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To display a typeset version of a journal * %_ * article easily, we suggest that you retrieve the article in DVI, * %_ * PostScript, or PDF format. * %_ ************************************************************************** \controldates{10-NOV-2003,10-NOV-2003,10-NOV-2003,10-NOV-2003} \RequirePackage[warning,log]{snapshot} \documentclass{era-l} \issueinfo{9}{14}{}{2003} \dateposted{November 13, 2003} \pagespan{111}{120} \PII{S 1079-6762(03)00118-5} \copyrightinfo{2003}{American Mathematical Society} \usepackage[english]{babel} \usepackage{amssymb} \usepackage{graphicx} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{prop}[theorem]{Proposition} \theoremstyle{definition} \newtheorem*{definition}{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem*{remark}{Remark} \begin{document} \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} \email{agrachev@ma.sissa.it} \author{A. Marigo} \address{IAC-CNR, Viale Policlinico 136, Roma, Italy} \email{marigo@iac.rm.cnr.it} \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. \end{abstract} \maketitle \section{Introduction} 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 $\Delta^l_q/\Delta^{l-1}_q$. \end{prop} 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, $e^{sT_q^{{\mathcal F}}f}\stackrel{def}{=}p^{l,1}(sf)$. \end{definition} 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 $\imath_*:T_q^{{\mathcal F}}M\to T_q^{{\mathcal G}}M$. \begin{prop}\label{prop:iso} 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 $\imath_*:\left(T_q{\mathcal P}^{{\mathcal F}},T_q^{{\mathcal F}}M\right)\longrightarrow \left(T_q{\mathcal P}^{\bar{\mathcal F}},T_q^{\bar{\mathcal F}}M\right)$ is an isomorphism. \end{prop} \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. \end{remark} \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: $\alpha=(\alpha_1,\ldots,\alpha_l)$, $\alpha_i=(\alpha_{i1},\ldots,\alpha_{ik_i})$, $|\alpha_i|=\sum_{j=1}^{k_i}\alpha_{ij}$, $i=1,\ldots,l$. 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 \[ \nu\left(x^\alpha\frac{\partial^{|\beta|}}{\partial x^\beta}\right)= \sum_{i=1}^l(|\alpha_i|-|\beta_i|)i, \] 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 $\mathbb R^n=\mathbb R^{k_1}\oplus\cdots\oplus\mathbb R^{k_l}$. We introduce a dilation $\delta_t:\mathbb R^n\to\mathbb R^n$, $t>0$, by setting \begin{equation} \delta_t(x_1,x_2,\ldots,x_l)=(tx_1,t^2x_2,\ldots,t^lx_l). \end{equation} 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 } i 0$. \end{definition} 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 $[v_i,v_j](q_0)\in\Delta_{q_0}^{i+j-1}$. \end{lemma} 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 $\Delta^1_{q_0}$. \begin{prop}\label{prop:5} 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 $\operatorname{Lie}\{T^{{\mathcal F}}_q f: f\in{\mathcal F}\}$. \end{prop} 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 $\ell_d(i)=d^i-\sum_{j|i}j\ell_d(j)$. 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 $i=1,\ldots,m_q-1$. \end{prop} 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. \end{theorem} \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 bidimensions. \end{remark} \section{Generalization} 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 $p^{l,k}(f)$, $f\in {\mathcal F}_k$, $k\ge 0$; \item ${\mathcal P}^l_\circ({\mathcal F})$, the normal subgroup of ${\mathcal P}^l({\mathcal F})$ generated by $p^{l,k}(f)$, $f\in {\mathcal F}_{k-1}$, $k>0$; \end{itemize} 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 $(q,0)$: \begin{equation} \dot x=f(x,u),\quad x\in M,\ u\in\mathbb R^r,\quad f(q,0)=0. \end{equation} We set ${\mathcal F}_\nu=\left\{\frac{\partial^{|\alpha|}}{\partial u^\alpha}f(\cdot,0): |\alpha|\le \nu\right\}$; 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. \begin{figure} \includegraphics[scale=.55]{era118el-fig-1} \caption{Rigid bidimensions with the indication of the number of isomorphism classes. The free bidimensions lie on the curves.}\label{fig:plot} \end{figure} \begin{thebibliography}{9} \bibitem{as} A.~A.~Agrachev, A.~V.~Sarychev, Filtrations of a Lie algebra of vector fields and nilpotent approximation of control systems. Dokl. Akad. Nauk SSSR 295 (1987); English transl., Soviet Math. Dokl. 36 (1988), 104--108. \MR{88j:93015} \bibitem{ags} A.~A.~Agrachev, R.~V.~Gamkrelidze, A.~V.~Sarychev, Local invariants of smooth control systems. Acta Appl. Math. 14 (1989), 191--237. \MR{90i:93033} \bibitem{be} A.~Bellaiche, The tangent space in sub-Riemannian geometry. 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