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\dateposted{November 13, 2003}
\PII{S 1079-6762(03)00118-5}
\copyrightinfo{2003}{American Mathematical Society}




\title[Nonholonomic tangent spaces]{Nonholonomic tangent spaces: intrinsic construction and rigid dimensions}

\author{A. Agrachev}
\address{Steklov Mathematical Institute, Moscow, Russia}
\curraddr{SISSA, Via Beirut 2--4, Trieste, Italy}
\author{A. Marigo}
\address{IAC-CNR, Viale Policlinico 136, Roma, Italy}

\subjclass[2000]{Primary 58A30; Secondary 58K50}

\date{March 25, 2003}

\commby{Svetlana Katok}

\keywords{Nonholonomic system, nilpotent approximation, Carnot group}

\begin{abstract} A nonholonomic space is a smooth manifold
equipped with a bracket generating family of vector fields. Its
infinitesimal version is a homogeneous space of a nilpotent Lie
group endowed with a dilation which measures the anisotropy of the
space. We give an intrinsic construction of these infinitesimal objects and
classify all rigid (i.e. not deformable) cases.



Let $M$ be a ($C^\infty$) smooth connected $n$-dimensional manifold and
${\mathcal F}\subset \operatorname{Vec}M$ 
a set of smooth vector fields on $M$. Given
$q\in M$ and an integer $l>0$ we set
\Delta_q^l=span\{[f_1,[\dots,[f_{i-1},f_i]\cdots](q) : f_j\in {\mathcal F},\
1\le j\le i,\ i\le l\}\subseteq T_qM.
Clearly, $\Delta_q^l\subseteq\Delta_q^m$ for $l0$,
 $\pi^l:T^{l,{\mathcal F}}_qM\to T^{l-1,{\mathcal F}}_qM$ is a fiber bundle
 with fiber $\Delta^l_q/\Delta^{l-1}_q$.
 In particular,  $T^{l,{\mathcal F}}_qM$ is diffeomorphic to
 $\Delta^l_q$ and  $T^{l,{\mathcal F}}_qM=T^{m_q,{\mathcal F}}_qM$
for $l\ge m_q$, where $m_q$ is the degree of nonholonomy.
Moreover, one can show that
$T_q{\mathcal P}^{l,{\mathcal F}}=T_q{\mathcal P}^{m_q,{\mathcal F}}$ for $l\ge
m_q$ as well.

\begin{definition} Let $l$ be greater than or equal to the degree of
nonholonomy, i.e. $\Delta^l_q=T_qM$. The nonholonomic tangent
space $T_q^{{\mathcal F}}M$ is the manifold $T^{l,{\mathcal
F}}_qM$ equipped with the transitive action of the group $
T_q{\mathcal P}^{{\mathcal F}}\stackrel{def}{=} T_q{\mathcal
P}^{l,{\mathcal F}}$. For any $f\in{\mathcal F}$, the vector field
$T_q^{\mathcal F}f\in\operatorname{Vec} T_q^{{\mathcal F}}M$ is the
generator of the one-parameter group $s\mapsto p^{l,1}(sf)$; in
other words, $e^{sT_q^{{\mathcal

Obviously, $f\mapsto T_q^{{\mathcal F}}f$ is a homomorphism
of Lie algebras of vector fields; the group $T_q{\mathcal
P}^{{\mathcal F}}$ is generated by the dilation and the one-parameter
subgroups $e^{sT_q^{{\mathcal F}}f}$, $s\in \mathbb R$, $f\in{\mathcal
F}$. Moreover, just from the fact that the definition of
$T_q^{{\mathcal F}}M$ is intrinsic, it follows that every diffeomorphism
$\Phi:M\to M$ automatically induces an equivariant mapping
$\Phi_*^{\mathcal F}:T_q^{{\mathcal F}}M\to
T_{\Phi(q)}^{\Phi_*{\mathcal F}}M$ such that
$(\Phi_1\circ\Phi_2)_*^{\mathcal F}= \Phi_{1*}^{\Phi_{2*}{\mathcal
F}}\circ\Phi_{2*}^{\mathcal F}$ for any pair of diffeomorphisms
$\Phi_1$, $\Phi_2$.  One more functorial property is as
follows. Assume that ${\mathcal F}\subset{\mathcal
G}\subset\operatorname{Vec}M$; the identity inclusion $\imath:{\mathcal
F}\to{\mathcal G}$ induces a homomorphism $\imath_*:T_q{\mathcal
P}^{{\mathcal F}}\to T_q{\mathcal P}^{{\mathcal G}}$ and an
equivariant smooth mapping $\imath_*:T_q^{{\mathcal F}}M\to
T_q^{{\mathcal G}}M$.
Let $\bar{\mathcal F}=\{\sum_{j=1}^ka_jf_j : f_j\in{\mathcal F},\
a_j\in C^\infty(M),\ k>0\}$ be the module over $C^\infty(M)$
generated by ${\mathcal F}$, and let $\imath:{\mathcal F}\to\bar{\mathcal F}$ be
the identity inclusion. Then
$\imath_*:\left(T_q{\mathcal P}^{{\mathcal F}},T_q^{{\mathcal
\left(T_q{\mathcal P}^{\bar{\mathcal F}},T_q^{\bar{\mathcal F}}M\right)$
is an isomorphism.

\begin{remark} The last proposition states that
nonholonomic tangent spaces depend on the submodule of
$\operatorname{Vec}M$ generated by ${\mathcal F}$ rather than on
${\mathcal F}$ itself. In fact, from the very beginning of the
paper, we could deal with submodules of $\operatorname{Vec}M$ instead of
subsets. This approach would provide slightly more general
functorial properties, but the whole construction would become even
more dry and abstract than it is now. Anyway, the algebraically
trained reader will easily recover the missing functorial properties.

\section{Coordinate presentation}

Given nonnegative integers $k_1,\ldots,k_l$, where $k_1+\cdots+k_l=n$,
we present $\mathbb R^n$ as the direct sum
$\mathbb R^{k_1}\oplus\cdots\oplus\mathbb R^{k_l}$.
Every vector $x\in\mathbb R^n$ can be written as
x=(x_1,\ldots,x_l),\quad x_i=(x_{i1},\ldots,x_{ik_i})\in \mathbb R^{k_i},
\ i=1,\ldots,l.
A differential operator on $\mathbb R^n$ with smooth coefficients has
the form $\sum_\alpha
a_\alpha(x)\frac{\partial^{|\alpha|}}{\partial x^\alpha}$, where
$a_\alpha\in C^\infty(\mathbb R^n)$ and $\alpha$ is a multiindex:
$|\alpha_i|=\sum_{j=1}^{k_i}\alpha_{ij}$, $i=1,\ldots,l$.  The
space of all differential operators with smooth coefficients is an
associative algebra with composition of operators as
multiplication. The set of differential operators with polynomial
coefficients is a subalgebra of this algebra with generators
$1,x_{ij},\frac\partial{\partial x_{ij}}$, $i=1,\ldots,l$,
$j=1,\ldots,k_i$. We introduce a $\mathbb Z$-grading of this
subalgebra by assigning weights $\nu$ to the generators: 
$\nu(1)=0$, $\nu(x_{ij})=i$, and $\nu(\frac\partial{\partial
x_{ij}})=-i$.  Accordingly
\nu\left(x^\alpha\frac{\partial^{|\beta|}}{\partial x^\beta}\right)=
where $\alpha$ and $\beta$ are multiindices.

A differential operator with polynomial coefficients is said to be
$\nu$-{\it ho\-mo\-ge\-ne\-ous} of weight $m$ if all monomials
occurring in it have weight $m$. It is easy to see that $\nu(A_1\circ
A_2)=\nu(A_1)+\nu(A_2)$ for any $\nu$-homogeneous differential
operators $A_1$ and $A_2$. The most important for us are the
differential operators of order 0 (functions) and of order 1 (vector
fields). We have $\nu(ga)=\nu(g)+\nu(a)$,
$\nu([g_1,g_2])=\nu(g_1)+\nu(g_2)$ for any $\nu$-homogeneous function
$a$ and vector fields $g$, $g_1$, $g_2$. A differential operator of
order $N$ has weight at least $-Nl$; in particular, the weight of
a nonzero vector field is at least $-l$. Vector fields of nonnegative
weights vanish at 0 while the values at 0 of fields of weight $-i$
belong to the subspace $\mathbb R^{k_i}$, the $i$th term in the
presentation $\mathbb R^n=\mathbb R^{k_1}\oplus\cdots\oplus\mathbb

We introduce a dilation $\delta_t:\mathbb R^n\to\mathbb R^n$,
$t>0$, by setting
The $\nu$-homogeneity means homogeneity with respect to this dilation.
In particular, we have that $a(\delta_tx)=t^{\nu(a)}a(x)$,
$\delta_{t*}g=t^{-\nu(g)}g$ for a $\nu$-homogeneous function $a$
and a vector field $g$.

Now let $g=\sum_{i,j}a_{ij}\frac\partial{\partial x_{ij}}$
be an arbitrary smooth vector field. Expanding the coefficients
$a_{ij}$ in a Taylor series in powers of $x_{ij}$ and
grouping  terms with the same weights, we get an expansion
$g\approx\sum_{m=-l}^{+\infty}g^{(m)}$, where $g^{(m)}$ is a
$\nu$-homogeneous field of weight $m$. This expansion enables us to
introduce a decreasing filtration in the Lie algebra of smooth
vector fields $\operatorname{Vec}\mathbb R^n$ by putting
\operatorname{Vec}^m(k_1,\ldots,k_l)=\{X\in \operatorname{Vec}\mathbb R^n : X^{(i)}=0
\mbox{ for } i0$.

Let ${\mathcal F}$ be regular at $q$ and $\dim\Delta^1_q=d$.
Take $f_1,\ldots,f_d\in{\mathcal F}$ such that the vectors
$f_1(q_0),\ldots,f_d(q_0)$ form a basis of $\Delta^1_{q_0}$. Then
$f_1(q),\ldots,f_d(q)$ form a basis of $\Delta^1_q$ for any $q$
in a neighborhood of $q_0$. Hence, for any $f\in{\mathcal F}$ there
exist smooth functions
$a_1,\ldots,a_d$ such that $f(q)=\sum_{i=1}^da_i(q)f_i(q)$
for any $q$ in  the same neighborhood.
It follows that
\Delta^l_q=span\{[f_{i_1},[\ldots,f_{i_l}]\cdots](q):1\le i_j\le
d\}+\Delta^{l-1}_q,\quad l=1,2,\dots\,.
By the regularity of ${\mathcal F}$ at $q$, one can select vector
fields from the collection \linebreak
$\{[f_{i_1},[\ldots,f_{i_l}]\cdots](q):1\le i_j\le d\}$ in such a way
that their values at $q$ form a basis of $\Delta^l_q/\Delta^{l-1}_q$
for all $q$ close enough to $q_0$. With these bases in hand we easily
obtain the following \pagebreak well-known fact:
\begin{lemma} Assume that ${\mathcal F}\subset\operatorname{Vec}\,M$ is regular
at $q_0$, $v_i,v_j\in\operatorname{Vec}\,M$, $v_i(q)\in\Delta^i_q,\
v_j(q)\in\Delta^j_q\ \forall q$, and $v_i(q_0)=0$. Then

It follows immediately from this lemma that the Lie brackets of 
vector fields with values in $\Delta_q^i,\ i=1,2,\dots$, induce
a structure of graded Lie algebra on the space
$\sum_{i>0}\Delta^i_{q_0}/\Delta^{i-1}_{q_0}$. We denote
this graded Lie algebra by $\operatorname{Lie}_{q_0}{\mathcal F}$. Obviously,
$\operatorname{Lie}_{q_0}{\mathcal F}$ is generated by $\Delta^1_{q_0}$.

Let ${\mathcal F}$ be regular and bracket generating at
$q\in M$. Then the mapping $f\mapsto T^{{\mathcal F}}_qf$, $f\in{\mathcal
F}$, induces a Lie algebra isomorphism of $\operatorname{Lie}_q{\mathcal
F}$ and $\operatorname{Lie}\{T^{{\mathcal F}}_q f: f\in{\mathcal F}\}$.

Recall that $\operatorname{Lie}\{T^{{\mathcal F}}_q f: f\in{\mathcal F}\}$ 
is the Lie
algebra of a codimension 1 normal subgroup of $T_q{\mathcal
P}^{{\mathcal F}}$, which acts transitively on $T_q^{\mathcal F}M$
(see Proposition \ref{prop:normal}).  The quotient of $T_q{\mathcal
P}^{{\mathcal F}}$ by this subgroup is the dilation. We thus have a
transitive action of the $n$-dimensional nilpotent Lie group generated
by the Lie algebra $\operatorname{Lie}_q{\mathcal F}$ on $T_qM$. Since
$T_q^{\mathcal F}M$ is diffeomorphic to $\mathbb R^n$ and has the
``origin" (the jet of the constant curve $q$), we obtain a canonical
isomorphism of $T_q^{\mathcal F}M$ with the simply connected Lie group
generated by $\operatorname{Lie}_q{\mathcal F}$.

\bigskip We now turn to the generic case. Let ${\mathcal L}_d$ be the
free Lie algebra with $d$ generators (all algebras in this paper
are over $\mathbb R$); in other words, ${\mathcal L}_d$ is the Lie
algebra of commutator polynomials of $d$ variables. We have ${\mathcal
L}_d=\bigoplus_{i=1}^\infty{\mathcal L}_d^i$, where ${\mathcal
L}_d^i$ is the space of degree $i$ homogeneous commutator
polynomials. We set $\ell_d(i)=\dim{\mathcal L}_d^i,\
\ell_d^{(i)}=\sum_{j=1}^i\ell_d(i)$. The classical
recursion expression of $\ell_d(i)$ is

Below we deal with the space of germs at $q\in M$ of $d$-tuples of
smooth vector fields $(f_1,\ldots,f_d)$ endowed with the standard
$C^\infty$ topology. The following statement is almost obvious.

\begin{prop}\label{prop:6} For an open, everywhere dense set of $d$-tuples
$(f_1,\ldots,f_d)$, the set ${\mathcal F}=\{f_1,\ldots,f_d\}$ is
regular and bracket generating at $q$ with degree of
nonholonomy $m_q=\min\{i :\ell_d^{(i)}\ge n\}$ and
$\dim(\Delta^i_q/\Delta^{i-1}_q)=\ell_d(i)$ for
Take ${\mathcal F}$ such that the module $\bar{\mathcal F}$ is generated
by a generic $d$-tuple of vector fields.
According to Propositions \ref{prop:5} and \ref{prop:6}, the classification of
$\left(T_q{\mathcal P}^{{\mathcal F}},T_q^{{\mathcal F}}M\right)$
for such ${\mathcal F}$ is reduced to the classification of generic
graded Lie algebras $\operatorname{Lie}_q{\mathcal F}$ or corresponding 
Lie groups.

Definitions of rigidity and of rigid bidimensions were given in
the Introduction (isomorphism in this case means just Lie
group isomorphism). In the next theorem we list all rigid
bidimensions. It is convenient to give special names to some
infinite series of bidimensions. For $d=2,3,4,\dots$, the
bidimensions $\left(d,\ell_d^{(i)}\right),\ i=1,2,3,\dots$, are
called {\it free}; the bidimension $(d,d+1)$ is called the {\it
Darboux bidimension}, and the bidimension $(d,(d-1)(d+2)/2)$ is
called the {\it dual Darboux \pagebreak bidimension}.

\begin{theorem} All free, Darboux, and dual Darboux bidimensions
are rigid; each of these bidimensions admits a unique up to
isomorphism rigid group. Besides, there are $16$ exceptional
rigid bidimensions:

$(2,4)_1,\ (2,6)_2,\ (2,7)_2,\ (4,6)_2,\ (4,7)_2,\ (4,8)_2$,

$(5,7)_1,\ (5,8)_2,\ (5,9)_3,\ (5,11)_3,\ (5,12)_2,\ (5,13)_1$,

$(6,8)_2,\ (6,19)_2,\ (7,9)_1,\ (7,26)_1$,\\
where the index $j$ in the expression $(d,n)_j$ indicates the
number of isomorphism classes of rigid groups for the
bidimension $(d,n)$.

There are no other rigid bidimensions.

\begin{remark} This theorem is based on a complete
classification of rigid groups, which will be published in a
separate paper. An interesting open problem is to classify all
``simple" $T_q^{\{f_1,\ldots,f_d\}}M$. The term comes from 
singularity theory: a nonholonomic tangent space
$T_q^{\{f_1,\ldots,f_d\}}M$ is called {\it simple} if its small
perturbations $T_q^{\{f'_1,\ldots,f'_d\}}M$ admit only a finite
number of isomorphism classes. The above-mentioned Martinet
distribution is an example of a nonrigid simple case.
Obviously, all simple cases must live in rigid


The construction of the functor $T_q^{{\mathcal F}}$ is easily
generalized to the case of an ${\mathcal F} $ with a fixed filtration
(when various fields from ${\mathcal F}$ have various ``weights''). More
precisely, let
{\mathcal F}_0\subseteq {\mathcal F}_1\subseteq\cdots\subseteq{\mathcal F}_\nu\subseteq\cdots={\mathcal F}
and $f_0(q)=0$ for any $f_0\in {\mathcal F}_0$. Then the (local)
diffeomorphism $e^{f_0}$ induces a
transformation $p^{l,0}(f)$ of $C^l_q$.
We now set:
\begin{itemize} \item
$ \Delta_q^l=span\{[f_1,[\dots,[f_{i-1},f_i]\cdots](q) : f_j\in
{\mathcal F}_{\nu_j},\ \nu_1+\cdots +\nu_i \le l,\ i>0\}; $
\item ${\mathcal P}^l({\mathcal F})$, the group of transformations
generated by $\delta^l_\alpha$, $\alpha\in\mathbb{R}\setminus 0$ and by
$p^{l,k}(f)$, $f\in {\mathcal F}_k$, $k\ge 0$;
${\mathcal P}^l_\circ({\mathcal F})$, the normal subgroup of
${\mathcal P}^l({\mathcal F})$ generated by $p^{l,k}(f)$, $f\in
{\mathcal F}_{k-1}$, $k>0$;
and repeat the whole construction with these modified definitions.
The presence of ${\mathcal F}_0$ brings some new phenomena. In particular,
the Lie algebra generated by $T_qf$, $f\in{\mathcal F}$, is not, in
general, nilpotent. Moreover, the substitution for ${\mathcal F}$ of
the module $\bar{\mathcal F}$ with the induced filtration may enlarge
this Lie algebra.

Typical examples are smooth nonlinear control systems with an
equilibrium at $(q,0)$:
\dot x=f(x,u),\quad x\in M,\
u\in\mathbb R^r,\quad f(q,0)=0.
We set ${\mathcal
u^\alpha}f(\cdot,0): |\alpha|\le \nu\right\}$; the induced
filtration of the module $\bar{\mathcal F}$ is feedback invariant.
If the linearization of system (2) at $(q,0)$ is controllable,
then the tangent functor provides exactly the linearization and
its module version admits a finite classification (Brunovsky normal forms).
The number of isomorphism classes equals the number of partitions of $r$.
It would be very interesting to study
the tangent functor for some classes of systems with
noncontrollable linearizations.

\caption{Rigid bidimensions with the
indication of the number of isomorphism classes. The free bidimensions
lie on the curves.}\label{fig:plot}

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