of elements in
$\ker(d_{0}F)$ there exists an extension with “polynomial coefficients” inside
$\mathrm{dom}(\mathscr{D}_{0}^{n}F)$ that is linear in the first factor
$\ker(d_{0}F)$.

######  Proposition 2.5.

Let $F\in C^{\infty}(X;\mathbb{R}^{m})$ be a smooth map such that
$\mathscr{D}_{0}^{h}F=0$ for all $2\leq h<n$, for some $n\geq 2$. For any
$v_{1}^{1},\ldots,v_{1}^{\ell}\in\ker(d_{0}F)\subset X$, $\ell\in\mathbb{N}$,
there exist vectors $v_{j}^{\beta}\in X$, $j=1,\dots,n-1$ and
$\beta\in\mathscr{I}_{\ell}^{0}$ with $|\beta|=j$, such that the function
$w\in C^{\infty}(\mathbb{R}^{\ell};X^{n-1})$ with coordinates

$w_{j}(t)=\sum_{\beta\in\mathscr{I}_{\ell}^{0},|\beta|=j}t^{\beta}v_{j}^{\beta},\quad
j=1,\ldots,n-1,$ (2.8)

satisfies $w(t)\in\mathrm{dom}(\mathscr{D}_{0}^{n}F)$ for all
$t\in\mathbb{R}^{\ell}$. In particular, when $\mathrm{e}^{i}$ is the $i$th
vector of the standard basis of $\mathbb{R}^{\ell}$ we have
$v_{1}^{\mathrm{e}^{i}}=v_{1}^{i}$.

###### Proof.

The proof is by induction on $n\geq 2$.

For $n=2$ the statement follows from the fact that
$\mathrm{dom}(\mathscr{D}_{0}^{2}F)=\ker(d_{0}F)$ is a vector space. In this
case, we have $j=1$ and $\beta=\mathrm{e}^{i}$ for some $i$. Fixing
$v_{1}^{\beta}=v_{1}^{i}$ with $v_{1}=(v_{1}^{1},\ldots,v_{1}^{\ell})$,
formula (2.8) gives a function $w_{1}$ with values in
$\mathrm{dom}(\mathscr{D}_{0}^{2}F)$.

We assume the claim for $n-1$ and we prove it for $n$. In particular, for
$j\leq n-2$, the vectors $v_{j}^{\beta}\in X$ are already fixed so that the
functions defined in (2.8) with $j\leq n-2$ satisfy
$(w_{1}(t),\dots,w_{n-2}(t))\in\mathrm{dom}(\mathscr{D}_{0}^{n-1}F)$ for all
$t\in\mathbb{R}^{\ell}$. Our goal is to find $v_{n-1}^{\beta}$, for
$\beta\in\mathscr{I}_{\ell}^{0}$ with $|\beta|=n-1$, such that
$w(t)=(w_{1}(t),\dots,w_{n-1}(t))\in\mathrm{dom}(\mathscr{D}_{0}^{n}F)$.

The condition $w(t)\in\mathrm{dom}(\mathscr{D}_{0}^{n}F)$ is equivalent to

  * 1)

$(w_{1}(t),\dots,w_{n-2}(t))\in\mathrm{dom}(\mathscr{D}_{0}^{n-1}F)$;

  * 2)

$D_{0}^{n-1}F(w(t))=0$.

The first condition is true by induction. By formula (2.7), the latter is
equivalent to

$d_{0}F(w_{n-1}(t))+\sum_{h=2}^{n-1}\sum_{\alpha\in\mathscr{I}_{h},|\alpha|=n-1}\frac{(n-1)!}{h!\alpha!}d_{0}^{h}F(w_{\alpha}(t))=0.$
(2.9)

We solve this equation to determine the vectors $v_{n-1}^{\beta}\in X$. By
linearity, we have

$d_{0}F(w_{n-1}(t))=\sum_{\beta\in\mathscr{I}_{\ell}^{0},|\beta|=n-1}t^{\beta}d_{0}F(v_{n-1}^{\beta}),$

and

$\displaystyle d_{0}^{h}F(w_{\alpha}(t))$
$\displaystyle=d_{0}^{h}F(w_{\alpha_{1}}(t),\dots,w_{\alpha_{1}}(t))$
$\displaystyle=\sum_{\beta^{1}\in\mathscr{I}_{\ell}^{0},|\beta^{1}|=\alpha_{1}}\dots\sum_{\beta^{h}\in\mathscr{I}_{\ell}^{0},|\beta^{h}|=\alpha_{h}}t^{\beta^{1}+\dots+\beta^{h}}d_{0}^{h}F(v_{\alpha_{1}}^{\beta^{1}},\dots,v_{\alpha_{h}}^{\beta^{h}}).$

By the identity principle of polynomials, solving equation (2.9) is equivalent
to solving the set of equations

$d_{0}F(v_{n-1}^{\beta})+\sum_{h=2}^{n-1}\sum_{\begin{subarray}{c}\alpha\in\mathscr{I}_{h},|\alpha|=n-1\\\
\beta^{1}+\dots+\beta^{h}=\beta\end{subarray}}\frac{(n-1)!}{h!\alpha!}d_{0}^{h}F(v_{\alpha_{1}}^{\beta^{1}},\dots,v_{\alpha_{h}}^{\beta^{h}})=0,$
(2.10)

for $\beta\in\mathscr{I}_{\ell}^{0}$ with $|\beta|=n-1$. This is possible
since, by assumption, we have $\mathscr{D}_{0}^{h}F=0$ for $1\leq h\leq n-1$.
This implies that
$\mathrm{pr}(d_{0}^{h}F(v_{\alpha_{1}}^{\beta^{1}},\dots,v_{\alpha_{h}}^{\beta^{h}}))=0$,
i.e.,
$d_{0}^{h}F(v_{\alpha_{1}}^{\beta^{1}},\dots,v_{\alpha_{h}}^{\beta^{h}})\in\mathrm{Im}(d_{0}F)$
and thus we can find $v_{n-1}^{\beta}\in X$ solving equation (2.10). ∎

As soon as the splitting $X=\ker(d_{0}F)\oplus X/\ker(d_{0}F)$ is fixed, the
differential $d_{0}F:X/\ker(d_{0}F)\to\mathrm{Im}(d_{0}F)$ is a linear
isomorphism. So the solutions $v_{n-1}^{\beta}$ of equations (2.10) are unique
in $X/\ker(d_{0}F)$. In this sense, Proposition 2.5 gives a unique extension
of the form (2.8) for vectors in $\ker(d_{0}F)$.

For every $t\in\mathbb{R}^{\ell}$ with $t_{i}\neq 0$ we set
$\mathrm{sgn}(t)=\big{(}\mathrm{sgn}(t_{1}),\dots,\mathrm{sgn}(t_{\ell})\big{)}$.
We call orthant each subset of $\mathbb{R}^{\ell}$ where $\mathrm{sgn}(t)$ is
constant. There are $2^{\ell}$ orthants. Given $2\ell$ elements
$v_{1}^{1,\pm},\dots,v_{1}^{\ell,\pm}\in\ker(d_{0}F)$, Proposition 2.5 gives
an extension $w^{\mathrm{sgn}(t)}\in\mathrm{dom}(\mathscr{D}_{0}^{n}F)$ of
$v_{1}^{1,\mathrm{sgn}(t_{1})},\dots,v_{1}^{\ell,\mathrm{sgn}(t_{\ell})}$ that
in each orthant has coordinates

$w_{j}^{\mathrm{sgn}(t)}(t)=\sum_{\beta\in\mathscr{I}_{\ell}^{0},|\beta|=j}t^{\beta}v_{j}^{\beta,\mathrm{sgn}(t)},\quad
j=1,\ldots,n-1.$ (2.11)

The vectors $v_{j}^{\beta,\mathrm{sgn}(t)}$ are the solutions to (2.10) in the
orthant of $t$.

######  Corollary 2.6.

Let $F\in C^{\infty}(X;\mathbb{R}^{m})$ be a smooth map such that
$\mathscr{D}_{0}^{h}F=0$ for all $2\leq h<n$. For any $2\ell$ elements
$v_{1}^{1,\pm},\dots,v_{1}^{\ell,\pm}\in\ker(d_{0}F)$, the function
$w^{\mathrm{sgn}(t)}$ defined for $t_{i}\neq 0$ in (2.11) extends to a
continuous function $w\in
C(\mathbb{R}^{\ell};\mathrm{dom}(\mathscr{D}_{0}^{n}F))$.

###### Proof.

We split the set $\\{1,\ldots,\ell\\}$ into two subsets
$1\leq{i_{1}}<\dots<{i_{k}}\leq\ell$ and $1\leq
j_{1}<\ldots<j_{\ell-k}\leq\ell$, for some $k\leq\ell$. We consider the
subspace $H=\\{t\in\mathbb{R}^{\ell}\mid t_{i_{1}}=\dots=t_{i_{k}}=0\\}$ and
define the set of multi-indices

$\mathscr{I}^{0}_{\ell,H}=\\{\beta\in\mathscr{I}^{0}_{\ell}\mid\beta_{i_{1}}=\ldots=\beta_{i_{k}}=0\big{\\}}.$

Let $\bar{t}\in H$ be such that $\bar{t}_{j_{1}}\cdots\bar{t}_{j_{\ell-k}}\neq
0$. We prove by induction on $j=1,\ldots,n-1$ that for
$\beta\in\mathscr{I}_{\ell,H}^{0}$ the vectors $v_{j}^{\beta,\mathrm{sgn}(t)}$
are constant when $t$ is close to $\bar{t}$. When $j=1$ we have
$\beta=\mathrm{e}^{j_{p}}$ and we can use the expression for
$v_{1}^{\beta,\mathrm{sgn}(t)}$ given by Proposition 2.5 to obtain
$v_{1}^{\beta,\mathrm{sgn}(t)}=v_{1}^{\mathrm{e}^{j_{p}},\mathrm{sgn}(t_{j_{p}})}=v_{1}^{j_{p},\mathrm{sgn}(\bar{t}_{j_{p}})},$
for $t$ close to $\bar{t}$.

We check the inductive step. The vectors $v_{n-1}^{\beta,\mathrm{sgn}(t)}$ are
the solutions to the equation

$d_{0}F(v_{n-1}^{\beta,\mathrm{sgn}(t)})+\sum_{h=2}^{n-1}\sum_{\begin{subarray}{c}\alpha\in\mathscr{I}_{h},|\alpha|=n-1\\\
\beta^{1}+\dots+\beta^{h}=\beta\end{subarray}}\frac{(n-1)!}{h!\alpha!}d_{0}^{h}F(v_{\alpha_{1}}^{\beta^{1},\mathrm{sgn}(t)},\dots,v_{\alpha_{h}}^{\beta^{h},\mathrm{sgn}(t)})=0,$
(2.12)

where the vectors
$v_{\alpha_{1}}^{\beta^{1},\mathrm{sgn}(t)},\dots,v_{\alpha_{h}}^{\beta^{h},\mathrm{sgn}(t)}$
are constant for $t$ close to $\bar{t}$, by inductive assumption. In fact,
$\beta^{1},\ldots,\beta^{h}\in\mathscr{I}_{\ell,H}^{0}$ if
$\beta\in\mathscr{I}_{\ell,H}^{0}$ .

Now the existence of the following limit concludes the proof:

$\lim_{t\to\bar{t}}w_{j}^{\mathrm{sgn}(t)}(t)=\lim_{t\to\bar{t}}\sum_{\beta\in\mathscr{I}_{\ell,H}^{0},|\beta|=j}v_{j}^{\beta,\mathrm{sgn}(t)},\quad
j=1,\ldots,n-1.$

∎

######  Definition 2.7 (Regular extension).

We call the function $w\in
C(\mathbb{R}^{\ell};\mathrm{dom}(\mathscr{D}_{0}^{n}F))$ of Corollary 2.6 the
_regular extension_ of $v^{1,\pm},\dots,v^{\ell,\pm}\in\ker(d_{0}F)$ to
$\mathrm{dom}(\mathscr{D}_{0}^{n}F)$.

We will use this extension to define the notion of regular $n$-differential.

##  3\. Open mapping theorem of order $n$

In this section, we prove an open mapping theorem with sufficient conditions
on the differentials of the map up to order $n\in\mathbb{N}$.

######  Definition 3.1.

Let $F\in C^{\infty}(X;\mathbb{R}^{m})$ be a smooth map. We say that $0\in X$
is a critical point of $F$ with corank $\ell\in\\{1,\dots,m\\}$ if
$\dim(\mathrm{coker}(d_{0}F))=\ell$.

######  Definition 3.2 (Regular $n$-differential).

Let $F\in C^{\infty}(X;\mathbb{R}^{m})$ be a smooth map such that $0\in X$ is
a critical point of corank $\ell\in\\{1,\dots,m\\}$ and
$\mathscr{D}_{0}^{h}F=0$ for all $2\leq h<n$, for some $n\geq 2$. We say that
$\mathscr{D}_{0}^{n}F:\mathrm{dom}(\mathscr{D}_{0}^{n}F)\to\mathrm{coker}(d_{0}F)$
is regular if:

  * i)

$n$ is even and there exist $2\ell$ elements
$v^{1,\pm},\dots,v^{\ell,\pm}\in\mathrm{dom}(\mathscr{D}_{0}^{n}F)$ such that
the function $f:\mathbb{R}^{\ell}\to\mathrm{coker}(d_{0}F)$

$f(t)=\ \mathscr{D}_{0}^{n}F(w(\varrho(t))),\quad t\in\mathbb{R}^{\ell},$
(3.1)

is a bijection with bounded inverse at zero, i.e., there exists $0<L<\infty$
such that

$|f^{-1}(\tau)|\leq L|\tau|,\quad\tau\in\mathrm{coker}(d_{0}F).$ (3.2)

Above,
$\varrho(t)=\left(\mathrm{sgn}(t_{1})|t_{1}|^{1/n},\dots,\mathrm{sgn}(t_{\ell})|t_{\ell}|^{1/n}\right)$
and $w$ is the regular extension of $v^{1,\pm},\dots,v^{\ell,\pm}$.

  * ii)

$n$ is odd and there exist
$v^{1},\dots,v^{\ell}\in\mathrm{dom}(\mathscr{D}_{0}^{n}F)$ such that the
function $f$ in (3.1) with $v^{i,\pm}=v^{i}$ is a bijection with bounded
inverse at zero.

When the corank is $\ell=1$ the notion of regular $n$-differential is
effective.

######  Proposition 3.3.

Let $F\in C^{\infty}(X;\mathbb{R}^{m})$ be a smooth map such that $0\in X$ is
a critical point of corank $\ell=1$ and $\mathscr{D}_{0}^{h}F=0$ for all
$2\leq h<n$, for some $n\geq 2$. Assume that:

  * i)

$n$ is even and there exist $2$ elements
$v^{\pm}\in\mathrm{dom}(\mathscr{D}_{0}^{n}F)$ such that
$\mathscr{D}_{0}^{n}F(v^{-})$ and $\mathscr{D}_{0}^{n}F(v^{+})$ have opposite
sign; or,

  * ii)

$n$ is odd and there exists $v\in\mathrm{dom}(\mathscr{D}_{0}^{n}F)$ such that
$\mathscr{D}_{0}^{n}F(v)\neq 0$ .

Then $\mathscr{D}_{0}^{n}F$ is regular.

###### Proof.

When $\ell=1$ and $n$ is even, the function $f$ in (3.1) is piece-wise linear.
The assumption $\mathscr{D}_{0}^{n}F(v^{-})\mathscr{D}_{0}^{n}F(v^{+})<0$
makes it injective from $\mathbb{R}$ to $\mathbb{R}$, and hence with bounded
inverse at zero.

When $n$ is odd, the function $f$ in (3.1) is linear and
$\mathscr{D}_{0}^{n}F(v)\neq 0$ makes it invertible.

∎

######  Theorem 3.4.

Let $F\in C^{\infty}(X;\mathbb{R}^{m})$ be a smooth map such that
$\mathscr{D}_{0}^{h}F=0$ for all $2\leq h<n$, for some $n\geq 2$, and with
regular $\mathscr{D}_{0}^{n}F$ at the critical point $0\in X$. Then $F$ is
open at $0$.

###### Proof.

We prove the theorem when $n$ is even. Let $0\in X$ be a critical point for
$F$ of corank $\ell\in\\{1,\dots,m\\}$. We identify
$\mathrm{coker}(d_{0}F)=\mathbb{R}^{\ell}$ and we split
$X=\mathbb{R}^{\ell-m}\oplus\ker(d_{0}F)$.

The regularity of $\mathscr{D}_{0}^{n}F$ means that there exist
$v^{1,\pm},\dots,v^{\ell,\pm}\in\mathrm{dom}(\mathscr{D}_{0}^{n}F)$, with
regular extension $w\in
C(\mathbb{R}^{\ell};\mathrm{dom}(\mathscr{D}_{0}^{n}F))$ as in Definition 2.7,
such that the function $f:\mathbb{R}^{\ell}\to\mathbb{R}^{\ell}$ in (3.1) is
bijective and satisfies (3.2).

By formula (2.8), the extension $w=(w_{1},\dots,w_{n-1})$ is of the form

$w_{j}(t)=\sum_{\beta\in\mathscr{I}_{\ell}^{0},|\beta|=j}t^{\beta}v_{j}^{\beta,\mathrm{sgn}(t)},\quad
j=1,\dots,n-1,$

where the vectors $v_{j}^{\beta,\mathrm{sgn}(t)}\in X$ are fixed in the proof
of Proposition 2.5.

We define the map $\Phi:\mathbb{R}^{m-\ell}\times\mathbb{R}^{\ell}\to X$ by

$\Phi(r,t)=r+\sum_{j=1}^{n-1}\frac{w_{j}(t)}{j!},\quad(r,t)\in\mathbb{R}^{m-\ell}\times\mathbb{R}^{\ell}.$

Above and hereafter, we identify $r\in\mathbb{R}^{\ell-m}$ with $(r,0)\in X$,
so that the sum $r+v$ with $v\in X$ is well defined. We claim that we have the
expansion

$F\left(\Phi(r,t)\right)=d_{0}F(r)+D_{0}^{n}F(w(t),0)+R(r,t),$ (3.3)

where the remainder satisfies

$\lim_{(r,t)\to 0}\frac{R(r,t)}{|r|+|t|^{n}}=0.$ (3.4)

For any positive scalar $s\geq 0$, using the homogeneity $w_{j}(st)=s^{j}w(t)$
we obtain the formula

$\Phi(0,st)=\sum_{j=1}^{n-1}\frac{s^{j}}{j!}w_{j}(t),$

and, for fixed $t$, the function $\varphi(s)=F(\Phi(0,st))$ has the Taylor
expansion

$\varphi(s)=\sum_{j=1}^{n}\frac{\varphi^{(j)}(0)}{j!}s^{j}+\frac{\varphi^{(n+1)}(\bar{s})}{(n+1)!}s^{n+1},\quad
s\in[.0,1],$ (3.5)

for some $\bar{s}\in[0,s]$.

By definition (2.2), we have
$\varphi^{(j)}(0)=D^{j}_{0}F(w_{1}(t),\dots,w_{j-1}(t))$ and since
$w(t)\in\mathrm{dom}(\mathscr{D}_{0}^{n}F)$, we deduce that
$\varphi^{(j)}(0)=0$ for $j=1,\dots,n-1$, while for $j=n$ we have

$\varphi^{(n)}(0)=D^{n}_{0}F(w_{1}(t),\dots,w_{n-1}(t),0)=D^{n}_{0}F(w(t),0).$

From the Taylor expansion (3.5) with $s=1$, we obtain

$F(\Phi(0,t))=D^{n}_{0}F(w(t),0)+E(t),\quad t\in\mathbb{R}^{\ell}$ (3.6)

where the error satisfies the estimate

$|E(t)|=\Big{|}\frac{\varphi^{(n+1)}(\bar{s})}{(n+1)!}\Big{|}\leq C|t|^{n+1},$
(3.7)

for some constant $C>0$.

Now, we obtain the expansion (3.3) adding a development in $r$ of the first
order. We are left with the proof of (3.4). Also by (3.7), the error $R(r,t)$
is estimated by a sum of the form

$|R(r,t)|\leq\sum_{0\leq i\leq 2,0\leq j\leq n+1}c_{ij}|r|^{i}|t|^{j}$

with constants $c_{ij}$ such that $c_{0j}=0$ if $j\leq n$ and $c_{10}=0$. So
we have $|R(r,t)|\leq C(|r|^{2}+|r||t|+|t|^{n+1})$ and the mixed term is
estimated by Young inequality:

$|r||t|\leq\frac{n}{n+1}|r|^{(n+1)/n}+\frac{1}{n+1}|t|^{n+1}.$

This finishes the proof of (3.4).

The map $F$ is open at $0$ if the map $F\circ\Phi$ is open at $0$. And
$F\circ\Phi$ is open at $0$ if and only if the map

$(r,t)\mapsto\Psi(r,t)=F(\Phi(r,\varrho(t)))=d_{0}F(r)+D_{0}^{n}F(w(\varrho(t)),0)+R(r,\varrho(t))$

is open at $(r,t)=0$. We will show that $\Psi$ is open at zero by a fixed
point argument.

With respect to the factorization
$(r,t)\in\mathbb{R}^{m-\ell}\times\mathbb{R}^{\ell}$, we introduce the norm
$\|(r,t)\|=\max\\{|r|,\lambda_{0}|t|\\}$ and the balls

$B_{\delta}=\\{(r,t)\in\mathbb{R}^{m-\ell}\times\mathbb{R}^{\ell}:\|(r,t)\|\leq\delta\\}$

for positive $\delta>0$. The balls $B_{\delta}$ are compact and convex. The
parameter $\lambda_{0}>0$ will be fixed later.

The map $\Psi$ is open at $0$ if for any (small) $\varepsilon>0$ there exists
$\delta>0$ such that $B_{\delta}\subset\Psi(B_{\varepsilon})$. We pick
$(\xi,\tau)\in B_{\delta}$ and we look for $(r,t)\in B_{\varepsilon}$ such
that $\Psi(r,t)=(\xi,\tau)$. We factorize

$D_{0}^{n}F(w(\varrho(t)),0)=(\mathscr{D}_{0}^{n}F(w(\varrho(t))),g(t))=(f(t),g(t)),$

and
$R(r,\varrho(t))=(R_{1}(r,t),R_{2}(r,t))\in\mathbb{R}^{m-\ell}\times\mathbb{R}^{\ell}$.
Here, with a slight abuse of notation, we are incorporating $\varrho$ into
$R_{1}$ and $R_{2}$.

Since $g$ is continuous and homogeneous of degree $1$, there exists a constant
$C_{1}>0$ such that

$|g(t)|\leq C_{1}|t|.$ (3.8)

By (3.4), for any $0<\sigma\leq 1$ there exists an $\varepsilon>0$ such that
for $|r|+|t|\leq\varepsilon$ (in particular for $(r,t)\in B_{\varepsilon}$) we
have

$|R_{1}(r,t)|+|R_{2}(r,t)|\leq\sigma(|r|+|t|).$ (3.9)

We will fix $\sigma$ in a while.

The equation $\Psi(r,t)=(\xi,\tau)$ is then equivalent to the system

$\left\\{\begin{array}[]{ll}d_{0}F(r)+g(t)+R_{1}(r,t)=\xi\\\
f(t)+R_{2}(r,t)=\tau,\end{array}\right.$ (3.10)

that reads as the following fixed-point system

$\left\\{\begin{array}[]{ll}r=d_{0}F^{-1}\big{(}\xi-g(t)-R_{1}(r,t)\big{)}=h_{1}(r,t)\\\
t=f^{-1}\big{(}\tau-R_{2}(r,t)\big{)}=h_{2}(r,t).\end{array}\right.$

We claim that the map $h=(h_{1},h_{2})$ maps $B_{\varepsilon}$ into itself,
provided that $\delta>0$, $\sigma>0$, and $\lambda_{0}>0$ are small enough.
Indeed, by (3.8) and (3.9) we have

$\begin{split}|h_{1}(r,t)|&\leq\|d_{0}F^{-1}\|(|\xi|+|g(t))|+|R_{1}(r,t)|)\\\
&\leq C_{2}(|\xi|+|t|+\sigma(|r|+|t|))\\\ &\leq
C_{2}(\delta+2\lambda_{0}\varepsilon+\sigma\varepsilon).\end{split}$

Choosing $\delta\leq\varepsilon/3C_{2}$, $\lambda_{0}=1/6C_{2}$, and
$\sigma\leq 1/3C_{2}$ we obtain $|h_{1}(r,t)|\leq\varepsilon$.

On the other hand, by (3.2) and (3.9)

$\begin{split}|h_{2}(r,t)|&\leq L(|\tau|+|R_{2}(r,t)|)\\\ &\leq
L(|\tau|+\sigma(|r|+|t|))\\\ &\leq
L(\lambda_{0}\delta+2\sigma\varepsilon).\end{split}$

Choosing $\delta\leq\varepsilon/2L$ and $\sigma\leq\lambda_{0}/4L$ we obtain
$|h_{2}(r,t)|\leq\lambda_{0}\varepsilon$. This finishes the proof that for
each $(\xi,\tau)\in B_{\delta}$ there exists $(r,t)\in B_{\varepsilon}$
solving the system (3.10).

∎

When the corank is $\ell=1$, by Proposition 3.2 we have the following:

######  Corollary 3.5.

Let $F\in C^{\infty}(X;\mathbb{R}^{m})$ be a smooth map such that $0\in X$ is
a critical point of corank $\ell=1$ and $\mathscr{D}_{0}^{h}F=0$ for all
$2\leq h<n$, for some $n\geq 2$. Assume that:

  * i)

$n$ is even and there exist $2$ elements
$v^{\pm}\in\mathrm{dom}(\mathscr{D}_{0}^{n}F)$ such that
$\mathscr{D}_{0}^{n}D(v^{-})$ and $\mathscr{D}_{0}^{n}(v^{+})$ have opposite
sign; or,

  * ii)

$n$ is odd and there exists $v\in\mathrm{dom}(\mathscr{D}_{0}^{n}F)$ such that
$\mathscr{D}_{0}^{n}F(v)\neq 0$ .

Then $F$ is open at $0$.

##  4\. Integrals on simplexes

In this section, we prove some elementary properties of integrals on simplexes
that will be used in the analysis of the end-point map. Here and hereafter,
$I=[0,1]$ denotes the unit interval. We fix $d\in\mathbb{N}$ (it will be the
rank of the distribution of vector fields on the manifold) and in the rest of
the paper we let

$X=L^{2}(I;\mathbb{R}^{d}).$

The tensor product
$\otimes:\mathbb{R}^{\ell}\times\mathbb{R}^{m}\to\mathbb{R}^{\ell m}$ is
defined by

$(v\otimes w)^{k}=v^{i}w^{j},\quad k=m(i-1)+j,$

where $1\leq i\leq\ell$ and $1\leq j\leq m$. Above, we are using the notation
$v=(v^{1},\dots,v^{\ell})\in\mathbb{R}^{\ell}$, etc. The map $\otimes$ is
associative but not commutative.

######  Definition 4.1.

For $n\in\mathbb{N}$ and $t,s\in I$ such that $t+s\leq 1$, we define the
$n$-dimensional simplex

$\Sigma_{n}(t,s)=\big{\\{}(t_{1},\dots,t_{n})\in I^{n}\mid
t<t_{n}<\dots<t_{1}<t+s\big{\\}}.$ (4.1)

When $t=0$ and $s=1$ we use the short notation $\Sigma_{n}=\Sigma_{n}(0,1)$.
We also let

$\Sigma_{n}^{\flat}(t,s)=\big{\\{}(t_{1},\dots,t_{n})\in I^{n}\mid
t<t_{1}<\dots<t_{n}<t+s\big{\\}},$ (4.2)

and $\Sigma_{n}^{\flat}=\Sigma_{n}^{\flat}(0,1).$

For $n\in\mathbb{N}$ we define the subset of $X$

$U_{n}=\left\\{v\in X\mid\int_{\Sigma_{n}}v(t_{n})\otimes\dots\otimes
v(t_{1})d\mathscr{L}^{n}=0\right\\},$ (4.3)

Here and in the following, we denote by $\mathscr{L}^{n}$ the Lebesgue measure
on $\mathbb{R}^{n}$. We also set

$V_{n}=\bigcap_{i=1}^{n}U_{i}.$ (4.4)

For any multi-index
$\alpha\in\mathscr{I}_{n,d}=\big{\\{}\alpha\in\mathscr{I}_{n}\mid\alpha_{i}\in\\{1,\dots,d\\}\big{\\}}$
and $v\in X$, we define the integral

$I_{n}^{\alpha}(v)=\int_{\Sigma_{n}}v^{\alpha_{n}}(t_{n})\dots
v^{\alpha_{1}}(t_{1})d\mathscr{L}^{n}.$

Then $v\in U_{n}$ if and only if $I_{n}^{\alpha}(v)=0$ for all
$\alpha\in\mathscr{I}_{n,d}$.

For $v\in X$, $n\in\mathbb{N}$ and $t,s\in I$ such that $t+s\leq 1$, we let

$\begin{split}I_{n}(t,s;v)&=\int_{\Sigma_{n}(t,s)}v(t_{n})\otimes\dots\otimes
v(t_{1})d\mathscr{L}^{n},\\\
I_{n}^{\flat}(t,s;v)&=\int_{\Sigma_{n}^{\flat}(t,s)}v(t_{n})\otimes\dots\otimes
v(t_{1})d\mathscr{L}^{n}.\end{split}$

######  Lemma 4.2.

For any $v\in V_{n}$ and $t\in I$ we have

$I_{n}^{\flat}(0,t;v)=(-1)^{n}I_{n}(t,1-t;v).$ (4.5)

###### Proof.

The proof is by induction on $n\in\mathbb{N}$. When $n=1$ the claim reads

$\int_{0}^{t}v(s)ds=-\int_{t}^{1}v(s)ds,\quad t\in I,$

that holds true because $v\in V_{1}$ means
$\displaystyle\int_{0}^{1}v(s)ds=0$.

We assume that formula (4.5) holds for $n-1$ and we prove it for $n$. Indeed,
using first $v\in U_{n}$ and then $v\in V_{n-1}$ we get

$\begin{split}I_{n}^{\flat}(0,t;v)&=\int_{0}^{t}v(t_{n})\otimes
I_{n-1}^{\flat}(0,t_{n};v)dt_{n}\\\ &=-\int_{t}^{1}v(t_{n})\otimes
I_{n-1}^{\flat}(0,t_{n};v)dt_{n}\\\ &=(-1)^{n}\int_{t}^{1}v(t_{n})\otimes
I_{n-1}(t_{n},1-t_{n};v)dt_{n}\\\ &=(-1)^{n}I_{n}(t,1-t;v).\end{split}$

∎

The reverse parametrization of a function $v\in X$ is the function
$v^{\flat}\in X$ defined by the formula

$v^{\flat}(t)=v(1-t),\quad t\in I.$

######  Corollary 4.3.

Let $v\in X$. Then $v\in V_{n}$ if and only if $v^{\flat}\in V_{n}$.

###### Proof.

If $v\in V_{n}$, by Lemma 4.2 with $t=0$ it follows that $v^{\flat}\in V_{n}$.
The opposite implication follows from $v^{\flat\flat}=v$. ∎

The set $V_{n}$ is stable also with respect to localization. Given $v\in X$,
$s>0$ and $t_{0}\in I$ such that $t_{0}+s\leq 1$, we define

$v_{t_{0},s}(t)=v\left(\frac{t-t_{0}}{s}\right)\chi_{[t_{0},t_{0}+s]}(t),\quad
t\in I.$ (4.6)

######  Lemma 4.4.

If $v\in V_{n}$ then $v_{t_{0},s}\in V_{n}$ for all $s>0$ and $t\in I$ such
that $t_{0}+s\leq 1$.

###### Proof.

The claim $v_{t_{0},s}\in X$ is clear. We prove that, for every $1\leq i\leq
n$,

$I_{i}(t_{0},s;v_{t_{0},s})=\int_{\Sigma_{i}(t_{0},s)}v_{t_{0},s}(t_{i})\otimes\dots\otimes
v_{t_{0},s}(t_{1})d\mathscr{L}^{n}=0.$ (4.7)

Indeed, by the change of variable
$(t_{1},\dots,t_{i})=(s\tau_{1}+t_{0},\dots,s\tau_{i}+t_{0})$, we get

$I_{i}(t_{0},s;v_{t_{0},s})=s^{i}I_{i}(0,1;v)=0.$

∎

The set $V_{n-1}\subset X$ is not a linear space and the map $v\mapsto
I_{n}(v)=I_{n}(0,1;v)$ is not additive. However, we can construct linear
subsets of $V_{n-1}$ of any finite dimension starting from one function. Given
$v\in X$, we define $v_{1},v_{2}\in X$ letting

$v_{1}=v_{0,1/2}\quad\textrm{and}\quad v_{2}=v_{1/2,1/2}.$

These are the localization of $v$ with $t_{0}=0,1/2$ and $s=1/2$.

######  Proposition 4.5.

If $v\in V_{n-1}$ then $v_{1},v_{2},v_{1}+v_{2}\in V_{n-1}$ and

$I_{n}(v_{1}+v_{2})=I_{n}(v_{1})+I_{n}(v_{2}).$

Moreover, we have $I_{n}(v_{1})=I_{n}(v_{2})=\frac{1}{2^{n}}I_{n}(v)$.

###### Proof.

The fact that $v_{1},v_{2}\in V_{n-1}$ is proved in Lemma 4.4. We show the
remaining claims. For any multi-index $\alpha\in\mathscr{I}_{h,2}$, $1\leq
h\leq n-1$, consider the integral

$I^{\alpha}(v_{1},v_{2})=\int_{\Sigma_{h}}v^{1}_{\alpha_{1}}(t_{1})\dots
v^{h}_{\alpha_{h}}(t_{h})d\mathscr{L}^{h}.$ (4.8)

Letting $I_{1}=[0,1/2]$ and $I_{2}=[1/2,1]$, the support of the function
$v^{1}_{\alpha_{1}}(t_{1})\dots v^{h}_{\alpha_{h}}(t_{h})$ is contained in the
product $I_{\alpha_{1}}\times\dots\times I_{\alpha_{h}}$. If there exist $i<j$
such that $\alpha_{i}=1<\alpha_{j}=2$, then $\Sigma_{h}\cap
I_{\alpha_{1}}\times\dots\times I_{\alpha_{h}}=\emptyset$, and then
$I^{\alpha}(v_{1}+v_{2})=0$.

The complementary case is when $\alpha_{1}=\dots=\alpha_{k}=2$ and
$\alpha_{k+1}=\dots=\alpha_{h}=1$ for some $k=0,1,\dots,h$. In this case, the
integral in (4.8) splits into the product of two integrals:

$I^{\alpha}(v_{1},v_{2})=\Big{(}\int_{\Sigma_{k}}v^{1}_{2}(t_{1})\dots
v^{k}_{2}(t_{k})d\mathscr{L}^{k}\Big{)}\Big{(}\int_{\Sigma_{h-k}}v^{k+1}_{1}(t_{k+1})\dots
v^{h}_{1}(t_{h})d\mathscr{L}^{h-k}\Big{)}.$

If $v_{1},v_{2}\in V_{n-1}$ this shows that $I^{\alpha}(v_{1},v_{2})=0$ for
all $\alpha\in\mathscr{I}_{h,2}$ and for all $h\leq n-1$. This proves that
$v_{1}+v_{2}\in V_{n-1}$.

When $h=n$ the argument above also shows that for all
$\alpha\in\mathscr{I}_{n,2}$ such that $\alpha\neq(1,\dots,1)$ and
$\alpha\neq(2,\dots,2)$ we have $I^{\alpha}(v_{1},v_{2})=0$. We conclude that

$I_{n}(v_{1}+v_{2})=\sum_{\alpha\in\mathscr{I}_{n,2}}I^{\alpha}(v_{1},v_{2})=I_{n}(v_{1})+I_{n}(v_{2}).$

∎

##  5\. Expansion of the end-point map

In this section we expand the end-point map and we compute its $n$th order
term. The computations use the language of chronological calculus for non-
autonomous vector fields. A detailed introduction to this formalism can be
found in [3, Chapter 20].

Let $M$ be a manifold with dimension $m=\mathrm{dim}(M)$. Since our analysis
is local, we shall without loss of generality identify $M$ with
$\mathbb{R}^{m}$. So $M$-valued maps will be in fact $\mathbb{R}^{m}$-valued,
fitting the framework of Section 2.

For vector fields $f_{1},\dots,f_{d}\in$ Vec$(M)$ and $u\in X$, we define the
time-dependent vector field $f_{u(t)}=\sum_{i=1}^{d}u_{i}(t)f_{i}$. For a
fixed initial point $q\in M$, the end-point map $F_{q}:X\to M$, is defined as

$F_{q}(u)=q\circ\overrightarrow{\exp}\int_{0}^{1}f_{u(t)}dt,\quad u\in X.$

Here, $\overrightarrow{\exp}\int_{0}^{1}f_{u(t)}dt$ denotes the right
exponential of a time-dependent vector field. As explained in [3, Chapter 20],
points of $M$, vector fields, and diffeomorphisms of $M$ are identified with
operators on $C^{\infty}(M)$. In this formalism, $\circ$ stands for a
composition of operators.

We denote by $\bar{q}=F_{q}(u)$ the end-point and we define the map
$G^{u}_{\bar{q}}:X\to M$ letting

$G^{u}_{\bar{q}}(v)=\bar{q}\circ\overrightarrow{\exp}\int_{0}^{1}g_{v(t)}^{u,t}dt,\quad
v\in X,$

where $g_{v(t)}^{u,t}$ is the time-dependent vector field

$g_{v(t)}^{u,t}=\mathrm{Ad}\left(\overrightarrow{\exp}\int_{1}^{t}f_{u(\tau)}d\tau\right)f_{v(t)}.$
(5.1)

The maps $F_{q}$ and $G^{u}_{\bar{q}}$ are related by the variation formula,
see [3, Formula $(2.28)$]. For $u,v\in X$ we have

$\displaystyle F_{q}(u+v)$
$\displaystyle=q\circ\overrightarrow{\exp}\int_{0}^{1}f_{u(t)+v(t)}dt$
$\displaystyle=q\circ\overrightarrow{\exp}\int_{0}^{1}f_{u(t)}dt\circ\overrightarrow{\exp}\int_{0}^{1}\mathrm{Ad}\left(\overrightarrow{\exp}\int_{1}^{t}f_{u(\tau)}d\tau\right)f_{v(t)}dt$
$\displaystyle=G^{u}_{\bar{q}}(v).$

For the definition of the $\mathrm{Ad}$ operator in chronological calculus see
[3, Chapter 2].

The control $u$ is a critical point of corank $\ell$ for $F$ if and only if 0
is a critical point of $G^{u}_{\bar{q}}$ of corank $\ell$. We shall omit the
subscript $\bar{q}$ and the superscript $u$ and write $G=G^{u}_{\bar{q}}$. We
call $G$ _variation map_.

Our next goal is to compute the Taylor’s expansion of the variation map. For
$k\in\mathbb{N}$ and $v\in X$, we define the vector field $W_{k}(v)$ as

$\begin{split}W_{k}(v)&=\int_{\Sigma_{k}}\mathrm{ad}g_{v(\tau_{k})}^{u,\tau_{k}}\circ\dots\circ\mathrm{ad}g_{v(\tau_{2})}^{u,\tau_{2}}\big{(}g_{v(\tau_{1})}^{u,\tau_{1}}\big{)}d\mathscr{L}^{k}\\\
&=\int_{\Sigma_{k}}[g_{v(\tau_{k})}^{u,\tau_{k}},\dots,g_{v(\tau_{1})}^{u,\tau_{1}}]d\mathscr{L}^{k}.\end{split}$
(5.2)

Here and hereafter, we use the following notation for the iterated bracket of
vector fields $g_{k},\dots,g_{1}$:

$[g_{k},\dots,g_{1}]=[g_{k},[\cdots,[g_{2},g_{1}]\cdots]]=\mathrm{ad}g_{k}\circ\cdots\circ\mathrm{ad}g_{2}(g_{1}).$

For a multi-index $\beta\in\mathscr{I}_{k}$ we define the operator
(composition of vector fields)

$W_{\beta}(v)=W_{\beta_{1}}(v)\circ\dots\circ W_{\beta_{k}}(v).$ (5.3)

The operator-valued map $v\mapsto W_{\beta}(v)$ introduced in (5.3) is
polynomial in $v$ with homogeneous degree $p=|\beta|$. Its $p$-polarization is
defined via the formula

$W_{\beta}(v_{1},\dots,v_{p})=\frac{1}{p!}\frac{\partial^{p}}{\partial
t_{1}\dots\partial
t_{p}}W_{\beta}\Big{(}\sum_{i=1}^{p}t_{i}v_{i}\Big{)}\bigg{|}_{t_{1}=\dots=t_{p}=0},$
(5.4)

where $v_{1},\dots,v_{p}\in X$. This definition is consistent with (2.1).

By the argument of Lemma 3.3 and Remark 3.4 in [7], for any $p\in\mathbb{N}$
and for any $v\in X$ the $p$-differential of $G$ has the representation

$d_{0}^{p}G(v)=\sum_{k=1}^{p}\sum_{\beta\in\mathscr{I}_{k},|\beta|=p}c_{\beta}W_{\beta}(v),$
(5.5)

where, for any $\beta\in\mathscr{I}_{k}$, we set

$c_{\beta}=|\beta|!\prod_{s=1}^{k}(\beta_{1}+\dots+\beta_{s})^{-1}\in\mathbb{R}.$

Using these formulas we obtain a representation for the differentials
$D^{h}_{0}G$.

######  Lemma 5.1.

For any $h\in\mathbb{N}$ and for all $v=(v_{1},\dots,v_{h})\in X^{h}$ we have

$D_{0}^{h}G(v)=\sum_{p=1}^{h}\sum_{\alpha\in\mathscr{I}_{p},|\alpha|=h}\frac{h!}{\alpha!p!}\sum_{k=1}^{p}\sum_{\beta\in\mathscr{I}_{k},|\beta|=p}c_{\beta}W_{\beta}(v_{\alpha}),$
(5.6)

where $v_{\alpha}=(v_{\alpha_{1}},\dots,v_{\alpha_{p}})$ for
$\alpha\in\mathscr{I}_{p}$.

###### Proof.

Formula (2.3) reads

$D_{0}^{h}G(v)=\sum_{p=1}^{h}\sum_{\alpha\in\mathscr{I}_{p},|\alpha|=h}\frac{h!}{\alpha!p!}d_{0}^{p}G(v_{\alpha}),$
(5.7)

and by (2.1), (5.5), and (5.4) we deduce that, for $w=(w_{1},\dots,w_{p})\in
X^{p}$,

$\begin{split}d_{0}^{p}G(w)&=\frac{1}{p!}\frac{\partial^{p}}{\partial
t_{1}\dots\partial
t_{p}}d_{0}^{p}G\Big{(}\sum_{i=1}^{p}t_{i}w_{i}\Big{)}\bigg{|}_{t_{1}=\dots=t_{p}=0}\\\
&=\frac{1}{p!}\sum_{k=1}^{p}\sum_{\beta\in\mathscr{I}_{k},|\beta|=p}c_{\beta}\frac{\partial^{p}}{\partial
t_{1}\dots\partial
t_{p}}W_{\beta}\Big{(}\sum_{h=1}^{p}t_{h}w_{h}\Big{)}\bigg{|}_{t_{1}=\dots=t_{p}=0}\\\
&=\sum_{k=1}^{p}\sum_{\beta\in\mathscr{I}_{k},|\beta|=p}c_{\beta}W_{\beta}(w).\end{split}$
(5.8)

∎

For a given $v\in X$ let us consider the localization $v_{t_{0},s}$ for some
$t_{0}\in[0,1)$ and small $s>0$, as in (4.6).

######  Proposition 5.2.

Let $h\in\mathbb{N}$, $v\in X$, and $t_{0}\in(0,1)$. For any $s\in(0,1-t_{0})$
we have

$W_{h}(v_{t_{0},s})=s^{h}\int_{\Sigma_{h}}[g_{v(t_{h})}^{t_{0}},\dots,g_{v(t_{1})}^{t_{0}}]d\mathscr{L}^{h}+O(s^{h+1}).$
(5.9)

Moreover, there exists a constant $C>0$ such that $|O(s^{h+1})|\leq Cs^{h+1}$
for all $v\in X$ with $\|v\|_{X}\leq 1$.

###### Proof.

With the notation introduced in (5.1) and omitting the superscript $u$, we
have

$\begin{split}g_{v_{t_{0},s}(t)}^{\tau}&=\mathrm{Ad}\left(\overrightarrow{\exp}\int_{1}^{\tau}f_{u(\sigma)}d\sigma\right)f_{v_{t_{0},s}(t)}\\\
&=\sum_{i=1}^{d}v_{t_{0},s}^{i}(t)\mathrm{Ad}\left(\overrightarrow{\exp}\int_{1}^{\tau}f_{u(\sigma)}d\sigma\right)f_{i}\\\
&=\sum_{i=1}^{d}v_{t_{0},s}^{i}(t)g^{\tau}_{i},\end{split}$

where $g^{\tau}_{i}$ is defined via the last identity. Letting, for
$\alpha\in\mathscr{I}_{h,d}$,

$J_{t_{0},s}^{\alpha}=\int_{\Sigma_{h}(t_{0},s)}v_{t_{0},s}^{\alpha_{h}}(\tau_{h})\dots
v_{t_{0},s}^{\alpha_{1}}(\tau_{1})[g^{\tau_{h}}_{\alpha_{h}},\dots,g^{\tau_{1}}_{\alpha_{1}}]d\mathscr{L}^{h},$
(5.10)

formula (5.2) reads

$W_{h}(v_{t_{0},s})=\sum_{\alpha\in\mathscr{I}_{h,d}}J_{t_{0},s}^{\alpha}.$

With the change of variable $\vartheta_{i}=\frac{\tau_{i}-t_{0}}{s}$, for
$i=1,\dots,n$, the integral in (5.10) becomes

$J_{t_{0},s}^{\alpha}=s^{h}\int_{\Sigma_{h}}v^{\alpha_{h}}(\vartheta_{h})\dots
v^{\alpha_{1}}(\vartheta_{1})[g^{s\vartheta_{h}+t_{0}}_{\alpha_{h}},\dots,g^{s\vartheta_{1}+t_{0}}_{\alpha_{1}}]d\mathscr{L}^{h},$
(5.11)

Since the maps

$t\mapsto
g^{t}_{i}=\mathrm{Ad}\left(\overrightarrow{\exp}\int_{1}^{t}f_{u(\sigma)}d\sigma\right)f_{i},\quad
i=1,\dots,d,$

are Lipschitz continuous, for every $i=1,\dots,d$ and $j=1,\dots,h$ we have
the expansion

$g_{i}^{s\vartheta_{j}+t_{0}}=g_{i}^{t_{0}}+O(s),$

with a uniform error $O(s)$ for $\vartheta_{j}\in I$. So we conclude that

$\displaystyle
J_{t_{0},s}^{\alpha}=s^{h}[g_{\alpha_{h}}^{t_{0}},\dots,g_{\alpha_{1}}^{t_{0}}]\int_{\Sigma_{h}}v^{\alpha_{h}}(\vartheta_{h})\dots
v^{\alpha_{1}}(\vartheta_{1})d\mathscr{L}^{h}+O(s^{h+1}).$

The claim (5.9) follows by summing over $\alpha\in\mathscr{I}_{h,d}$. ∎

######  Corollary 5.3.

Let $v\in V_{h}$, for some $h\in\mathbb{N}$, and $t_{0}\in(0,1)$. For any
$s\in(0,1-t_{0})$ we have
$d_{0}^{h}G(v_{t_{0},s},\dots,v_{t_{0},s})=O(s^{h+1})$.

###### Proof.

By formula (5.8), the $h$-differential of $G$ has the representation

$d_{0}^{h}G(w)=\sum_{k=1}^{h}\sum_{\beta\in\mathscr{I}_{k},|\beta|=h}c_{\beta}W_{\beta}(v_{t_{0},s},\dots,v_{t_{0},s}).$
(5.12)

Let $\beta\in\mathscr{I}_{k}$ with $|\beta|=h$. We claim that the coefficient
of $s^{h}$ in the expansion of $s\mapsto
W_{\beta}(v_{t_{0},s},\dots,v_{t_{0},s})$ vanishes. Indeed, consider the
coordinate $j=\beta_{i}$. By Proposition 5.2 we have

$\begin{split}W_{j}(v_{t_{0},s})&=s^{j}\sum_{\alpha\in\mathscr{I}_{j,d}}[g_{\alpha_{j}}^{t_{0}},\dots,g_{\alpha_{1}}^{t_{0}}]\int_{\Sigma_{j}}v^{\alpha_{j}}(\vartheta_{j})\dots
v^{\alpha_{1}}(\vartheta_{1})d\mathscr{L}^{j}+O(s^{j+1})\\\
&=O(s^{j+1}),\end{split}$

because for $j\leq h$ we have

$\int_{\Sigma_{j}}v^{\alpha_{j}}(\vartheta_{h})\dots
v^{\alpha_{1}}(\vartheta_{1})d\mathscr{L}^{j}=0,$

by our assumption $v\in V_{h}$ and by Lemma 4.2. The claim follows.

∎

Let $v\in X$ and $s>0$. Assume that the function $w_{1;t_{0},s}=v_{t_{0},s}$
belongs to $\ker(d_{0}G)$. By Proposition 2.5 with $\ell=1$ and $t=1$, there
exist functions $w_{j;t_{0},s}\in X$, $j=2,\dots,n-1$, such that
$w_{t_{0},s}=(w_{1;t_{0},s},\dots,w_{n-1;t_{0},s})\in\mathrm{dom}(\mathscr{D}_{0}^{n}G)$.

######  Lemma 5.4.

If $v\in V_{n-1}$ then $\|w_{j;t_{0},s}\|_{X}=O(s^{j+1})$ for all
$j=2,\dots,n-1$.

###### Proof.

The proof is by induction on $j=2,\dots,n-1$. We start with $j=2$. Since
$(w_{1;t_{0},s},w_{2;t_{0},s})\in\mathrm{dom}(\mathscr{D}_{0}^{3}G)$ we have
$D_{0}^{2}G(w_{1;t_{0},s},w_{2;t_{0},s})=0$, and by (2.3) this equation reads

$d_{0}G(w_{2;t_{0},s})=-d_{0}^{2}G(v_{t_{0},s},v_{t_{0},s})=O(s^{3}),$

by Corollary 5.3. The claim follows composing with the inverse of $d_{0}G$.

Now we assume that the claim holds for $j\leq n-2$ and we prove it for
$j=n-1$. Since $w_{t_{0},s}\in\mathrm{dom}(\mathscr{D}_{0}^{n}G)$ we have
$D_{0}^{n-1}G(w_{t_{0},s})=0$ and, by (2.3), this equation reads

$d_{0}G(w_{n-1;t_{0},s})=-d_{0}^{n-1}G(v_{t_{0},s},\dots,v_{t_{0},s})-\sum_{h=2}^{n-2}\sum_{\alpha\in\mathscr{I}_{h},|\alpha|=n-1}\frac{(n-1)!}{\alpha!h!}d_{0}^{h}G\big{(}(w_{t_{0},s})_{\alpha}\big{)}.$

We clearly have $d_{0}^{n-1}G(v_{t_{0},s},\dots,v_{t_{0},s})=O(s^{n})$, by
Corollary 5.3.

We estimate the terms in the sum. When $2\leq h\leq n-2$ and
$\alpha\in\mathscr{I}_{h}$ with $|\alpha|=n-1$, the multi-index $\alpha$
contains at least one coordinate different from $1$ and does not contain the
coordinate $n-1$, and so

$\mathrm{Card}\\{j\mid\alpha_{j}=1\\}+\sum_{i=2}^{n-2}(i+1)\,\mathrm{Card}\\{j\mid\alpha_{j}=i\\}>|\alpha|=n-1.$

Then, from our inductive assumption it follows that
$d_{0}^{h}G\big{(}(w_{t_{0},s})_{\alpha}\big{)}=O(s^{n})$.

∎

######  Lemma 5.5.

Let $v\in X$, $s>0$ and $t_{0}\in[0,1)$ be such that
$v_{t_{0},s}\in\ker(d_{0}G)$ and let
$w_{t_{0},s}\in\mathrm{dom}(\mathscr{D}_{0}^{n}G)$ be the extension of
$v_{t_{0},s}$. If $v\in V_{n-1}$ then we have

$D_{0}^{n}G(w_{t_{0},s})=c_{n}s^{n}\int_{\Sigma_{n}}[g_{v(t_{n})}^{t_{0}},\dots,g_{v(t_{1})}^{t_{0}}]d\mathscr{L}^{n}+O(s^{n+1}).$
(5.13)

###### Proof.

By formula (5.7),

$D_{0}^{n}G(w_{t_{0},s})=\sum_{p=1}^{n}\sum_{\alpha\in\mathscr{I}_{p},|\alpha|=n}\frac{n!}{\alpha!p!}d_{0}^{p}G((w_{t_{0},s})_{\alpha}).$

If $\alpha\in\mathscr{I}_{p}$ has one entry different from 1, then
$d_{0}^{p}G((w_{t_{0},s})_{\alpha})=O(s^{n+1})$ by Lemma 5.4. The leading term
is given by $p=n$ and $\alpha\in\mathscr{I}_{n}$ with $\alpha=(1,\dots,1)$.
The expansion of this term is given by formula (5.9) with $h=n$ and this
yields formula (5.13).

∎

##  6\. Open mapping property for the extended end-point map

In this section, we study the open mapping property for the extended end-point
map at critical points of corank $1$. As in Section 5, we denote by
$\bar{q}=F_{q}(u)$ the end-point and we consider the variation map
$G=G_{\bar{q}}^{u}$. The cokernel $\mathrm{coker}(d_{0}G)$ is a subset of the
tangent space $T_{\bar{q}}M$. We identify $M$ and $T_{\bar{q}}M$ with
$\mathbb{R}^{m}$.

Let $f_{1},\dots,f_{d}\in\mathrm{Vec}(M)$ be smooth vector fields on the
manifold $M$ spanning the distribution $\Delta$ and satisfying the Hörmander
condition (1.1). If $f_{1},\dots,f_{d}$ are declared orthonormal, the length
of a horizontal curve $\gamma\in AC(I;M)$, $\dot{\gamma}=f_{u}(\gamma)$, is
$L(\gamma)=\|u\|_{L^{1}(I;\mathbb{R}^{d})}$ while its energy is given by the
functional $J:X\to[0,\infty)$

$J(u)=\frac{1}{2}\|u\|^{2}_{L^{2}(I;\mathbb{R}^{d})}.$ (6.1)

The minimizers of $J$ coincide with minimizers of the length by standard
arguments. The extended end-point map is the map $F_{J}:X\to
M\times\mathbb{R}$ given by $F_{J}(u)=(F(u),J(u))$.

######  Definition 6.1 (regular, singular, strictly singular).

A critical point $u\in X$ of $F_{J}$ is regular (singular) if there exists
$(\lambda,\lambda_{0})\in\mathrm{Im}(d_{u}F_{J})^{\perp}\subset
T_{F(u)}^{*}M\times\mathbb{R}$ such that $\lambda_{0}\neq 0$ ($\lambda_{0}=0$,
respectively). A critical point $u\in X$ is strictly singular if for every
$(\lambda,\lambda_{0})\in\mathrm{Im}(d_{u}F_{J})^{\perp}$ we have
$\lambda_{0}=0$.

We can also define the extended variation map $G_{J}(v)=(F(u+v),J(u+v))$.
Then, $0\in X$ is a regular, singular, strictly singular critical point of
$G_{J}$ if and only if $u$ is a regular, singular, strictly singular critical
point of $F_{J}$, respectively.

We are interested in strictly singular critical points of $F_{J}$. In this
case, $\mathrm{Im}(d_{u}F_{J})=\mathrm{Im}(d_{u}F)\oplus\mathbb{R}$, that is,
$\mathrm{coker}(d_{u}F_{J})$ and $\mathrm{coker}(d_{u}F)$ are isomorphic and
can be identified. The differential analysis of the extended map $F_{J}$ can
be consequently reduced to the analysis of the end-point map $F$. In fact, for
any $h\geq 2$ we have
$\mathscr{D}_{u}^{h}F_{J}=\mathscr{D}_{u}^{h}F\big{|}_{\ker(d_{u}F_{J})}$,
where the kernel $\ker(d_{u}F_{J})=\ker(d_{u}F)\cap\ker(d_{u}J)$ is finitely
complemented in $X$, and the restriction to $\ker(d_{u}F_{J})$ means
$\mathrm{dom}(\mathscr{D}_{0}^{h}F_{J})=\\{v\in\mathrm{dom}(\mathscr{D}_{0}^{h}F)\,|\,v_{1}\in\ker(d_{u}J)\\}$.
Similarly, we have

$\mathscr{D}_{0}^{h}G_{J}=\mathscr{D}_{0}^{h}G\big{|}_{\ker(d_{0}G_{J})},\qquad
h\geq 2,$ (6.2)

with $\ker(d_{0}G_{J})$ finitely complemented in $X$, and

$\mathrm{dom}(\mathscr{D}_{0}^{h}G_{J})=\\{v\in\mathrm{dom}(\mathscr{D}_{0}^{h}G)\,|\,v_{1}\in\ker(d_{u}J)\\}.$
(6.3)

Finally, $0\in X$ is a critical point for $G_{J}$ of corank $\ell=1$ if and
only if $u$ is a critical point for $G$ of corank $\ell=1$.

Thanks to the previous remarks, we can without loss of generality consider the
situation where $0$ is a corank-one critical point for $G$. This means that
$\mathrm{coker}(d_{0}G)$ has dimension $1$. We fix a nonzero dual vector
$\lambda\in\mathrm{coker}(d_{0}G)^{*}$ such that
$\langle\lambda,w\rangle=\mathrm{pr}(w)$, $w\in\mathbb{R}^{m}$, where
$\mathrm{pr}$ is the projection on $\mathrm{coker}(d_{0}G)$.

For $n\geq 2$ and $t_{0}\in[0,1)$, we consider the function
$\mathscr{G}^{n}_{t_{0}}:X\to\mathbb{R}$

$\mathscr{G}^{n}_{t_{0}}(v)=\int_{\Sigma_{n}}\langle\lambda,[g_{v(t_{n})}^{t_{0}},\dots,g_{v(t_{1})}^{t_{0}}]\rangle
d\mathscr{L}^{n},\quad v\in X.$ (6.4)

This is the coefficient of the leading term in the expansion of
$D_{0}^{n}G(w_{t_{0},s})$ in (5.13), up to the constant $c_{n}$. Here and
hereafter, vector fields are evaluated at the end-point $\bar{q}$, with
notation as in the previous section.

For a multi-index $\alpha\in\mathscr{I}_{n,d}$ let us introduce the short
notation

$[g_{\alpha}^{t_{0}}]=[g_{\alpha_{n}}^{t_{0}},\dots,g_{\alpha_{1}}^{t_{0}}],$
(6.5)

where the entries $\alpha_{1},\dots,\alpha_{n}$ appear in the bracket with
reversed order, and

$I^{\alpha}(v)=\int_{\Sigma_{n}}v^{\alpha_{n}}(t_{n})\dots
v^{\alpha_{1}}(t_{1})d\mathscr{L}^{n}.$ (6.6)

Then formula (6.4) reads

$\mathscr{G}^{n}_{t_{0}}(v)=\sum_{\alpha\in\mathscr{I}_{n,d}}\langle\lambda,[g_{\alpha}^{t_{0}}]\rangle
I^{\alpha}(v).$ (6.7)

######  Lemma 6.2.

If $v\in V_{n-1}$ then $v^{\flat}\in V_{n-1}$ and
$\mathscr{G}^{n}_{t_{0}}(v^{\flat})=(-1)^{n-1}\mathscr{G}^{n}_{t_{0}}(v)$.

###### Proof.

We have $v^{\flat}\in V_{n-1}$ by Corollary 4.3. By Lemma 4.2 – here we use
the assumption $v\in V_{n-1}$, – the integrals $I^{\alpha}(v)$ can be
transformed in the following way:

$\begin{split}I^{\alpha}(v)&=\int_{0}^{1}v^{\alpha_{n}}(t_{n})\Big{(}\int_{\Sigma_{n-1}(t_{n},1-t_{n})}v^{\alpha_{n-1}}(t_{n-1})\dots
v^{\alpha_{1}}(t_{1})d\mathscr{L}^{n-1}\Big{)}dt_{n}\\\
&=(-1)^{n-1}\int_{0}^{1}v^{\alpha_{n}}(t_{n})\Big{(}\int_{\Sigma_{n-1}^{\flat}(0,t_{n})}v^{\alpha_{n-1}}(t_{n-1})\dots
v^{\alpha_{1}}(t_{1})d\mathscr{L}^{n-1}\Big{)}dt_{n}\\\
&=(-1)^{n-1}\int_{\Sigma_{n}^{\flat}}v^{\alpha_{n}}(t_{n})\dots
v^{\alpha_{1}}(t_{1})d\mathscr{L}^{n}\\\
&=(-1)^{n-1}I^{\alpha}(v^{\flat}).\end{split}$

The last identity follows by the change of variable $t_{i}=1-s_{i}$. This
proves our claim
$\mathscr{G}^{n}_{t_{0}}(v^{\flat})=(-1)^{n-1}\mathscr{G}^{n}_{t_{0}}(v)$.

∎

In the next step, we show that if $\mathscr{G}^{n}_{t_{0}}$ is positive and
additive on a suitable subspace of $V_{n-1}$ then the extended map $G_{J}$ is
open at zero. As usual, $m=\mathrm{dim}(M)$ is the dimension of the manifold.

######  Theorem 6.3.

Let $0$ be a strictly singular critical point of $G_{J}$ with corank $1$ and
assume that $\mathscr{D}_{0}^{h}G=0$ for $2\leq h\leq n-1$ with $n\geq 2$.
Also, assume that there exist $t_{0}\in[0,1)$ and $v_{1},\dots,v_{k}\in
V_{n-1}$ such that:

  * i)

$k\geq m+1$ when $n$ is even and $k\geq m+2$ when $n$ is odd;

  * ii)

$\mathscr{G}^{n}_{t_{0}}(v_{i})=1$ for $i=1,\dots,k$;

  * iii)

$v_{1},\dots,v_{k}$ span a vector space $Y\subset V_{n-1}$ of dimension $k$;

  * iv)

$\mathscr{G}^{n}_{t_{0}}$ is additive on $v_{1},\dots,v_{k}$, in the sense
that

$\mathscr{G}^{n}_{t_{0}}\Big{(}\sum_{i=1}^{k}\tau_{i}v_{i}\Big{)}=\sum_{i=1}^{k}\mathscr{G}^{n}_{t_{0}}(\tau_{i}v_{i})$

for any $\tau_{1},\ldots,\tau_{k}\in\mathbb{R}$.

Then the extended map $G_{J}$ is open at $0$.

###### Proof.

The kernel $\ker(d_{0}G)$ has codimension $m-1$ in $X$ and thus
$\ker(d_{0}G_{J})=\ker(d_{0}G)\cap\ker(d_{0}J)$ has codimension at most $m$ in
$X$.

For $s>0$, let $L_{s}:X=L^{1}(I;\mathbb{R}^{d})\to
X_{s}:L^{1}([t_{0},t_{0}+s];\mathbb{R}^{d})$ be the linear isomorphism
$L_{s}(v)=v_{t_{0},s}$. Since $\ker(d_{0}G_{J})\cap X_{s}$ has codimension at
most $m$ in $X_{s}$, then $L_{s}^{-1}(\ker(d_{0}G_{J})\cap X_{s})\subset X$
has codimension at most $m$ and thus

$\displaystyle\dim(Y\cap L_{s}^{-1}(\ker(d_{0}G_{J})\cap X_{s}))\geq
1\qquad\textrm{when $n$ is even,}$ (6.8) $\displaystyle\dim(Y\cap
L_{s}^{-1}(\ker(d_{0}G_{J})\cap X_{s}))\geq 2\qquad\textrm{when $n$ is odd.}$
(6.9)

We discuss the case when $n$ is even. By iv), $n$-homogeneity and ii), for
$\tau\in\mathbb{R}^{k}$, $\tau\neq 0$, we have

$\mathscr{G}^{n}_{t_{0}}\Big{(}\sum_{i=1}^{k}\tau_{i}v_{i}\Big{)}=\sum_{i=1}^{k}\tau_{i}^{n}>0.$

Thus, the function $\mathscr{G}^{n}_{t_{0}}$ attains a positive minimum on the
sphere $K=\\{v\in Y:\|v\|_{X}=1\\}$: there exists $\delta>0$ such that

$\mathscr{G}^{n}_{t_{0}}(v)\geq\delta>0\quad\textrm{for all }v\in\ K.$ (6.10)

By (6.8), for any $s>0$ there exists $v\in K$ such that
$v_{t_{0},s}\in\ker(d_{0}G_{J})$. This $v\in K$ depends on $s$. Let
$w_{t_{0},s}$ be the extension of $v_{t_{0},s}$ given by Proposition 2.5
applied with $\ell=1$ and $t=1$. We have
$w_{t_{0},s}\in\mathrm{dom}(\mathscr{D}_{0}^{n}G_{J})$ for all $s>0$. By (6.2)
and by formula (5.13)

$\mathscr{D}_{0}^{n}G_{J}(w_{s,t_{0}})=\mathscr{D}_{0}^{n}G(w_{s,t_{0}})=s^{n}c_{n}\mathscr{G}^{n}_{t_{0}}(v)+O(s^{n+1}),$

where $|O(s^{n+1})|\leq C_{1}s^{s+1}$ for a constant $C_{1}>0$ independent of
$v$ with $\|v\|_{X}\leq 1$. Choosing $0<s<\frac{1}{2}\delta/C_{1}c_{n}$, from
(6.10) we deduce that

$\mathscr{D}_{0}^{n}G_{J}(w_{s,t_{0}})\geq
s^{n}\big{(}c_{n}\mathscr{G}^{n}_{t_{0}}(v)-C_{1}s\big{)}\geq
s^{n}\frac{\delta}{2}>0.$

By Lemma 6.2, for $n$ even we have
$\mathscr{G}^{n}_{t_{0}}(v^{\flat})=-\mathscr{G}^{n}_{t_{0}}(v)$. Repeating
the above argument starting from $v_{1}^{\flat},\dots,v_{k}^{\flat}$, we
conclude that for all $s>0$ small enough there exists
$w_{t_{0},s}^{\flat}\in\mathrm{dom}(\mathscr{D}_{0}^{n}G_{J})$ such that
$\mathscr{D}_{0}^{n}G_{J}(w_{s,t_{0}}^{\flat})=-1$. By Corollary 3.5 part i),
we conclude that $G_{J}$ is open at $0$.

We pass to the case when $n$ is odd. Let $H\subset Y$ be the set $H=\\{v\in
Y:\mathscr{G}^{n}_{t_{0}}(v)=1\\}.$ We claim that there exists $v\in H\cap
T_{s}^{-1}(\ker(d_{0}G_{J})\cap X_{s})$ such that $\|v\|_{X}\leq C_{2}$ for a
constant $C_{2}$ independent of $s>0$.

We may assume that $k=m+2$ and on $Y$ we fix the coordinates
$\tau\in\mathbb{R}^{k}$ with respect to the basis $v_{1},\dots,v_{k}$, i.e.,
$v=\tau_{1}v_{1},+\dots+\tau_{k}v_{k}\in Y$. The equation
$\mathscr{G}^{n}_{t_{0}}(v)=1$ reads

$\tau_{1}^{n}+\dots+\tau_{k}^{n}=1.$ (6.11)

The set $Y\cap L_{s}^{-1}(\ker(d_{0}G_{K})\cap X_{s})$ contains a hyperplane
of the form $b_{1}\tau_{1}+\dots+b_{k}\tau_{k}=0$ where
$(b_{1},\dots,b_{k})\neq 0$ are coefficients depending on $s>0$. Without loss
of generality we can assume that $b_{k}=1$ and $|b_{1}|,\dots,|b_{k-1}|\leq
1$. Then last equation reads
$\tau_{k}=-(b_{1}\tau_{1}+\dots+b_{k-1}\tau_{k-1})$, and equation (6.11)
becomes

$p(\tau_{1},\dots,\tau_{k-1})=-(b_{1}\tau_{1}+\dots+b_{k-1}\tau_{k-1})^{n}+\tau_{1}^{n}+\dots+\tau_{k}^{n}=1.$
(6.12)

The polynomial $p$ is not the zero polynomial and has homogeneous degree $n$,
that is odd. So the equation $p(\tau_{1},\dots,\tau_{k-1})=1$ has solutions,
for any $b_{1}\dots,b_{k-1}$. We are left to show that there are solutions
bounded by a constant independent of $b_{1},\dots,b_{k-1}$.

For $\delta\in(0,1)$ consider the set

$Q_{\delta}=\big{\\{}b=(b_{1},\dots,b_{k-1})\in\mathbb{R}^{k-1}\mid\min\\{|b_{1}|,\dots,|b_{k-1}|\\}\geq\delta,\,\max\\{|b_{1}|,\dots,|b_{k-1}|\\}\leq
1\big{\\}}.$

When $b\in Q_{\delta}$, we look for a solution
$\tau=(\tau_{1},\dots,\tau_{k-1})$ of equation (6.12) of the form
$\tau=tb/|b|$ for some $t\in\mathbb{R}$, where
$|b|=\sqrt{b_{1}^{2}+\dots+b_{k-1}^{2}}$. The equation reads

$t^{n}\Big{[}\frac{b_{1}^{n}+\dots+b_{k-1}^{n}}{|b|^{n}}-|b|^{n}\Big{]}=1.$
(6.13)

On $Q_{\delta}$, the quantity $(b_{1}^{n}+\dots+b_{k-1}^{n})/{|b|^{n}}$
attains the maximum at the points where all the coordinates are $\delta$
except one that is $1$, while $-|b|^{n}$ attains the maximum at the vertex
$(\delta,\dots,\delta)$. So we get

$\frac{b_{1}^{n}+\dots+b_{k-1}^{n}}{|b|^{n}}-|b|^{n}\leq\frac{1+(k-2)\delta^{n}}{(1+(k-2)\delta^{2})^{n/2}}-((k-1)\delta^{2})^{n/2}=\psi(\delta).$

Since $\psi(1)=(k-1)^{1-n/2}-(k-1)^{n/2}<0$ for $k>1$, there exist
$\varepsilon_{0}>0$ and $0<\delta<1$ such that
$\psi(\delta)\leq-\varepsilon_{0}$. Now $\delta=\delta(n)$ is fixed. The
solution $t\in\mathbb{R}$ of equation (6.13) is negative and satisfies
$|t|\leq\varepsilon_{0}^{-1/n}$. This proves the existence of a solution
$(\tau_{1},\ldots,\tau_{k-1})\in\mathbb{R}^{k-1}$ of (6.12) that is uniformly
bounded when $b\in Q_{\delta}$.

In the case that $\min\\{|b_{1}|,\dots,|b_{k-1}|\\}\leq\delta$ we argue as
follows. Assume for instance that $|b_{1}|\leq\delta$. With
$\tau_{2}=\dots\tau_{k-1}=0$ we have the equation
$-(b_{1}\tau_{1})^{n}+\tau_{1}^{n}=1$ that has a solution $\tau_{1}>0$ bounded
by

$\tau_{1}\leq\Big{(}\frac{1}{1-\delta^{n}}\Big{)}^{1/n}.$

Now the proof can be concluded as in the case of even $n$.

∎

In fact, in order $G_{J}$ to be open it is sufficient that
$\mathscr{G}^{n}_{t_{0}}$ is positive at one element of $V_{n-1}$.

######  Theorem 6.4.

Let $0$ be a strictly singular critical point of $G_{J}$ with corank $1$ and
assume that $\mathscr{D}_{0}^{h}G=0$ for $2\leq h\leq n-1$ with $n\geq 2$. If
there exist $t_{0}\in[0,1)$ and $v\in V_{n-1}$ such that
$\mathscr{G}^{n}_{t_{0}}(v)\neq 0$, then $G_{J}$ is open at $0$.

###### Proof.

For any $k\in\mathbb{N}$, we apply iteratively Proposition 4.5 to find $2^{k}$
functions $v_{1},\dots,v_{2^{k}}$ with mutually disjoint support, spanning a
linear space in $V_{n-1}$ and such that

$\mathscr{G}^{n}_{t_{0}}\Big{(}\sum_{i=1}^{2^{k}}v_{i}\Big{)}=\sum_{i=1}^{2^{k}}\mathscr{G}^{n}_{t_{0}}(v_{i}),$

and
$\mathscr{G}^{n}_{t_{0}}(v_{i})=\frac{1}{2^{kn}}\mathscr{G}^{n}_{t_{0}}(v)$.
The claim follows from Theorem 6.3. ∎

If $G_{J}$ is not open at $0$ we have $\mathscr{G}^{n}_{t_{0}}(v)=0$ for all
$t_{0}\in[0,1)$ and $v\in V_{n-1}$. Even though $V_{n-1}$ is not a linear
space, we polarize the map $T=\mathscr{G}^{n}_{t_{0}}$.

The polarization of $T:X\to\mathbb{R}$ is the multilinear map
$\mathscr{T}:X^{n}\to\mathbb{R}$ defined in one of the two equivalent ways

$\begin{split}\mathscr{T}(v_{1},\dots,v_{n})&=\frac{\partial^{n}}{\partial
t_{1}\dots\partial
t_{n}}T\Big{(}\sum_{h=1}^{n}t_{h}v_{h}\Big{)}\bigg{|}_{t_{1}=\dots=t_{n}=0}\\\
&=\frac{1}{n!}\sum_{\sigma\in
S_{n}}\sum_{\alpha\in\mathscr{I}_{n,d}}\langle\lambda,[g^{t_{0}}_{\alpha}]\rangle\int_{\Sigma_{n}}v^{\alpha_{n}}_{\sigma_{n}}(t_{n})\dots
v^{\alpha_{1}}_{\sigma_{1}}(t_{1})d\mathscr{L}^{n}.\end{split}$ (6.14)

If $Y\subset X$ is a linear subspace such that $T=0$ on $Y$ then
$\mathscr{T}=0$ on $Y^{n}$. This follows easily from the differential
definition of polarization. Linear spaces $Y\subset V_{n-1}$ where $T=0$ can
be obtained with the following construction.

Fix $w_{1},\dots,w_{n}\in X$, with coordinates
$w_{i}=(w_{i}^{1},\dots,w_{i}^{d})$, and fix a selection function
$s:\\{1,\dots,n\\}\to\\{1,\dots,d\\}$, $s(i)=s_{i}$. This function will be
used to select (with multiplicity) which vector-fields from
$f_{1},\dots,f_{d}$ appear in the bracket $[g_{\alpha}^{t_{0}}]$.

We can define $v_{1},\dots,v_{n}\in X$ setting, for $i=1,\dots,n$,

$v_{i}=(0,\dots,0,u^{i},0,\dots,0),\quad\textrm{with }u^{i}=w_{i}^{s_{i}},$
(6.15)

where $u^{i}$ is the $i$th coordinate. Then we define $u\in X$ as
$u=(u^{1},\dots,u^{n})$. The function $u$ depends on the selection function
$s$, but we do not keep track of this dependence in our notation. As in (6.6),
for a permutation $\sigma\in S_{n}$ we let

$I^{\sigma}(u)=\int_{\Sigma_{n}}u^{\sigma_{n}}(t_{n})\cdots
u^{\sigma_{1}}(t_{1})d\mathscr{L}^{n}.$

When $u\in V_{n-1}$, where now $V_{n-1}$ is defined as in (4.4) and (4.3) but
with $d=n$, the polarization $\mathscr{T}$ takes the following form.

######  Lemma 6.5.

For any selection function $s$, if $u\in V_{n-1}$ then $v_{1},\dots,v_{n}$
span a linear subspace of $V_{n-1}$ and

$\begin{split}\mathscr{T}(v_{1},\dots,v_{n})&=\frac{1}{n!}\sum_{\sigma\in
S_{n}}\langle\lambda,[g^{t_{0}}_{s\sigma}]\rangle I^{\sigma}(u).\end{split}$
(6.16)

###### Proof.

Given $\sigma\in S_{n}$ and $\alpha\in\mathscr{I}_{n,d}$, by the structure
(6.15) of $v_{1},\dots,v_{n}$, there holds

$\int_{\Sigma_{n}}v^{\alpha_{n}}_{\sigma_{n}}(t_{n})\dots
v^{\alpha_{1}}_{\sigma_{1}}(t_{1})d\mathscr{L}^{n}=0$

as soon as there exists $i$ such that $\alpha_{i}\neq s(\sigma_{i})$. For the
surviving terms it must be $\alpha=s\sigma$ and in this case

$\int_{\Sigma_{n}}v^{\alpha_{n}}_{\sigma_{n}}(t_{n})\dots
v^{\alpha_{1}}_{\sigma_{1}}(t_{1})d\mathscr{L}^{n}=\int_{\Sigma_{n}}w^{s_{\sigma_{n}}}_{\sigma_{n}}(t_{n})\dots
w^{s_{\sigma_{1}}}_{\sigma_{1}}(t_{1})d\mathscr{L}^{n}=I^{\sigma}(u).$

The claim follows from the combinatorial definition of polarization in (6.14).

∎

##  7\. Generalized Jacobi identities and integrals on simplexes

In this section we fix a selection function
$s:\\{1,\dots,n\\}\to\\{1,\dots,d\\}$ and $u=(u^{1},\dots,u^{n})$, as in
(6.15). For varying $\sigma\in S_{n}$, the brackets
$[g_{\sigma}^{t_{0}}]=[g_{\sigma_{n}}^{t_{0}},\dots,g_{\sigma_{1}}^{t_{0}}]$
satisfy several linear relations. Using generalized Jacobi identities, we
clean up formula (6.16) getting rid of these relations. Our goal is to prove
the following formula, where we work with permutations $\sigma\in S_{n}$
fixing $1$

$S_{n}^{1}=\\{\sigma\in S_{n}\mid\sigma_{1}=1\\}.$

######  Theorem 7.1.

For any selection function $s$ and for any $v_{1},\dots,v_{n}$ as in (6.15),
we have the identity

$\mathscr{T}(v_{1},\dots,v_{n})=\frac{1}{(n-1)!}\sum_{\sigma\in
S_{n}^{1}}\langle\lambda,[g^{t_{0}}_{s\sigma}]\rangle I^{\sigma}(u).$ (7.1)

The fact that in (7.1) the sum is restricted to permutations fixing $1$ will
be important in the next section. The proof relies upon the generalized Jacobi
identities proved in [6]. For $n,j\in\mathbb{N}$ with $1\leq j\leq n$, let us
consider the sets of permutations

$X_{nj}=\\{\xi\in S_{n}\,|\,\xi_{1}>\xi_{2}>\dots>\xi_{j}=1\textrm{ and
}\,\xi_{j}<\xi_{j+1}<\dots<\xi_{n}\\},$ (7.2)

and

$X_{n}=\bigcup_{j=1}^{n}X_{nj}.$ (7.3)

The set $X_{n1}$ contains only the identity permutation, while $X_{nn}$
contains only the order reversing permutation. We are denoting elements of
$X_{n}$ by $\xi$, while in [6] they are denoted by $\pi$.

Let $g_{1},\dots,g_{n}$ be elements of a Lie algebra. The action of a
permutation $\sigma\in S_{n}$ on the iterated bracket
$[g_{n},\dots,g_{1}]=[g_{n},[\dots,[g_{2},g_{1}]\dots]]$ is denoted by

$\sigma[g_{n},\dots,g_{1}]=[g_{\sigma_{n}},\dots,g_{\sigma_{1}}].$

The selection function $s$ acts similarly,
$s[g_{n},\dots,g_{1}]=[g_{s_{n}},\dots,g_{s_{1}}]$, and so in the notation
used above we have $[g^{t_{0}}_{s\sigma}]=s[g^{t_{0}}_{\sigma}]$.

The generalized Jacobi identities of order $n$ that we need are described in
the next theorem.

######  Theorem 7.2.

For any Lie elements $g_{1},\dots,g_{n}$ and for any permutation $\sigma\in
S_{n}$ such that $\sigma_{1}\neq 1$,

$\Big{(}\sigma+\sum_{\xi\in
X_{n},\,\sigma\xi(1)=1}(-1)^{\xi^{-1}(1)}\sigma\xi\Big{)}[g_{n},\dots,g_{1}]=0,$
(7.4)

where $X_{n}$ is the set of permutations introduced in (7.3).

###### Proof.

The proof of formula (7.4) is contained in [6] on pages 117 and 119. ∎

######  Lemma 7.3.

For $v_{1},\dots,v_{n}$ as in (6.15), we have the identity

$\mathscr{T}(v_{1},\dots,v_{n})=\frac{1}{n!}\sum_{\sigma\in
S_{n}^{1}}\langle\lambda,[g^{t_{0}}_{s\sigma}]\rangle\sum_{\xi\in
X_{n}}c_{\xi}I^{\sigma\xi^{-1}}(u),$ (7.5)

where $c_{\xi}=(-1)^{1+\xi^{-1}(1)}$.

###### Proof.

Starting from (6.16) and using (7.4), we get

$\begin{split}n!\,\mathscr{T}(v_{1},\dots,v_{n})&=\sum_{\sigma\in
S_{n}^{1}}\langle\lambda,[g_{s\sigma}^{t_{0}}]\rangle
I^{\sigma}(u)+\sum_{\sigma\in S_{n},\,\sigma_{1}\neq
1}\langle\lambda,[g_{s\sigma}^{t_{0}}]\rangle I^{\sigma}(u)\\\
&=\sum_{\sigma\in S_{n}^{1}}\langle\lambda,[g_{s\sigma}^{t_{0}}]\rangle
I^{\sigma}(u)+\sum_{\begin{subarray}{c}\xi\in X_{n},\,\sigma\in S_{n}\\\
\sigma_{1}\neq
1,\,\sigma\xi(1)=1\end{subarray}}c_{\xi}\langle\lambda,[g_{s\sigma\xi}^{t_{0}}]\rangle
I^{\sigma}(u)\\\ &=\sum_{\sigma\in
S_{n}^{1}}\langle\lambda,[g_{s\sigma}^{t_{0}}]\rangle\Big{(}I^{\sigma}(u)+\sum_{{\xi\in
X_{n},\,\sigma\xi^{-1}(1)\neq 1}}c_{\xi}I^{\sigma\xi^{-1}}(u)\Big{)}\\\
&=\sum_{\sigma\in
S_{n}^{1}}\langle\lambda,[g_{s\sigma}^{t_{0}}]\rangle\sum_{\xi\in
X_{n}}c_{\xi}I^{\sigma\xi^{-1}}(u).\end{split}$ (7.6)

In the last line, we used the fact that, when $\sigma_{1}=1$, we have
$\sigma\xi^{-1}(1)=1$ if and only if $\xi$ is the identity.

∎

A permutation $\sigma\in S_{n}$ acts on the integrals $I^{\xi^{-1}}(u)$ as
$\sigma I^{\xi^{-1}}(u)=I^{\sigma\xi^{-1}}(u)$. So, the sum over $\xi\in
X_{n}$ appearing in (7.5) reads

$\sum_{\xi\in X_{n}}c_{\xi}I^{\sigma\xi^{-1}}(u)=\sigma\Big{(}\sum_{\xi\in
X_{n}}c_{\xi}I^{\xi^{-1}}(u)\Big{)},$

where the action is extended linearly. Our next task is to compute the sum in
the round brackets.

A permutation $\sigma\in S_{n}$ acts on the simplexes $\Sigma_{n}(t,s)$, with
$0\leq t<t+s\leq 1$, as

$\sigma\Sigma_{n}(t,s)v=\big{\\{}(t_{1},\ldots,t_{n})\in\mathbb{R}^{n}\mid
t<t_{\sigma_{n}}<\ldots<t_{\sigma_{1}}<t+s\big{\\}}.$ (7.7)

In particular, if $\bar{\sigma}\in S_{n}$ is the reversing order permutation,
$\bar{\sigma}(i)=n-i+1$, then
$\Sigma_{n}^{\flat}(t,s)=\bar{\sigma}\Sigma_{n}(t,s)$. We also let
$\Sigma_{n}^{\sigma}(t,s)=\sigma\Sigma_{n}(t,s)$ and
$\Sigma_{n}^{\sigma}=\Sigma_{n}^{\sigma}(0,1)$.

Finally, for $k=1,\ldots,n$ we let

$\begin{split}I_{k}^{\flat}(u)&=\int_{\Sigma_{k}^{\flat}}u^{1}(s_{1})\dots
u^{k}(s_{k})d\mathscr{L}^{k},\\\
I_{n-k}(u)&=\int_{\Sigma_{n-k}}u^{k+1}(s_{k+1})\dots
u^{n}(s_{n})d\mathscr{L}^{n-k}.\end{split}$

######  Lemma 7.4.

For any $j=2,\dots,n$ we have the identity

$\sum_{\xi\in
X_{nj}}I^{\xi^{-1}}(u)=I_{j-1}^{\flat}(u)I_{n-j+1}(u)-\sum_{\xi\in
X_{n,j-1}}I^{\xi^{-1}}(u).$ (7.8)

###### Proof.

Fix a permutation $\xi\in X_{nj}$, so that $\xi_{j}=1$. In the integral
$I^{\xi^{-1}}(u)$ we perform the change of variable $t_{\xi_{k}}=s_{k}$. The
integration domain $\Sigma_{n}$ is transformed into the new domain
$\Sigma_{n}^{\xi^{-1}}=\\{0<s_{\xi^{-1}_{n}}<\dots<s_{\xi^{-1}_{1}}<1\\}$:

$\begin{split}I^{\xi^{-1}}(u)&=\int_{\Sigma_{n}}u^{\xi^{-1}_{n}}(t_{n})\dots
u^{\xi^{-1}_{1}}(t_{1})d\mathscr{L}^{n}\\\
&=\int_{\Sigma_{n}}u^{n}(t_{\xi_{n}})\dots
u^{1}(t_{\xi_{1}})d\mathscr{L}^{n}\\\
&=\int_{\Sigma_{n}^{\xi^{-1}}}u^{n}(s_{n})\dots
u^{1}(s_{1})d\mathscr{L}^{n}.\end{split}$ (7.9)

We denote by $\widehat{s}_{j}$ the variables
$(s_{1},\dots,s_{j-1},s_{j+1},\dots,s_{n})$. Since $s_{\xi^{-1}1}=s_{j}$, the
set $\Sigma_{n}^{\xi^{-1}}$ is

$\Sigma_{n}^{\xi^{-1}}=I\times\Sigma_{n-1;j}^{\xi^{-1}}(0,s_{j}),$

where $s_{j}\in I$ and

$\Sigma_{n-1;j}^{\xi^{-1}}(0,s_{j})=\big{\\{}\widehat{s}_{j}\in\mathbb{R}^{n-1}\mid
0<s_{\xi^{-1}_{n}}<\dots<s_{\xi^{-1}_{2}}<s_{j}\big{\\}}.$

Since $\xi\in X_{nj}$, here we have $s_{n}<\dots<s_{j+1}<s_{j}$ and
$s_{1}<\dots<s_{j-1}<s_{j}$. For varying $\xi\in X_{nj}$, we obtain all the
shuffles of $s_{1}<\dots<s_{j-1}$ into $s_{n}<\dots<s_{j+1}$ and thus we have

$\bigcup_{\xi\in
X_{nj}}\Sigma_{n-1;j}^{\xi^{-1}}(0,s_{j})=A_{j-1}(s_{j})\times
B_{n-j}(s_{j}),$

where $A_{j-1}(s_{j})=\Sigma_{j-1}^{\flat}(0,s_{j})$ and
$B_{n-j}(s_{j})=\Sigma_{n-j}(0,s_{j})$.

Summing (7.9) over $\xi\in X_{nj}$ we get

$\begin{split}\sum_{\xi\in X_{nj}}I^{\xi^{-1}}(u)&=\sum_{\xi\in
X_{nj}}\int_{0}^{1}\Big{(}\int_{\Sigma_{n-1}^{\xi^{-1}}(0,s_{j})}\prod_{k\neq
j}u^{k}(s_{k})d\mathscr{L}^{n-1}(\widehat{s}_{j})\Big{)}u^{j}(s_{j})ds_{j}\\\
&=\int_{0}^{1}\Big{(}\int_{A_{j-1}(s_{j})\times B_{n-j}(s_{j})}\prod_{k\neq
j}u^{k}(s_{k})d\mathscr{L}^{n-1}(\widehat{s}_{j})\Big{)}u^{j}(s_{j})ds_{j}.\end{split}$
(7.10)

The inner integral is the product of two integrals. Namely, letting

$\begin{split}f(s_{j})&=u^{j}(s_{j})\int_{B_{n-j}(s_{j})}u^{j+1}(s_{j+1})\dots
u^{n}(s_{n})d\mathscr{L}^{n-j},\\\
g(s_{j})&=\int_{A_{j-1}(s_{j})}u^{j-1}(s_{j-1})\dots
u^{1}(s_{1})d\mathscr{L}^{j-1},\end{split}$ (7.11)

formula (7.10) becomes

$\begin{split}\sum_{\xi\in
X_{nj}}I^{\xi^{-1}}(u)=\int_{0}^{1}f(s_{j})g(s_{j})ds_{j}.\end{split}$ (7.12)

A primitive for $f$ is the function
$h(s_{j})=\int_{0}^{s_{j}}f(\sigma)d\sigma$, and an integration by parts gives

$\begin{split}\int_{0}^{1}f(s_{j})g(s_{j})ds_{j}&=h(1)g(1)-\int_{0}^{1}h(s_{j})g^{\prime}(s_{j})ds_{j},\end{split}$

where the boundary term is easily computed:

$\begin{split}h(1)&=\int_{B_{n-j+1}(1)}u^{j}(s_{j})\dots
u^{n}(s_{n})d\mathscr{L}^{n-j+1}=I_{n-j+1}(u),\\\
g(1)&=\int_{A_{j-1}(1)}u^{j-1}(s_{j-1})\dots
u^{1}(s_{1})d\mathscr{L}^{j-1}=I^{\flat}_{j-1}(u).\end{split}$

In order to compute the integral, notice that

$g^{\prime}(s_{j})=u^{j-1}(s_{j})\int_{A_{j-2}(s_{j})}u^{j-2}(s_{j-2})\dots
u^{1}(s_{1})d\mathscr{L}^{j-2},$

and thus, by (7.10) but for $j-1$, we have

$\begin{split}\int_{0}^{1}h(s_{j})g^{\prime}(s_{j})ds_{j}&=\int_{0}^{1}\Big{(}\int_{A_{j-2}(\sigma)\times
B_{n-j+1}(\sigma)}\prod_{k\neq
j-1}u^{k}(s_{k})d\mathscr{L}^{n-1}(\widehat{s}_{j-1})\Big{)}u^{j-1}(\sigma)d\sigma\\\
&=\sum_{\xi\in X_{n,j-1}}I^{\xi^{-1}}(u).\end{split}$

∎

######  Corollary 7.5.

For any $u\in V_{n-1}$ there holds

$\sum_{\xi\in X_{n}}c_{\xi}I^{\xi^{-1}}(u)=n\int_{\Sigma_{n}}u^{1}(t_{1})\dots
u^{n}(t_{n})d\mathscr{L}^{n}.$

###### Proof.

When $u\in V_{n-1}$ we have $I^{\flat}_{j-1}(u)=I_{n-j+1}(u)=0$ for
$j=2,\dots,n$. Taking into account the constants
$c_{\xi}=(-1)^{1+\xi^{-1}_{1}}$, formula (7.8) reads

$\displaystyle\sum_{\xi\in X_{nj}}c_{\xi}I^{\xi^{-1}}(u)$
$\displaystyle=\sum_{\xi\in X_{n,j-1}}c_{\xi}I^{\xi^{-1}}(u).$

Applying iteratively this identity, we obtain

$\displaystyle\sum_{\xi\in X_{n}}c_{\xi}I^{\xi^{-1}}(u)$
$\displaystyle=\sum_{\xi\in X_{nn}}c_{\xi}I^{\xi^{-1}}(u)+\sum_{\xi\in
X_{n,n-1}}c_{\xi}I^{\xi^{-1}}(u)+\dots+\sum_{\xi\in
X_{n1}}c_{\xi}I^{\xi^{-1}}(u)$ $\displaystyle=2\sum_{\xi\in
X_{n,n-1}}c_{\xi}I^{\xi^{-1}}(u)+\sum_{\xi\in
X_{n,n-2}}c_{\xi}I^{\xi^{-1}}(u)+\dots+\sum_{\xi\in
X_{n1}}c_{\xi}I^{\xi^{-1}}(u)$ $\displaystyle=3\sum_{\xi\in
X_{n,n-2}}c_{\xi}I^{\xi^{-1}}(u)+\sum_{\xi\in
X_{n,n-3}}c_{\xi}I^{\xi^{-1}}(u)+\dots+\sum_{\xi\in
X_{n1}}c_{\xi}I^{\xi^{-1}}(u)$ $\displaystyle=\dots$
$\displaystyle=n\sum_{\xi\in X_{n1}}c_{\xi}I^{\xi^{-1}}(u).$

Since $X_{n1}$ contains only the identity permutation, the claim follows. ∎

This corollary concludes the proof of Theorem 7.1.

##  8\. Non-singularity via trigonometric functions

We start the study of equation $\mathscr{T}(v_{1},\dots,v_{n})=0$ for the
polarization map $\mathscr{T}$ in (7.1). We will work with functions $v_{i}$
as in (6.15) of trigonometric-type.

For each permutation fixing 1, $\sigma\in S_{n}^{1}$, we introduce a real
unknow $x_{\sigma}$. There are $(n-1)!=\mathrm{Card}(S_{n}^{1})$ unknowns. We
are interested in the linear system of equations

$\sum_{\sigma\in S_{n}^{1}}I^{\sigma}(u_{\tau})x_{\sigma}=0,\quad\tau\in
S_{n}^{1},$ (8.1)

where $I^{\sigma}(u_{\tau})$ are regarded as coefficients depending on
$u_{\tau}\in V_{n-1}$. In this section, we prove the following preparatory
result.

######  Theorem 8.1.

There exist $u_{\tau}\in V_{n-1}$, $\tau\in S_{n}^{1}$, such that
$\det(I^{\sigma}(u_{\tau}))_{\sigma,\tau\in S_{n}^{1}}\neq 0$.

With a choice of coefficients as in Theorem 8.1, the linear system (8.1) has
only the zero solution, implying $x_{\sigma}=0$ for all $\sigma\in S_{n}^{1}$.
This fact will be used in Section 9.

For $z=a+ib\in\mathbb{C}$ and $k\in\mathbb{N}$ we let

$w_{z;k}(t)=a\cos(2k\pi t)+b\sin(2k\pi t),\quad t\in I.$

We call $w_{z;k}$ a $w$-type function of parameters $z$ and $k$, and we call
$k$ frequence of $w_{z;k}$. We need exact computations for iterated integrals
on $n$-simplexes of $w$-type functions. In particular, we are interested in
the case when every linear combination with coefficients $\pm 1$ of at most
$n-1$ frequences out of a set of $n$ frequences is not zero, see (8.4) below.
This condition will ensure assumption $u\in V_{n-1}$ in Lemma 6.5.

Any $w$-type function satisfies the integration formula

$\int_{t}^{1}w_{z;k}(s)ds=\frac{1}{2k\pi}\big{(}w_{iz;k}(1)-w_{iz;k}(t)\big{)},\quad
k\neq 0,$ (8.2)

and a pair of $w$-type functions satisfies the multiplication formula
(Werner’s identities)

$w_{z;k}w_{\zeta;h}=\frac{1}{2}\big{(}w_{z\zeta;k+h}+w_{\bar{z}\zeta;h-k}\big{)}.$
(8.3)

For $h\in\mathbb{N}$, we let $\mathscr{J}_{h}=\\{1,\dots,h\\}$ and

$\mathscr{A}_{h}=\big{\\{}\alpha:\mathscr{J}_{h}\to\\{1,-1\\}\mid\alpha_{1}=1\big{\\}}.$

Here we are denoting $\alpha(j)=\alpha_{j}$ and, with a slight abuse of
notation, we identify $\alpha\in\mathscr{A}_{h}$ with
$\alpha=(\alpha_{1},\dots,\alpha_{h})\in\\{1,-1\\}^{h}$. Letting
$\mathrm{z}_{h}=(z_{1},\dots,z_{h})\in\mathbb{C}^{h}$, for each
$\alpha\in\mathscr{A}_{h}$ we define the multiplicative function
$p_{\alpha}:\mathbb{C}^{h}\to\mathbb{C}$

$p_{\alpha}(\mathrm{z}_{h})=\prod_{\ell\in\mathscr{J}_{h},\alpha_{\ell}=1}z_{\ell}\prod_{j\in\mathscr{J}_{h},\alpha_{j}=-1}\bar{z}_{j}.$

Also, letting $\mathrm{k}_{h}=(k_{1},\dots,k_{h})\in\mathbb{N}^{h}$, we define
the additive function $s_{\alpha}:\mathbb{N}^{h}\to\mathbb{N}$

$s_{\alpha}(\mathrm{k}_{h})=\sum_{j=1}^{h}\alpha_{j}k_{j}.$

Notice that, since $\alpha_{1}=1$, $\bar{z}_{1}$ never appears in
$p_{\alpha}(\mathrm{z}_{h})$ and $k_{1}$ has always positive sign in
$s_{\alpha}(\mathrm{k}_{h})$.

Finally, for $\ell,h\in\mathbb{N}$ with $\ell\leq h$ we let
$\mathscr{B}_{\ell}^{h}=\\{\beta:\mathscr{J}_{\ell}\to\mathscr{J}_{h}\mid\beta\textrm{
injective}\\}$ and for $\mathrm{k}_{h}=(k_{1},\dots,k_{h})\in\mathbb{N}^{h}$
and $\beta\in\mathscr{B}_{\ell}^{h}$ we set
$\mathrm{k}_{h}^{\beta}=(k_{\beta_{1}},\dots,k_{\beta_{\ell}})\in\mathbb{N}^{\ell}$.
Here, we are using the math-roman font for vectors and italics for
coordinates.

######  Theorem 8.2.

Let $\mathrm{k}_{n}=(k_{1},\dots,k_{n})\in\mathbb{N}^{n}$, $n\in\mathbb{N}$,
be a vector of frequences such that

$s_{\alpha}(\mathrm{k}_{n}^{\beta})\neq 0\quad\textrm{for all
}\alpha\in\mathscr{A}_{h}\textrm{ and }\,\beta\in\mathscr{B}_{h}^{n},\,\,\
1\leq h\leq n-1.$ (8.4)

Then for any $1\leq h\leq n-1$, for all
$\mathrm{z}_{h}=(z_{1},\dots,z_{h})\in\mathbb{C}^{h}$ and for all $t\in I$ we
have

$\int_{\Sigma_{h}(t,1-t)}w_{z_{h};k_{h}}(t_{h})\dots
w_{z_{1};k_{1}}(t_{1})d\mathscr{L}^{h}=g_{\mathrm{z}_{h};\mathrm{k}_{h}}^{h}(t)-\sum_{\alpha\in\mathscr{A}_{h}}c_{\alpha}^{h}(\mathrm{k}_{h})w_{p_{\alpha}(i\mathrm{z}_{h});s_{\alpha}(\mathrm{k}_{h})}(t),$
(8.5)

where $c_{\alpha}^{h}(\mathrm{k}_{h})\neq 0$ and the function
$g_{\mathrm{z}_{h};\mathrm{k}_{h}}^{h}$ satisfies

$g_{\mathrm{z}_{h};\mathrm{k}_{h}}^{h}(0)=\sum_{\alpha\in\mathscr{A}_{h}}c_{\alpha}^{h}(\mathrm{k}_{h})w_{p_{\alpha}(i\mathrm{z}_{h});s_{\alpha}(\mathrm{k}_{h})}(0),$
(8.6)

and

$\int_{\Sigma_{n-h}}w_{z_{n};k_{n}}(t_{n})\dots
w_{z_{h+1};k_{h+1}}(t_{h+1})g_{\mathrm{z}_{h};\mathrm{k}_{h}}^{h}(t_{h+1})d\mathscr{L}^{n-h}=0.$
(8.7)

###### Proof.

The proof is by induction on $n\in\mathbb{N}$, $n\geq 2$. The constants
$c_{\alpha}^{h}(\mathrm{k}_{h})$ are given by the formula

$c_{\alpha}^{h}(\mathrm{k}_{h})=\frac{2}{4^{h}\pi^{h}}\prod_{\ell=1}^{h}\frac{\alpha_{\ell}}{s_{(\alpha_{1},\dots,\alpha_{\ell})}(\mathrm{k}_{\ell})}.$
(8.8)

The formula is well-defined because
$s_{(\alpha_{1},\dots,\alpha_{\ell})}(\mathrm{k}_{\ell})\neq 0$ by assumption
(8.4).

When $n=2$ we only have $h=1$ and $\alpha=1$, so that
$c_{1}^{1}(\mathrm{k}_{1})=1/2\pi k_{1}$. The integration formula in (8.2)
gives (8.5) with
$g_{\mathrm{z}_{1};\mathrm{k}_{1}}^{1}=c_{1}^{1}(\mathrm{k}_{1})w_{iz_{1};k_{1}}(1)$,
a constant. Identity (8.6) is satisfied and also identity (8.7):

$\int_{0}^{1}w_{z_{2};k_{2}}(t_{2})g_{\mathrm{z}_{1};\mathrm{k}_{1}}^{1}dt_{2}=g_{\mathrm{z}_{1};\mathrm{k}_{1}}^{1}\int_{0}^{1}w_{z_{2};k_{2}}(t_{2})dt_{2}=0,$

because $k_{2}\neq 0$, again by (8.4).

Now we assume the theorem holds for $n-1$ and we prove it for $n$. In
particular, from (8.5) with $t=0$ and (8.6) we have the inductive assumption

$\int_{\Sigma_{h}}w_{z_{h};k_{h}}(t_{h})\dots
w_{z_{1};k_{1}}(t_{1})d\mathscr{L}^{h}=0,\quad h=1,\dots,n-2.$ (8.9)

We distinguish the cases $h=1$ and $2\leq h\leq n-1$. When $h=1$, (8.5) is
exactly the integration formula (8.2) with

$g^{1}_{z_{1},k_{1}}=\frac{1}{2k_{1}\pi}w_{z_{1};k_{1}}(1).$

The 1-periodicity of $w$-type functions also prove (8.6). In order to prove
(8.7), we claim

$\int_{\Sigma_{n-1}}w_{z_{n};k_{n}}(t_{n})\dots
w_{z_{2};k_{2}}(t_{2})d\mathscr{L}^{n-1}=0.$

In fact,

$\displaystyle\int_{\Sigma_{n-1}}w_{z_{n};k_{n}}(t_{n})\dots
w_{z_{2};k_{2}}(t_{2})d\mathscr{L}^{n-1}=\;g^{1}_{z_{2},k_{2}}\int_{\Sigma_{n-2}}w_{z_{n};k_{n}}(t_{n})\dots
w_{z_{3};k_{3}}(t_{3})$ $\displaystyle d\mathscr{L}^{n-2}$
$\displaystyle-\frac{1}{2k_{2}\pi}\int_{\Sigma_{n-2}}w_{z_{n};k_{n}}(t_{n})\dots
w_{z_{3};k_{3}}(t_{3})w_{z_{2};k_{2}}(t_{3})$ $\displaystyle
d\mathscr{L}^{n-2}$

By the multiplication formula (8.3), in the second integral are involved the
vectors of frequences $(k_{2}\pm k_{3},k_{4},\dots,k_{n})$. Both of them
satisfies (8.4), by assumption (8.4) itself. Then both the summands vanish
thanks to the inductive assumption (8.9), proving our claim.

For $2\leq h\leq n-1$, set

$D_{h}(t)=\int_{\Sigma_{h}(t,1-t)}w_{z_{h};k_{h}}(t_{h})\dots
w_{z_{1};k_{1}}(t_{1})d\mathscr{L}^{h}.$

When $h\geq 2$, we use the inductive assumption (8.5) for $h-1$ and the
multiplication formula (8.3) to obtain

$\displaystyle D_{h}(t)$
$\displaystyle=\int_{t}^{1}w_{z_{h};k_{h}}(t_{h})\Big{(}\int_{\Sigma_{h-1}(t_{h},1-t_{h})}w_{z_{h-1};k_{h-1}}(t_{h-1})\dots
w_{z_{1};k_{1}}(t_{1})d\mathscr{L}^{h-1}\Big{)}dt_{h}$
$\displaystyle=\int_{t}^{1}w_{z_{h};k_{h}}(t_{h})\Big{(}g^{h-1}_{\mathrm{z}_{h-1};\mathrm{k}_{h-1}}-\sum_{\alpha\in\mathscr{A}_{h-1}}c_{\alpha}^{h-1}(\mathrm{k}_{h-1})w_{p_{\alpha}(i\mathrm{z}_{h-1});s_{\alpha}(\mathrm{k}_{h-1})}\Big{)}dt_{h}$
$\displaystyle=\int_{t}^{1}w_{z_{h};k_{h}}g^{h-1}_{\mathrm{z}_{h-1};\mathrm{k}_{h-1}}dt_{h}-\frac{1}{2}\sum_{\alpha\in\mathscr{A}_{h-1}}c_{\alpha}^{h-1}(\mathrm{k}_{h-1})\int_{t}^{1}\big{(}w_{**}+w_{\dagger\dagger}\big{)}dt_{h},$

where
$**=z_{h}p_{\alpha}(i\mathrm{z}_{h-1});s_{\alpha}(\mathrm{k}_{h-1})+k_{h}$ and
$\dagger\dagger=\bar{z}_{h}p_{\alpha}(i\mathrm{z}_{h-1});s_{\alpha}(\mathrm{k}_{h-1})-k_{h}$
satisfy

$\begin{split}w_{**}&=-w_{ip_{(\alpha,1)}(i\mathrm{z}_{h});s_{(\alpha,1)}(\mathrm{k}_{h})},\\\
w_{\dagger\dagger}&=w_{ip_{(\alpha,-1)}(i\mathrm{z}_{h});s_{(\alpha,-1)}(\mathrm{k}_{h})}.\end{split}$

By the integration formula (8.2), the function $D_{h}$ equals

$\begin{split}D_{h}&=g^{h}_{\mathrm{z}_{h},\mathrm{k}_{h}}-\frac{1}{4\pi}\sum_{\alpha\in\mathscr{A}_{h-1}}c_{\alpha}^{h-1}(\mathrm{k}_{h-1})\Big{(}\frac{w_{p_{(\alpha,1)}(i\mathrm{z}_{h});s_{(\alpha,1)}(\mathrm{k}_{h})}}{s_{(\alpha,1)}(\mathrm{k}_{h})}-\frac{w_{p_{(\alpha,-1)}(i\mathrm{z}_{h});s_{(\alpha,-1)}(\mathrm{k}_{h})}}{s_{(\alpha,-1)}(\mathrm{k}_{h})}\Big{)},\end{split}$

where

$g^{h}_{\mathrm{z}_{h},\mathrm{k}_{h}}(t)=\int_{t}^{1}w_{z_{h};k_{h}}g^{h-1}_{\mathrm{z}_{h-1};\mathrm{k}_{h-1}}dt_{h}+c(\mathrm{z}_{h},\mathrm{k}_{h}),$
(8.10)

with $c(\mathrm{z}_{h},\mathrm{k}_{h})$ a constant that we are going to fix in
a moment. Using
$\mathscr{A}_{h}=\\{(\alpha,1)\mid\alpha\in\mathscr{A}_{h-1}\\}\cup\\{(\alpha,-1)\mid\alpha\in\mathscr{A}_{h-1}\\}$,
and the relations

$\frac{1}{4\pi}c_{\alpha}^{h-1}(\mathrm{k}_{h-1})\frac{1}{s_{(\alpha,1)}(\mathrm{k}_{h})}=c_{(\alpha,1)}^{h}(\mathrm{k}_{h})\quad\text{and}\quad\frac{1}{4\pi}c_{\alpha}^{h-1}(\mathrm{k}_{h-1})\frac{1}{s_{(\alpha,-1)}(\mathrm{k}_{h})}=-c_{(\alpha,-1)}^{h}(\mathrm{k}_{h}),$

we conclude that

$D_{h}(t)=g^{h}_{\mathrm{z}_{h},\mathrm{k}_{h}}(t)-\sum_{\alpha\in\mathscr{A}_{h}}c_{\alpha}^{h}(\mathrm{k}_{h})w_{p_{\alpha}(i\mathrm{z}_{h});s_{\alpha}(\mathrm{k}_{h})}(t).$

This proves (8.5).

We are left with the proof of (8.6) and (8.7). The constant above is

$c(\mathrm{z}_{h},\mathrm{k}_{h})=\sum_{\alpha\in\mathscr{A}_{h}}c_{\alpha}^{h}(\mathrm{k}_{h})w_{p_{\alpha}(i\mathrm{z}_{h});s_{\alpha}(\mathrm{k}_{h})}(1).$

By the $1$-periodicity of $t\mapsto g^{h}_{\mathrm{z}_{h},\mathrm{k}_{h}}(t)$,
we have

$g^{h}_{\mathrm{z}_{h},\mathrm{k}_{h}}(0)=g^{h}_{\mathrm{z}_{h},\mathrm{k}_{h}}(1)=c(\mathrm{z}_{h},\mathrm{k}_{h}),$

and this shows (8.6).

Finally, we check (8.7). By (8.10) it is sufficient to show that

$\int_{\Sigma_{n-h}}w_{z_{n};k_{n}}(t_{n})\dots
w_{z_{h+1};k_{h+1}}(t_{h+1})\Big{(}\int_{t_{h+1}}^{1}w_{z_{h};k_{h}}g^{h-1}_{\mathrm{z}_{h-1};\mathrm{k}_{h-1}}dt_{h}\Big{)}d\mathscr{L}^{n-h}=0.$
(8.11)

and

$c(\mathrm{z}_{h},\mathrm{k}_{h})\int_{\Sigma_{n-h}}w_{z_{n};k_{n}}(t_{n})\dots
w_{z_{h+1};k_{h+1}}(t_{h+1})d\mathscr{L}^{n-h}=0.$ (8.12)

Identity (8.12) holds by (8.9) and identity (8.11) holds by the inductive
validity of (8.7).

∎

The explicit formula (8.8) for the constants $c_{\alpha}^{h}(\mathrm{k}_{h})$
will be crucial in Lemma 8.5.

######  Corollary 8.3.

Let $\mathrm{k}_{n}=(k_{1},\dots,k_{n})\in\mathbb{N}^{n}$, $n\in\mathbb{N}$,
be a vector of frequences satisfying (8.4) and assume there exists a unique
$\bar{\alpha}\in\mathscr{A}_{n}$ of the form $\bar{\alpha}=(\alpha,-1)$ with
$\alpha\in\mathscr{A}_{n-1}$ such that $s_{\bar{\alpha}}(\mathrm{k}_{n})=0$.
Then we have

$\int_{\Sigma_{n}}w_{z_{n};k_{n}}(t_{n})\dots
w_{z_{1};k_{1}}(t_{1})d\mathscr{L}^{n}=-\frac{1}{2}c_{\alpha}^{n-1}(\mathrm{k}_{n-1})\mathrm{Re}\big{(}\bar{z}_{n}p_{\alpha}(i\mathrm{z}_{n-1})\big{)}.$
(8.13)

###### Proof.

Using formulas (8.5) and (8.7) we obtain

$\begin{split}\int_{\Sigma_{n}}w_{z_{n};k_{n}}\dots
w_{z_{1};k_{1}}&d\mathscr{L}^{n}=\int_{0}^{1}w_{z_{n};k_{n}}\Big{(}\int_{\Sigma(t_{n},1-t_{n})}w_{z_{n-1};k_{n-1}}\dots
w_{z_{1};k_{1}}d\mathscr{L}^{n-1}\Big{)}dt_{n}\\\
&=\int_{0}^{1}w_{z_{n};k_{n}}\Big{(}g_{\mathrm{z}_{n-1};\mathrm{k}_{n-1}}^{n-1}-\sum_{\alpha\in\mathscr{A}_{n-1}}c_{\alpha}^{n-1}(\mathrm{k}_{n-1})w_{p_{\alpha}(i\mathrm{z}_{n-1});s_{\alpha}(\mathrm{k}_{n-1})}\Big{)}dt_{n}\\\
&=-c_{\alpha}^{n-1}(\mathrm{k}_{n-1})\int_{0}^{1}w_{z_{n};k_{n}}w_{p_{\alpha}(i\mathrm{z}_{n-1});s_{\alpha}(\mathrm{k}_{n-1})}dt_{n}.\\\
\end{split}$

Now we use the multiplication formula (8.3). Only the one term with a
resulting zero frequence contributes to the integral, and we get

$\begin{split}\int_{\Sigma_{n}}w_{z_{n};k_{n}}\dots
w_{z_{1};k_{1}}d\mathscr{L}^{n}=-\frac{1}{2}c_{\alpha}^{n-1}(\mathrm{k}_{n-1})\mathrm{Re}(\bar{z}_{n}p_{\alpha}(i\mathrm{z}_{n-1})).\end{split}$

∎

######  Remark 8.4.

Let $\mathrm{k}_{n}=(k_{1},\dots,k_{n})\in\mathbb{N}^{n}$ be a vector of
frequences such that

$\begin{split}k_{1}=\sum_{j=2}^{n}k_{j}\quad\textrm{and}\quad
k_{j}>\sum_{\ell=j+1}^{n}k_{\ell},\quad 2\leq j\leq n-1.\end{split}$ (8.14)

Then $\mathrm{k}_{n}$ stisfies (8.4) and there exists a unique
$\bar{\alpha}=(\alpha,-1)\in\mathscr{A}_{n}$ such that
$s_{\bar{\alpha}}(\mathrm{k}_{n})=0$, and namely
$\bar{\alpha}=(1,-1,\dots,-1)$.

######  Lemma 8.5.

There exists $\mathrm{k}_{n}=(k_{1},\dots,k_{n})\in\mathbb{N}^{n}$ as in
(8.14) such that, with $\bar{\alpha}=(1,-1,\dots,-1)$, there holds

$|c_{\bar{\alpha}}^{n-1}(\mathrm{k}_{n-1})|>\sum_{\sigma\in
S_{n}^{1},\sigma\neq
id}|c_{\bar{\alpha}}^{n-1}(k_{\sigma_{1}},\dots,k_{\sigma_{n-1}})|.$ (8.15)

###### Proof.

Setting, for $\ell=3,\dots,n$,

$q(k_{\ell},\dots,k_{n})=\prod_{i=\ell}^{n}\frac{1}{k_{\ell}+\dots+k_{n}},$

by formula (8.8) and by the choice of $k_{1}$ in (8.14), we obtain

$\begin{split}|c_{\bar{\alpha}}^{n-1}(\mathrm{k}_{n-1})|&=\frac{2}{4^{n-1}\pi^{n-1}k_{1}}q(k_{3},\dots,k_{n}),\end{split}$

and so inequality (8.15) is equivalent to

$q(k_{3},\dots,k_{n})>\sum_{\sigma\in S_{n}^{1},\sigma\neq
id}q(k_{\sigma_{3}},\dots,k_{\sigma_{n}}).$ (8.16)

Notice that $k_{1}$ does not appear in (8.16), whereas $k_{2}$ may appear in
the right hand side.

For $i=1,\dots,n$, consider the set of permutations fixing $1,\dots,i$:

$S_{n}^{i}=\\{\sigma\in S_{n}\mid\sigma_{1}=1,\dots,\sigma_{i}=i\\}.$

We claim that there exist $k_{2},\dots,k_{n}$ as in (8.14), such that for any
$\ell=3,\dots,n$ there holds

$q(k_{\ell},\dots,k_{n})>\sum_{i=\ell-2}^{n-2}\sum_{\sigma\in
S_{n}^{i},\sigma(i+1)\neq i+1}q(k_{\sigma_{\ell}},\dots,k_{\sigma_{n}}).$
(8.17)

Claim (8.17) for $\ell=3$ is exactly (8.16).

We prove (8.17) by induction on $\ell$ starting from $\ell=n$ and descending.
When $\ell=n$, the sums in the right hand side of (8.17) reduce to the sum on
one element, the permutation switching $n$ and $n-1$. So, inequality (8.17)
reads in this case

$\frac{1}{k_{n}}=q(k_{n})>q(k_{n-1})=\frac{1}{k_{n-1}},$

that holds as soon as $0<k_{n}<k_{n-1}$.

By induction, assume that $k_{\ell}>\dots>k_{n}$ are already fixed in such a
way that (8.17) holds with $\ell+1$ replacing $\ell$. Notice that $k_{\ell-1}$
may and indeed does appear in the right hand side. We claim that there exists
$k_{\ell-1}>k_{\ell}$ such that (8.17) holds.

We split the sum in (8.17) obtaining

$q(k_{\ell},\dots,k_{n})>\sum_{i=\ell-1}^{n-2}\sum_{\sigma\in
S_{n}^{i},\sigma(i+1)\neq
i+1}q(k_{\sigma_{\ell}},\dots,k_{\sigma_{n}})+\sum_{\sigma\in
S_{n}^{\ell-2},\sigma(\ell-1)\neq\ell-1}q(k_{\sigma_{\ell}},\dots,k_{\sigma_{n}}),$
(8.18)

and we consider the quantity

$\displaystyle
Q(k_{\ell},\dots,k_{n})=q(k_{\ell},\dots,k_{n})-\sum_{i=\ell-1}^{n-2}\sum_{\sigma\in
S_{n}^{i},\sigma(i+1)\neq i+1}q(k_{\sigma_{\ell}},\dots,k_{\sigma_{n}})$

A permutation $\sigma\in S_{n}^{i}$ with $i\geq\ell-1$ fixes all the $k_{i}$s
with $i\leq\ell-1$ and then we have

$q(k_{\sigma\ell},\dots,k_{\sigma
n})=\frac{1}{k_{\ell}+\dots+k_{n}}q(k_{\sigma_{\ell+1}},\dots,k_{\sigma_{n}}),$

and thus

$\displaystyle
Q(k_{\ell},\dots,k_{n})=\frac{1}{k_{\ell}+\dots+k_{n}}\Big{(}q(k_{\ell+1},\dots,k_{n})-\sum_{i=\ell-1}^{n-2}\sum_{\sigma\in
S_{n}^{i},\sigma(i+1)\neq
i+1}q(k_{\sigma_{\ell+1}},\dots,k_{\sigma_{n}})\Big{)}.$

By our inductive assumption, we have $Q(k_{\ell},\dots,k_{n})>0$. Notice that
$Q(k_{\ell},\dots,k_{n})$ does not depend on $k_{\ell-1}$.

Conversely, every summand in the second sum in (8.18), i.e., every
$q(k_{\sigma_{\ell}},\dots,k_{\sigma_{n}})$, depends on $k_{\ell-1}$ and tends
to $0$ as $k_{\ell-1}\to\infty$. We conclude that for all large enough
$k_{\ell-1}>k_{\ell}$ claim (8.18) holds. This ends the proof of (8.17).

∎

###### Proof of Theorem 8.1.

We claim that for each $\tau\in S_{n}^{1}$ there exists $u_{\tau}\in V_{n-1}$
such that the matrix $(I^{\sigma}(u_{\tau}))_{\sigma,\tau\in S_{n}^{1}}$ is
strictly diagonally dominant and thus invertible.

Let $\mathrm{k}_{n}=(k_{1},\dots,k_{n})\in\mathbb{N}^{n}$ be a vector of
frequences as in (8.14) and satisfying the claim of Lemma 8.5 and choose
complex numbers $\mathrm{z}_{n}=(z_{1},\dots,z_{n})\in\mathbb{C}^{n}$ such
that

$-\frac{1}{2}\mathrm{Re}(\bar{z}_{n}p_{\alpha}(i\mathrm{z}_{n-1}))=1,$

where $\alpha=(1,-1,\dots,-1)\in\mathscr{A}_{n-1}$. The function
$u=(w_{z_{1};k_{1}},\dots,w_{z_{n};k_{n}})$ is in $V_{n-1}$, by Theorem 8.2,
formulas (8.5) and (8.6). By formula (8.13) and by Lemma 8.5

$|I^{{id}}(u)|=|c_{\bar{\alpha}}^{n-1}(\mathrm{k}_{n-1})|>\sum_{\sigma\in
S_{n}^{1},\sigma\neq
id}|c_{\bar{\alpha}}^{n-1}(k_{\sigma_{1}},\dots,k_{\sigma_{n-1}})|=\sum_{\sigma\in
S_{n}^{1},\sigma\neq id}|I^{\sigma}(u)|$

For each $\tau\in S_{n}^{1}$, we define
$u_{\tau}=(w_{z_{\bar{1}};k_{\bar{1}}},\dots,w_{z_{\bar{n}};k_{\bar{n}}})$,
where $\bar{\ell}=\tau^{-1}(\ell)$. As above, we have

$|I^{\tau}(u_{\tau})|>\sum_{\sigma\in
S_{n}^{1},\sigma\neq\tau}|I^{\sigma}(u_{\tau})|.$

This concludes the proof that $(I^{\sigma}(u_{\tau}))_{\sigma,\tau\in
S_{n}^{1}}$ is strictly diagonally dominant.

∎

##  9\. Goh conditions of order $n$ in the corank 1 case

Let $\Delta\subset TM$ be the distribution spanned point-wise by the vector
fields $f_{1},\dots,f_{d}$. For any $n\in\mathbb{N}$ and $q\in M$ we let

$\Delta_{n}(q)=\mathrm{span}_{\mathbb{R}}\big{\\{}[f_{s_{n}},\dots,f_{s_{1}}](q)\mid
s_{1},\dots,s_{n}\in\\{1,\dots,d\\}\big{\\}}\subset T_{q}M.$

For a given $q\in M$, the annihilator of $\Delta_{n}$ is

$\Delta_{n}^{\perp}(q)=\big{\\{}\lambda\in T_{q}^{*}M\mid\lambda(v)=0\textrm{
for all $v\in\Delta_{n}(q)$}\big{\\}}.$

A horizontal curve $\gamma\in AC(I;M)$ is regular (singular or strictly
singular) if its control is regular (singular or strictly singular,
respectively). The corank of $\gamma$ is the corank of its control.

Let $\gamma:I\to M$ be a horizontal curve with control $u$, $\gamma(0)=q$ and
$\gamma(1)=\bar{q}$. We denote by $P_{0}^{t}$ the flow of the non-autonomous
vector field $f_{u}=\sum_{i=1}^{d}u_{i}f_{i}$. Then we have
$\gamma(t)=P_{0}^{t}(q)$ for $t\in I$ and the differential
$(P_{t}^{1})_{*}:T_{\gamma(t)}M\to T_{\bar{q}}M$ is given by

$(P_{t}^{1})_{*}=\mathrm{Ad}\left(\overrightarrow{\exp}\int_{1}^{t}f_{u(\tau)}d\tau\right).$

We refer the reader to [3, Chapter 2] for a detailed introduction to the
formalism of chronological calculus. We denote by
$(P_{t}^{1})^{*}:T^{*}_{\bar{q}}M\to T_{\gamma(t)}^{*}M$ the adjoint map and
for every $\lambda\in\mathrm{Im}(d_{0}G)^{\perp}$, the curve of covectors
defined by

$\lambda(t)=(P_{t}^{1})^{*}\lambda\in T_{\gamma(t)}^{*}M,\quad t\in I,$ (9.1)

is called the _adjoint curve_ to $\gamma$ relative to $\lambda$. In the corank
1 case, this curve is unique up to normalization of $\lambda\neq 0$.

######  Theorem 9.1.

Let $(M,\Delta,g)$ be a sub-Riemannian manifold, $\gamma\in AC(I;M)$ be a
horizontal curve with control $u\in X$, and $n\in\mathbb{N}$ be an integer
with $n\geq 3$. Assume that:

  * i)

$\mathscr{D}^{h}_{u}F=0$ for $h=2,\dots,n-1$, where $F$ is the end-point map
starting from $\gamma(0)$;

  * ii)

$\gamma$ is a strictly singular length minimizing curve with corank $1$.

Then any adjoint curve $\lambda\in AC(I;T^{*}M)$ satisfies

$\lambda(t)\in\Delta_{n}^{\perp}(\gamma(t))\quad\textrm{for all $t\in I$}.$
(9.2)

###### Proof.

If $\gamma$ is length minimizing then the extended end-point map $F_{J}$ is
not open at $u$, i.e., the extend variation map $G_{J}$ is not open at $0$. By
Theorem 6.4 we consequently have $\mathscr{G}_{t_{0}}^{n}(v)=0$ for all
$t_{0}\in I$ and for all $v\in V_{n-1}$. In order to use Theorem 6.4 we need
both assumptions i) and ii). The map $\mathscr{G}_{t_{0}}^{n}$ is introduced
in (6.4) and incorporates $\lambda$. The strict singularity of $\gamma$ is
used to translate the differential analysis from $G_{J}$ to $G$.

We polarize the equation $\mathscr{G}_{t_{0}}^{n}(v)=0$, as explained at the
end of Section 6. The polarization, denoted by $\mathscr{T}$, is introduced in
(6.14). We have $\mathscr{T}=0$ on _linear_ spaces contained in $V_{n-1}$. We
fix any selection function $s:\\{1,\dots,n\\}\to\\{1,\dots,d\\}$ and we
translate our claim (9.2) into the new claim

$\langle\lambda(t),[f_{s_{n}},\dots,f_{s_{1}}](\gamma(t))\rangle=0,\quad t\in
I.$ (9.3)

By formula (7.1) for $\mathscr{T}$ proved in Theorem 7.1, if we choose
$v_{1},\dots,v_{n}\in X$ as in (6.15) and such that the corresponding $u$
satisfies $u\in V_{n-1}$, then the equation $\mathscr{T}(v_{1},\dots,v_{n})=0$
reads

$\sum_{\sigma\in S_{n}^{1}}\langle\lambda,[g^{t_{0}}_{s\sigma}]\rangle
I^{\sigma}(u)=0,\quad t_{0}\in I.$ (9.4)

We regard (9.4) as a linear equation in the unknowns
$\langle\lambda,[g^{t_{0}}_{s\sigma}]\rangle$ with coefficients
$I^{\sigma}(u)$.

By Theorem 8.1, for any $\tau\in S_{n}^{1}$ there exists $u_{\tau}\in V_{n-1}$
such that the matrix $(I^{\sigma}(u_{\tau}))_{\sigma,\tau\in S_{n}^{1}}$ is
invertible. From (9.4), the definition (9.1) of adjoint curve, and (5.1) we
deduce that for any $\sigma\in S_{n}^{1}$ and $t_{0}\in I$

$\begin{split}0=\langle\lambda,[g^{t_{0}}_{s\sigma}]\rangle&=\langle\lambda,[g^{t_{0}}_{s\sigma_{n}},\dots,g^{t_{0}}_{s\sigma_{1}}]\rangle\\\
&=\langle\lambda,[(P^{1}_{t_{0}})_{*}f_{s\sigma_{n}},\dots,(P^{1}_{t_{0}})_{*}f_{s\sigma_{1}}]\rangle\\\
&=\langle\lambda,(P^{1}_{t_{0}})_{*}[f_{s\sigma_{n}},\dots,f_{s\sigma_{1}}]\rangle\\\
&=\langle(P^{1}_{t_{0}})^{*}\lambda,[f_{s\sigma_{n}},\dots,f_{s\sigma_{1}}](\gamma(t_{0}))\rangle\\\
&=\langle\lambda(t_{0}),[f_{s\sigma_{n}},\dots,f_{s\sigma_{1}}](\gamma(t_{0}))\rangle.\end{split}$
(9.5)

This identity with $\sigma=id$ is (9.3).

∎

######  Remark 9.2.

We comment on assumption i), $\mathscr{D}^{h}_{u}F=0$ for $h=2,\dots,n-1$.
This is equivalent to $\mathscr{D}^{h}_{0}G=0$. Using formula (5.13), for
$v\in V_{h-1}$,

$0=\mathscr{D}^{h}_{0}G(w_{t_{0},s})=\langle\lambda,D_{0}^{h}G(w_{t_{0},s})\rangle=c_{h}s^{h}\int_{\Sigma_{h}}\langle\lambda,[g_{v(t_{h})}^{t_{0}},\dots,g_{v(t_{1})}^{t_{0}}]\rangle
d\mathscr{L}^{h}+O(s^{h+1}).$

Dividing by $s^{h}$ and letting $s\to 0$ we deduce that
$\mathscr{G}_{t_{0}}^{h}(v)=0$. Now, as in the proof of Theorem 9.1 we
conclude that
$\langle\lambda(t_{0}),[f_{s_{h}},\dots,f_{s_{1}}](\gamma(t_{0}))\rangle=0$,
that is

$\lambda(t)\in\Delta_{h}^{\perp}(\gamma(t)),\quad\text{for all
}h=2,\dots,n-1.$ (9.6)

In particular, as a combination of (9.2) and (9.6), if $\gamma$ is a strictly
singular length minimizer satisfying i), then the associated adjoint curve
annihilates all the brackets up to length $n$.

######  Remark 9.3.

The inverse implication in Theorem 9.1 does not hold. Namely, a strictly
singular curve satisfying assumption i) and (9.2) in Theorem 9.1 needs not be
length-minimizing. A counterexample is given in the next section.

##  10\. An example of singular extremal

On the manifold $M=\mathbb{R}^{3}$, with coordinates
$x=(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}$, we consider the rank $2$
distribution $\Delta^{n}=\mathrm{span}\\{f_{1},f_{2}\\}$, where $f_{1}$ and
$f_{2}$ are the vector-fields in (1.6). The vector-field $f_{2}$ depends on
the parameter $n\in\mathbb{N}$. We fix on $\Delta^{n}$ the metric $g$ making
$f_{1}$ and $f_{2}$ orthonormal.

In this section, we study the (local) length-minimality in
$(\mathbb{R}^{3},\Delta^{n},g)$ of the curve $\gamma:I=[0,1]\to\mathbb{R}^{3}$

$\gamma(t)=(0,t,0),\quad t\in I.$ (10.1)

The curve $\gamma$ is in fact defined for all $t\in\mathbb{R}$.

Our results rely upon the analysis of the variation map $G$ introduced in
Section 5, $G(v)=G_{\bar{q}}^{u}(v)=F_{0}(u+v)$, where $F=F_{0}$ is the end-
point map with starting point $q=0$, $\bar{q}=\gamma(1)=(0,1,0)$ is the end-
point, and $u=(0,1)\in L^{1}(I;\mathbb{R}^{2})$ is the control of $\gamma$.
The extended maps $F_{J},G_{J}$ are defined as in Sections 5-6.

The minimality properties of $\gamma$ are described in the following theorem.

######  Theorem 10.1.

For $n\in\mathbb{N}$, let us consider the sub-Riemannian manifold
$(\mathbb{R}^{3},\Delta^{n},g)$ and the curve $\gamma$ in (10.1).

  * i)

For any $n\geq 2$, $\gamma$ is the unique strictly singular extremal in
$(\mathbb{R}^{3},\Delta^{n})$ passing through the origin, up to
reparameterization.

  * ii)

If $n\geq 2$ is even, $\gamma$ is locally length minimizing in
$(\mathbb{R}^{3},\Delta^{n},g)$.

  * iii)

If $n\geq 3$ is odd, $\gamma$ is not length minimizing in
$(\mathbb{R}^{3},\Delta^{n},g)$, not even locally.

Above, “locally length minimizing” means that short sub-arcs of $\gamma$ are
length minimizing for fixed end-points. Claims i) and ii) are well-known. In
particular, claim ii) can be proved with a straightforward adaptation of Liu-
Sussmann’s argument for $n=2$ in [17]. For $n=3$, claim iii) is proved in [7]
and here we prove the general case.

We compute the intrinsic differentials of $G$ and we show that, for odd $n$,
they satisfy the hypotheses of Theorem 3.4. This implies that the extended
variation map $G_{J}$ is open and, as a consequence, the non-minimality of
$\gamma$.

We denote by $\gamma^{x}$ the horizontal curve with control $u=(0,1)$ and
$\gamma^{x}(1)=x$, so that $\gamma^{\bar{q}}=\gamma$. By the formulas in (1.6)
for the vector fields $f_{1}$ and $f_{2}$, we find

$\gamma_{1}^{x}(t)=x_{1},\quad\gamma_{2}^{x}(t)=(t-1)(1-x_{1})+x_{2},\quad\gamma_{3}^{x}(t)=(t-1)x_{1}^{n}+x_{3}.$

The “optimal flow” associated with $\gamma$ is the $1$-parameter family of
diffeomorphisms $P_{1}^{t}\in C^{\infty}(\mathbb{R}^{3};\mathbb{R}^{3})$,
$t\in\mathbb{R}$, defined by $P_{1}^{t}(x)=\gamma^{x}(t)$. For fixed
$x\in\mathbb{R}^{3}$, the inverse of the differential of $P_{1}^{t}$ is the
map $P_{1}^{t}(x)_{*}^{-1}=P^{1}_{t}(x)_{*}:T_{\gamma^{x}(t)}\mathbb{R}^{3}\to
T_{x}\mathbb{R}^{3}$

$P^{1}_{t}(x)_{*}=\begin{pmatrix}1&0&0\\\ t-1&1&0\\\
-n(t-1)x_{1}^{n-1}&0&1\end{pmatrix}.$

As explained in Section 5, see formula (5.8), the differential of $G$ at 0 is

$d_{0}G(v)=\int_{0}^{1}g^{t}_{v(t)}(\bar{q})dt,\quad v=(v^{1},v^{2})\in
L^{2}(I;\mathbb{R}^{2}).$

Above, we set $g^{t}_{v(t)}=v^{1}(t)g_{1}^{t}+v^{2}(t)g_{2}^{t}$ where the
vector fields $g_{1}^{t}$ and $g_{2}^{t}$ are

$\displaystyle g_{1}^{t}$
$\displaystyle=P_{t}^{1}(x)_{*}f_{1}=\frac{\partial}{\partial
x_{1}}+(t-1)\frac{\partial}{\partial
x_{2}}-n(t-1)x_{1}^{n-1}\frac{\partial}{\partial x_{3}},$ $\displaystyle
g_{2}^{t}$
$\displaystyle=P_{t}^{1}(x)_{*}f_{2}=f_{2}=(1-x_{1})\frac{\partial}{\partial
x_{2}}+x_{1}^{n}\frac{\partial}{\partial x_{3}}.$

So the differential is given by the formula

$d_{0}G(v)=\begin{pmatrix}\displaystyle\int_{0}^{1}v^{1}(t)dt\\\
\displaystyle\int_{0}^{1}\big{\\{}(t-1)v^{1}(t)+v^{2}(t)\big{\\}}dt\\\
0\end{pmatrix}.$ (10.2)

We deduce that a generator for $\mathrm{Im}(d_{0}G)^{\perp}$ is the covector
$\lambda=(0,0,1)$, and that $v\in\ker(d_{0}G)$ if and only if

$\begin{split}\int_{0}^{1}v^{1}(t)dt=0\qquad\mathrm{and}\qquad\int_{0}^{1}(tv^{1}(t)+v^{2}(t))dt=0.\end{split}$
(10.3)

In the computation of the differentials $\mathscr{D}_{0}^{h}G$, $h\geq 2$, we
need the following lemma. For $y\in L^{2}(I;\mathbb{R})$ and $n\geq 2$, we let

$\Gamma_{y}^{n}=\int_{\Sigma_{n}}y(t_{1})\dots
y(t_{n})(t_{2}-t_{1})d\mathscr{L}^{n}.$ (10.4)

######  Lemma 10.2.

For $n\geq 2$ and $y\in L^{2}(I;\mathbb{R})$ such that
$\displaystyle\int_{0}^{1}y(t)dt=0$ we have

$\Gamma_{y}^{n}=\frac{1}{n!}\int_{0}^{1}\left(\int_{t}^{1}y(\tau)d\tau\right)^{n}dt.$
(10.5)

###### Proof.

We first observe that, integrating by parts, we have

$\displaystyle\int_{t_{2}}^{1}y(t_{1})(t_{2}-t_{1})dt_{1}$
$\displaystyle=t_{2}\int_{t_{2}}^{1}y(t_{1})dt_{1}-\int_{t_{2}}^{1}t_{1}y(t_{1})dt_{1}$
$\displaystyle=t_{2}\int_{t_{2}}^{1}y(t_{1})dt_{1}-\left[s\int_{s}^{1}y(t_{1})dt_{1}\right]_{s=t_{2}}+\int_{t_{2}}^{1}\int_{s}^{1}y(t_{1})dt_{1}ds$
$\displaystyle=\int_{t_{2}}^{1}\int_{s}^{1}y(t_{1})dt_{1}ds.$

Applying this identity to $\Gamma_{y}^{n}$ and integrating by parts again, we
get

$\displaystyle\Gamma_{y}^{n}$
$\displaystyle=\int_{0}^{1}y(t_{n})\int_{t_{n}}^{1}\dots\int_{t_{2}}^{1}\int_{s}^{1}y(t_{1})dt_{1}dsdt_{2}\dots
dt_{n}$
$\displaystyle=\int_{0}^{1}y(t_{n})dt_{n}\int_{0}^{1}y(t_{n-1})\int_{t_{n}}^{1}\dots\int_{t_{2}}^{1}\int_{s}^{1}y(t_{1})dt_{1}dsdt_{2}\dots
dt_{n-1}$
$\displaystyle\quad\;-\int_{0}^{1}\left(\int_{t_{n}}^{1}y(\tau)d\tau\right)y(t_{n})\int_{t_{n}}^{1}y(t_{n-2})\dots\int_{t_{2}}^{1}\int_{s}^{1}y(t_{1})dt_{1}dsdt_{2}\dots
dt_{n-2}dt_{n}$
$\displaystyle=\frac{1}{2}\int_{0}^{1}\frac{d}{dt_{n}}\left(\int_{t_{n}}^{1}y(\tau)d\tau\right)^{2}\int_{t_{n}}^{1}y(t_{n-2})\dots\int_{t_{2}}^{1}\int_{s}^{1}y(t_{1})dt_{1}dsdt_{2}\dots
dt_{n-2}dt_{n}.$

In the last identity, we used our assumption
$\displaystyle\int_{0}^{1}y(t)dt=0$. Now our claim follows by iterating this
integration by parts argument. ∎

######  Theorem 10.3.

Let $n\in\mathbb{N}$. The variation map $G$ in $(\mathbb{R}^{3},\Delta^{n})$
satisfies:

  * i)

$\mathscr{D}_{0}^{h}G=0$, for $h<n$;

  * ii)

for any $v=(v_{1},\dots,v_{n-1})\in\mathrm{dom}(\mathscr{D}_{0}^{n}G)$,

$\mathscr{D}_{0}^{n}G(v)=\int_{0}^{1}\left(\int_{t}^{1}v_{1}^{1}(\tau)d\tau\right)^{n}dt,$
(10.6)

where $v_{1}^{1}$ is the first coordinate of $v_{1}$.

###### Proof.

The Lie brackets of the vector fields $g_{1}^{t}$ and $g_{2}^{t}=f_{2}$ are,
at different times,

$\displaystyle[g_{1}^{t},g_{1}^{s}]$
$\displaystyle=n(n-1)(t-s)x_{1}^{n-2}\frac{\partial}{\partial x_{3}},$
$\displaystyle[g_{1}^{t},g_{2}^{s}]$ $\displaystyle=-\frac{\partial}{\partial
x_{2}}+nx_{1}^{n-1}\frac{\partial}{\partial x_{3}}.$

Notice that the bracket in the latter line is time-independent. Then, for
$3\leq h\leq n$ and $i_{1},\dots,i_{h}\in\\{1,2\\}$, the iterated brackets of
length $h$ are

$[g^{t_{h}}_{i_{h}},\dots,g^{t_{1}}_{i_{1}}]=\begin{cases}n\dots(n-h+1)(t_{2}-t_{1})x_{1}^{n-h}\frac{\partial}{\partial
x_{3}},&\text{ if }i_{1}=\dots=i_{h}=1,\\\
n\dots(n-h+2)(t_{2}-t_{1})x_{1}^{n-h+1}\frac{\partial}{\partial x_{3}},&\text{
if }i_{2}=\dots=i_{h}=1,\text{ and }i_{1}=2,\\\ 0,&\text{
otherwise}.\end{cases}$

For $h<n$, the coefficient of $\partial/\partial x_{3}$ in the formulas above
vanishes at the point $\bar{q}=(0,1,0)$ and thus the projection of these
brackets along the covector $\lambda=(0,0,1)$ vanishes, for any $2\leq h<n$,

$\langle\lambda,[g^{t_{h}}_{i_{h}},\dots,g^{t_{1}}_{i_{1}}](\bar{q})\rangle=0.$

Using formulas (5.2) and (5.8), we deduce that for any
$(v_{1},\dots,v_{h-1})\in\mathrm{dom}(\mathscr{D}_{0}^{h}G)$ we have

$\mathscr{D}_{0}^{h}G(v_{1},\dots,v_{h-1})=\langle\lambda,D_{0}^{h}G(v_{1},\dots,v_{h-1},*)(\bar{q})\rangle=0,$

proving claim i).

For $h=n$, the coefficient of $\partial/\partial x_{3}$ vanishes at $\bar{q}$
except for the case $i_{1}=\dots=i_{n}=1$, that is

$\displaystyle\langle\lambda,[g^{t_{n}}_{i_{n}},\dots,g^{t_{1}}_{i_{1}}](\bar{q})\rangle$
$\displaystyle=0,\quad\text{if }i_{j}\neq 1\text{ for some }j,$
$\displaystyle\langle\lambda,[g^{t_{n}}_{1},\dots,g^{t_{1}}_{1}](\bar{q})\rangle$
$\displaystyle=n!(t_{2}-t_{1}).$

Then, for any $v=(v_{1},\dots,v_{n-1})\in\mathrm{dom}(\mathscr{D}_{0}^{n}G)$
we have

$\displaystyle\mathscr{D}_{0}^{n}G(v)$
$\displaystyle=\langle\lambda,D_{0}^{n}G(v_{1},\dots,v_{n-1},*)\rangle=n!\int_{\Sigma_{n}}v_{1}^{1}(t_{1})\dots
v_{1}^{1}(t_{n})(t_{2}-t_{1})d\mathscr{L}^{n}$
$\displaystyle=\int_{0}^{1}\left(\int_{t}^{1}v_{1}^{1}(\tau)d\tau\right)^{n}dt.$

In the last identity we used Lemma 10.2. This proves claim ii). ∎

Before proving claim iii) of Theorem 10.1, we recall that
$\ker(d_{0}G_{J})=\ker(d_{0}G)\cap\ker(d_{u}J)$, where $J$ is the energy
functional (see Section 6). In particular, for any $v\in
L^{2}(I;\mathbb{R}^{2})$ we have

$d_{u}J(v)=\int_{0}^{1}\big{(}u^{1}(t)v^{1}(t)dt+u^{2}(t)v^{2}(t)\big{)}dt=\int_{0}^{1}v^{2}(t)dt.$
(10.7)

###### Proof of Theorem 10.1 \- claim iii).

Let $n\in\mathbb{N}$ be an odd integer. We claim that there exists
$v=(v_{1},\dots,v_{n-1})\in\mathrm{dom}(\mathscr{D}_{0}^{n}G_{J})\subset\mathrm{dom}(\mathscr{D}_{0}^{n}G)$
such that $\mathscr{D}_{0}^{n}G(v)\neq 0$. The inclusion of domains is ensured
by (6.3). By Theorem 10.3 we have $\mathscr{D}_{0}^{h}G=0$ for any $h<n$. Then
from (6.2) it follows that also the extended map satisfies
$\mathscr{D}_{0}^{n}G_{J}(v)\neq 0$ and $\mathscr{D}_{0}^{h}G_{J}=0$ for
$h<n$.

By Proposition 3.3 the differential $\mathscr{D}_{0}^{n}G_{J}$ is regular;
here, we are using the fact that $n$ is odd. By Theorem 3.4, $G_{J}$ is open
at zero and thus $F_{J}$ is open at $u$. This implies that $\gamma$ is not
length minimizing.

So, the proof of our claim reduces to find a function
$v_{1}\in\ker(d_{0}G_{J})$ such that

$\int_{0}^{1}\left(\int_{t}^{1}v_{1}^{1}(\tau)d\tau\right)^{n}dt\neq 0.$
(10.8)

If such a control $v_{1}$ exists, then by Proposition 2.5 there also exist
$v_{2},\dots,v_{n-1}\in L^{2}(I;\mathbb{R}^{2})$ such that
$v=(v_{1},\dots,v_{n-1})\in\mathrm{dom}(\mathscr{D}_{0}^{n}G_{J})$ and by
(10.6) it follows $\mathscr{D}_{0}^{n}G(v)\neq 0$.

The condition $v_{1}\in\ker(d_{0}G_{J})$ is equivalent to $d_{0}G(v_{1})=0$
and $d_{u}J(v_{1})=0$. By (10.2) and (10.7) this means

$\int_{0}^{1}v_{1}^{1}(t)dt=\int_{0}^{1}tv_{1}^{1}(t)dt=\int_{0}^{1}v_{1}^{2}(t)dt=0.$
(10.9)

We choose any funtion $v_{1}^{2}$ with vanishing mean. Also choosing
$v_{1}^{1}(t)=\chi_{\left[0,\frac{1}{2}\right]}(t)-5\chi_{\left[\frac{1}{2},\frac{3}{4}\right]}(t)+3\chi_{\left[\frac{3}{4},1\right]}(t)$,
all the conditions in (10.9) are satisfied. Moreover, we have

$\displaystyle\int_{t}^{1}v_{1}^{1}(\tau)d\tau=-t\chi_{\left[0,\frac{1}{2}\right]}(t)+(5t-3)\chi_{\left[\frac{1}{2},\frac{3}{4}\right]}(t)-3(t-1)\chi_{\left[\frac{3}{4},1\right]}(t),$

and then, after a short computation,

$\displaystyle\int_{0}^{1}\left(\int_{t}^{1}v_{1}^{1}(\tau)d\tau\right)^{n}dt=\frac{6}{5(n+1)4^{n}}\big{(}3^{n-1}-2^{n-1}\big{)}.$

The last quantity is different from 0 for any odd $n\geq 3$, completing the
proof.

∎

######  Remark 10.4.

We briefly comment on claim ii) of Theorem 10.1. By formula (10.6), when $n$
is even we have $\mathscr{D}_{0}^{n}G(v)\geq 0$ for any
$v\in\mathrm{dom}(\mathscr{D}_{0}^{n}G)$. So condition i) in Proposition 3.3
cannot be satisfied. The differential $\mathscr{D}_{0}^{n}G$ is not regular in
the sense of Definition 3.2 part i) and so the open mapping Theorem 3.4 does
not work. Though not sufficient to prove the local minimality of $\gamma$,
this is consistent with claim ii) of Theorem 10.1.

######  Remark 10.5.

Claim iii) of Theorem 10.1 answers the question raised in Remark 9.3. By
Theorem 10.3, the curve $\gamma$ satisfies assumption i) of Theorem 9.1 for
any $n\in\mathbb{N}$. On the other hand, the non-vanishing Lie brackets of the
vector fields $f_{1}$ and $f_{2}$ are

$\displaystyle[f_{1},f_{2}]$ $\displaystyle=-\frac{\partial}{\partial
x_{2}}+nx^{n-1}\frac{\partial}{\partial x_{3}},$
$\displaystyle[\underbrace{f_{1},\dots,f_{1}}_{\text{$(h-1)$-times}},f_{2}]$
$\displaystyle=n(n-1)\dots(n-h+2)x_{1}^{n-h+1}\frac{\partial}{\partial
x_{3}}.$

Then, for any $h\leq n$ we have

$\langle\lambda,[\underbrace{f_{1},\dots,f_{1}}_{\text{$(h-1)$-times}},f_{2}](\gamma(t))\rangle=0,\quad\text{for
any }t\in I.$ (10.10)

For $2\leq h<n$, this is consistent with Remark 9.2: the vanishing of the
$h$-th differential implies (10.10). When $h=n$, identity (10.10) is the Goh
condition of order $n$ in (9.3).

Thus, if $n$ is odd the curve $\gamma$ satisfies both assumption i) of Theorem
9.1 and condition (9.3). However, it is a strictly singular curve which is not
length minimizing.

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