ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 9, Pages 111–120 (November 13, 2003) S 1079-6762(03)00118-5

NONHOLONOMIC TANGENT SPACES: INTRINSIC CONSTRUCTION AND RIGID DIMENSIONS

A. AGRACHEV AND A. MARIGO

(Communicated by Svetlana Katok)

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.

1. Introduction

Let M be a (C^{∞}) smooth connected n-dimensional manifold and F ⊂ Vec M a set of smooth vector fields on M. Given q ∈ M and an integer l > 0 we set

Δ_{q}^{l} = span{[f_{1}, […, [f_{i−1}, f_{i}] ⋯](q) : f_{j} ∈ F, 1 ≤ j ≤ i, i ≤ l} ⊆ T_{q}M.

Clearly, Δ_{q}^{l} ⊆ Δ_{q}^{m} for l < m. The set F is called bracket generating (or completely nonholonomic) at q if there exists m_{q} such that Δ_{q}^{m_{q}} = T_{q}M. The minimum among these m_{q} is called the degree of nonholonomy of F at q. The set F is called bracket generating if it is bracket generating at every point.

We treat the pair (M, F) as a “nonholonomic space”, i.e. we assume that information propagates in M only along the integral curves of fields from F and the concatenations of these curves. If F is bracket generating, then any point of M is “reachable” from any other, i.e. points can be connected by an admissible curve. This is the Rashevskii–Chow theorem [5, 7]. Actually, one can derive more from this theorem. Let t ↦ e^{tf} be the (local) one-parameter subgroup of diffeomorphisms generated by a vector field f ∈ F so that t ↦ e^{tf}(q), q ∈ M, are integral trajectories of f. The natural “nonholonomic topology” is the strongest topology on M such that the mappings

(t_{1}, …, t_{k}) ↦ e^{t_{1}f_{1}} ∘ ⋯ ∘ e^{t_{k}f_{k}}(q), f_{i} ∈ F q ∈ M, k = 1, 2, …,

are continuous. It follows from the arguments of Rashevskii and Chow that this nonholonomic topology coincides with the standard topology on M.

Thus, the nonholonomic space (M, F) is homeomorphic to the manifold M, but Analysis and Geometry on (M, F) are very different from those on M. The main difference comes from the anisotropy of nonholonomic spaces: information propagates in all directions but with very different rates.

\begin{matrix}c \\ ◯ \end{matrix}2003 American Mathematical Society

Received by the editors March 25, 2003.

2000 Mathematics Subject Classification. Primary 58A30; Secondary 58K50.

Key words and phrases. Nonholonomic system, nilpotent approximation, Carnot group.

In this note we focus on tangent spaces to nonholonomic spaces. Let q ∈ M; we denote by T_{q}^{F}M the nonholonomic tangent space at q to be defined. We should make some remarks before we go to its construction, since this kind of spaces, burdened by extraneous structures, already appeared in various contexts. First of all, the nonholonomic tangent functor is a generalization of the usual one: if F = Vec M then T_{q}^{F}M coincides with T_{q}M. Actually the tangent space T_{q}^{F}M is homeomorphic to ℝ^{n} for any bracket generating F. Moreover, T_{q}^{F}M is a homogeneous space of a nilpotent Lie group, but in general, it has no intrinsic linear structure. On the other hand, it is intrinsically equipped with a canonical dilation which measures the anisotropy of the space.

Sub-Riemannian context. Assume that F is the space of sections of a vector subbundle of T M (a vector distribution) and that M is endowed with a Riemannian structure. The Carnot–Caratheodory distance between two points in M is defined to be the infimum of all the lengths of admissible curves connecting these points. It is well known that the Carnot–Caratheodory metric spaces admit tangent metric spaces in the Gromov–Hausdorff sense. These tangent spaces are homeomorphic to ℝ^{n} and their metrics are homogeneous with respect to certain dilations (see [3] for details). It turns out that the space and the dilation are defined independently on any metric and just represent T_{q}^{F}M.

Other contexts are mathematical control theory and hypoelliptic operators. Nonholonomic tangent functors appear here via nilpotent or graded approximations of the anisotropic objects to be studied (see [1, 2, 4, 6, 8, 9, 10]). The cited papers contain effective constructions and algorithms which allow one to compute things explicitly. A weak point of these constructions is their heavy dependence on the choice of coordinates. Because of that, the approximation looks like an auxiliary technical tool rather than a fundamental functorial operation; the geometric insight and the application of geometric machinery are highly impeded.

In this paper we introduce the nonholonomic tangent functor which performs the above-mentioned graded approximation in a natural coordinate-free way and demonstrates the intrinsic meaning of the dilation. One more goal of the paper is to classify all “rigid” cases. What does “rigid” mean? Obviously, T_{q}^{F}M depends only on the germ of F at q. Moreover, it depends only on the module generated by F over the algebra of germs at q of smooth scalar functions. Assume that this module has d generators f_{1}, …, f_{d}; then T_{q}^{F}M = T_{q}^{{f_{1}...f_{d}}}M. We say that T_{q}^{{f_{1}...f_{d}}}M is

rigid if T_{q}^{{f_{1}...f_{d}}}M is isomorphic to T_{q}^{{f′_{1}...f′_{d}}}M for all f′_{i} close enough to f_{i} in the C^{∞} topology. The pair (d, n) is called rigid bidimension if there exists at least one set of germs, {f_{1}, …, f_{d}} ⊂ Vec_{0} ℝ^{n}, such that T_{0}^{{f_{1}...f_{d}}}ℝ^{n} is rigid. Given a rigid bidimension (d, n), it is not hard to show that a generic germ {f_{1}, …, f_{d}} is rigid and there is only a finite number of mutually nonisomorphic rigid T_{0}^{{f_{1}...f_{d}}}ℝ^{n}. In this paper we completely characterize the rigid bidimensions and, for each rigid bidimension, we indicate the number of different (up to isomorphism) T_{0}^{{f_{1}...f_{d}}}ℝ^{n}. It turns out that this number can only be 1, 2, or 3. Normal forms and proofs will be given in a forthcoming long paper.

2. Tangent functor

A flow in M is a one-parameter family of diffeomorphisms P_{t} ∈ Diff M satisfying the condition P_{0} = id. The set of flows is a group with respect to pointwise composition: the product of P_{t} and P̂_{t} is the flow t ↦ P_{t} ∘ P̂_{t}. Let q ∈ M. The tangent space T_{q}M consists of 1-jets at 0 of smooth curves γ(t) in M such that γ(0) = q. To construct the nonholonomic tangent space T_{q}^{F}M we need jets of order m_{q}, where m_{q} is the degree of nonholonomy of F at q. The group of flows naturally acts on the space of smooth curves starting from q: the flow P_{t} sends the curve γ(t) to the curve P_{.}(γ) : t ↦ P_{t}(γ(t)). Of course, for any positive integer l, the l-jet at 0 of P_{.}(γ) depends only on P_{t} and on the l-jet of γ. We thus obtain the action of the group of flows on the space C_{q}^{l} of l-jets of smooth curves in M starting from q. We keep the notation q for the constant curve γ(t) ≡ q and its jets; indeed, the trajectory t ↦ P_{t}(q) of the flow P_{t} is the image of the constant curve under the natural action. The canonical dilation

δ_{α}^{l} : C_{q}^{l} → C_{q}^{l}, δ_{α_{1}α_{2}}^{l} = δ_{α_{1}}^{l} ∘ δ_{α_{2}}^{l}, α ∈ ℝ, δ_{0}(C_{q}^{l}) = q,

is induced by the standard dilation δ_{α} on the space of smooth curves: (δ_{α}γ)(t) = γ(αt) for any curve γ.

We denote by p^{l,k}(f) : C_{q}^{l} → C_{q}^{l} the transformation induced by the flow t ↦ e^{t^{k}f}, where f ∈ Vec M. Let P^{l,F} be the group of transformations generated by δ_{α}^{l} with α ≠ 0, and by p^{l,k}(f) with f ∈ F and k = 1, 2, 3, …. Dilations do not commute with flows, even at the jet level of P^{l,F}. However, given a flow P_{t}, one can see that lim_{s→0}(δ_{s}^{l} ∘ P_{.} ∘ δ_{s^{−1}}^{l}) = id on the space of curves. This motivates the definition D(ϕ) = lim_{s→0}(δ_{s}^{l}∘ϕ∘δ_{s^{−1}}^{l}), ϕ ∈ P^{l,F}. Then the image of the group homomorphism D is the dilation group {δ_{α} : α ≠ 0}.

We denote by O^{l,F} ⊂ C_{q}^{l} the orbit of the jet q under the action of the group P^{l,F}. Let us also consider the normal subgroup P_{∘}^{l,F} of P^{l,F} generated by p^{l,k}(f), f ∈ F, k = 2, 3, … (the difference from P^{l,F} is that p^{l,1}(f) and δ_{α}^{l} are absent). Finally, we set:

• T_{q}^{l,F}M, the quotient space of O^{l,F} by the action of the group P_{∘}^{l,F};

• T_{q}P^{l,F}, the group of transformations of T_{q}^{l,F}M induced by the action of

P^{l,F}/P_{∘}^{l,F}.

We keep the symbols δ_{α}^{l} and p^{l,k}(f) for the induced transformations of T_{q}^{l,F}M, and the symbol D for the induced homomorphism of T_{q}P^{l,F} on {δ_{α} : α ≠ 0}. See Section 3 for further understanding of these objects.

Proposition 2.1. The codimension 1 normal subgroup D^{−1}(id) ⊂ T_{q}P^{l,F} is a nilpotent Lie group generated by the one-parameter subgroups s ↦ p^{l,1}(sf), f ∈ F. The group D^{−1}(id) acts transitively on T_{q}^{l,F}M.

Let π^{l} : C_{q}^{l} → C_{q}^{l−1} be the standard projection. Obviously, π^{l} ∘ p^{l,k}(f) = p^{l−1,k}(f) ∘ π^{l}, π^{l} ∘ δ_{α}^{l} = δ_{α}^{l−1} ∘ π^{l}. Hence π^{l} sends orbits of the groups P^{l,F} and P_{∘}^{l,F} to orbits of the groups P^{l−1,F} and P_{∘}^{l−1,F} and induces a mapping on the quotient spaces. We keep the symbol π^{l} for the induced mapping so that π^{l} : T_{q}^{l,F}M → T_{q}^{l−1,F}M.

Proposition 2.2. For any l > 0, π^{l} : T_{q}^{l,F}M → T_{q}^{l−1,F}M is a fiber bundle with fiber Δ_{q}^{l}/Δ_{q}^{l−1}.

In particular, T_{q}^{l,F}M is diffeomorphic to Δ_{q}^{l} and T_{q}^{l,F}M = T_{q}^{m_{q},F}M for l ≥ m_{q}, where m_{q} is the degree of nonholonomy. Moreover, one can show that T_{q}P^{l,F} = T_{q}P^{m_{q},F} for l ≥ m_{q} as well.

Definition. Let l be greater than or equal to the degree of nonholonomy, i.e. Δ_{q}^{l} = T_{q}M. The nonholonomic tangent space T_{q}^{F}M is the manifold T_{q}^{l,F}M equipped with

the transitive action of the group T_{q}P^{F} =^{def} T_{q}P^{l,F}. For any f ∈ F, the vector field T_{q}^{F}f ∈ Vec T_{q}^{F}M is the generator of the one-parameter group s ↦ p^{l,1}(sf); in other

words, e^{sT_{q}^{F} f} =^{def} p^{l,1}(sf).

Obviously, f ↦ T_{q}^{F}f is a homomorphism of Lie algebras of vector fields; the group T_{q}P^{F} is generated by the dilation and the one-parameter subgroups e^{sT_{q}^{F} f}, s ∈ ℝ, f ∈ F. Moreover, just from the fact that the definition of T_{q}^{F}M is intrinsic, it follows that every diffeomorphism Φ : M → M automatically induces an equivariant mapping Φ_{∗}^{F} : T_{q}^{F}M → T_{Φ(q)}^{Φ_{∗}F}M such that (Φ_{1} ∘ Φ_{2})_{∗}^{F} = Φ_{1∗}^{Φ_{2∗}F} ∘ Φ_{2∗}^{F} for any pair of diffeomorphisms Φ_{1}, Φ_{2}. One more functorial property is as follows. Assume that F ⊂ G ⊂ Vec M; the identity inclusion ı : F → G induces a homomorphism ı_{∗} : T_{q}P^{F} → T_{q}P^{G} and an equivariant smooth mapping ı_{∗} : T_{q}^{F}M → T_{q}^{G}M.

Proposition 2.3. Let F̄ = {∑_{j=1}^{k} a_{j}f_{j} : f_{j} ∈ F, a_{j} ∈ C^{∞}(M), k > 0} be the module over C^{∞}(M) generated by F, and let ı : F → F̄ be the identity inclusion.

Then ı_{∗} : (T_{q}P^{F}, T_{q}^{F}M) ⟶ (T_{q}PF̄, T F̄ ; qM) is an isomorphism.

Remark. The last proposition states that nonholonomic tangent spaces depend on the submodule of Vec M generated by F rather than on F itself. In fact, from the very beginning of the paper, we could deal with submodules of 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.

3. Coordinate presentation

Given nonnegative integers k_{1}, …, k_{l}, where k_{1} + ⋯ + k_{l} = n, we present ℝ^{n} as the direct sum ℝ^{k_{1}} ⊕ ⋯ ⊕ ℝ^{k_{l}}. Every vector x ∈ ℝ^{n} can be written as

x = (x_{1}, …, x_{l}), x_{i} = (x_{i1}, …, x_{ik_{i}}) ∈ ℝ^{k_{i}}, i = 1, …, l.

A differential operator on ℝ^{n} with smooth coefficients has the form ∑_{α} a_{α}(x)\frac{∂^{|α|}}{∂x^{α}}, where a_{α} ∈ C^{∞}(ℝ^{n}) and α is a multiindex: α = (α_{1}, …, α_{l}), α_{i} = (α_{i1}, …, α_{ik_{i}}), |α_{i}| = ∑_{j=1}^{k_{i}} α_{ij}, i = 1, …, 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{∂}{∂x_{ij}}, i = 1, …, l, j = 1, …, k_{i}. We introduce a ℤ-grading of this subalgebra by assigning weights ν to the generators: ν(1) = 0, ν(x_{ij}) = i, and ν(\frac{∂}{∂x_{ij}}) = −i. Accordingly

ν (x^{α} \frac{∂^{|β|}}{∂x^{β}} ) = ∑_{i=1}^{l}(|α_{i}| − |β_{i}|)i,

where α and β are multiindices.

A differential operator with polynomial coefficients is said to be ν-homogeneous of weight m if all monomials occurring in it have weight m. It is easy to see that ν(A_{1}∘A_{2}) = ν(A_{1})+ν(A_{2}) for any ν-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 ν(ga) = ν(g)+ν(a), ν([g_{1}, g_{2}]) = ν(g_{1})+ν(g_{2}) for any ν-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 ℝ^{k_{i}}, the ith term in the presentation ℝ^{n} = ℝ^{k_{1}} ⊕ ⋯ ⊕ ℝ^{k_{l}}.

We introduce a dilation δ_{t} : ℝ^{n} → ℝ^{n}, t > 0, by setting

(1)  δ_{t}(x_{1}, x_{2}, …, x_{l}) = (tx_{1}, t^{2}x_{2}, …, t^{l}x_{l}).

The ν-homogeneity means homogeneity with respect to this dilation. In particular, we have that a(δ_{t}x) = t^{ν(a)}a(x), δ_{t∗}g = t^{−ν(g)}g for a ν-homogeneous function a and a vector field g.

Now let g = ∑_{i,j} a_{ij} \frac{∂}{∂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 ≈ ∑_{m=−l}^{+∞} g^{(m)}, where g^{(m)} is a ν-homogeneous field of weight m. This expansion enables us to introduce a decreasing filtration in the Lie algebra of smooth vector fields Vec ℝ^{n} by putting

Vec^{m}(k_{1}, …, k_{l}) = {X ∈ Vec ℝ^{n} : X^{(i)} = 0 for i < m}, −l ≤ m < +∞.

It is easy to see that

[Vec^{m_{1}}(k_{1}, …, k_{l}), Vec^{m_{2}}(k_{1}, …, k_{l})] ⊆ Vec^{m_{1}+m_{2}}(k_{1}, …, k_{l}).

It happens that this class of filtrations is in a sense universal. The following theorem is a special case of the general results proved in [2, 4].

Theorem 3.1. Suppose that dim(Δ_{q}^{i}/Δ_{q}^{i−1}) = k_{i}, i = 1, …, l. Then there exists a neighborhood O_{q} of q in M and a coordinate mapping χ : O_{q} → ℝ^{n} such that

χ(q) = 0, χ_{∗}(Δ_{q}^{i}) = ℝ^{k_{1}} ⊕ ⋯ ⊕ ℝ^{k_{i}}, 1 ≤ i ≤ l,

and χ_{∗}(F) ⊂ Vec^{−1}(k_{1}, …, k_{l}).

A mapping χ : O_{q} → ℝ^{n} which satisfies the hypothesis of the above theorem is called an adapted coordinate map for F. All constructions have a simple explicit presentation in adapted coordinates. Unfortunately, an adapted coordinate map for a given F is far from being unique and it is not clear how to select a canonical one.

In any coordinates, the l-jet of a curve is identified with its degree l Taylor polynomial. In particular, χ_{∗}(C_{q}^{l}) = {∑_{i=1}^{l} t^{i}ξ_{i} : ξ_{i} ∈ ℝ^{n}}.

Proposition 3.2. Let χ be an adapted coordinate map for F; then

χ_{∗}(O^{l,F}) = {∑_{i=1}^{l} t^{i}ξ_{i} : ξ_{i} ∈ ℝ^{k_{1}} ⊕ ⋯ ⊕ ℝ^{k_{i}}, 1 ≤ i ≤ l}

and, for any γ ∈ O^{l,F},

χ_{∗}(P_{∘}^{l,F}(γ)) = χ_{∗}(γ) + {∑_{i=1}^{l} t^{i}ξ_{i} : ξ_{i} ∈ ℝ^{k_{1}} ⊕ ⋯ ⊕ ℝ^{k_{i−1}}, 1 ≤ i ≤ l} .

The proof of this proposition is based on the expansions

χ_{∗}(f) ≈ χ_{∗}(f)^{(−1)} + ∑_{j=0}^{∞} χ_{∗}(f)^{(j)} ∀f ∈ F,

χ(e^{τf}(q)) ≈ ∑_{i=1}^{∞} \frac{τ^{i}}{i!} \underbrace{(χ_{∗}f) ∘ ⋯ ∘ (χ_{∗}f)}_{i times} Id (0),

where Id (x) ≡ x, x ∈ ℝ^{n}, and on the fact that all ν-homogeneous functions of positive weight vanish at 0.

Proposition 2.3 implies the identification of T_{0}^{χ_{∗}(F)}ℝ^{n} with {∑_{i=1}^{l} t^{i}ξ_{i} : ξ_{i} ∈ ℝ^{k_{i}}, 1 ≤ i ≤ l}. Similarly, the vector field T_{0}^{χ_{∗}(F)}f is identified with t(χ_{∗}(f))^{(−1)}. In particular, the fields T_{q}^{F}f, f ∈ F, are represented by ν-homogeneous vector fields of weight −1 in adapted coordinates.

4. Regularity and rigidity

In general, the nilpotent Lie algebra generated by the fields T_{q}^{F}f, f ∈ F, may have a dimension strictly greater than dim T_{q}^{F}M = dim M, and T_{q}^{F}M may not have an intrinsic Lie group structure. A simple example is provided by the Martinet distribution F = span\{\frac{∂}{∂x_{1}} , \frac{∂}{∂x_{2}} + x_{1}^{2} \frac{∂}{∂x_{3}} \} in ℝ^{3}, where T_{q}^{F}f, f ∈ F, generate a 4-dimensional Lie algebra. This phenomenon does not occur in the so-called regular case, first studied by Vershik and Gershkovich (see [8, 9, 10]).

Definition. We say that F ⊂ Vec M is regular at q_{0} ∈ M if dim Δ_{q}^{i} is constant in a neighborhood of q_{0}, ∀i > 0.

Let F be regular at q and dim Δ_{q}^{1} = d. Take f_{1}, …, f_{d} ∈ F such that the vectors f_{1}(q_{0}), …, f_{d}(q_{0}) form a basis of Δ_{q_{0}}^{1}. Then f_{1}(q), …, f_{d}(q) form a basis of Δ_{q}^{1} for any q in a neighborhood of q_{0}. Hence, for any f ∈ F there exist smooth functions a_{1}, …, a_{d} such that f(q) = ∑_{i=1}^{d} a_{i}(q)f_{i}(q) for any q in the same neighborhood. It follows that

Δ_{q}^{l} = span{[f_{i_{1}}, […, f_{i_{l}}] ⋯](q) : 1 ≤ i_{j} ≤ d} + Δ_{q}^{l−1}, l = 1, 2, ….

By the regularity of F at q, one can select vector fields from the collection {[f_{i_{1}}, […, f_{i_{l}}] ⋯](q) : 1 ≤ i_{j} ≤ d} in such a way that their values at q form a basis of Δ_{q}^{l}/Δ_{q}^{l−1} for all q close enough to q_{0}. With these bases in hand we easily obtain the following well-known fact:

Lemma 4.1. Assume that F ⊂ Vec M is regular at q_{0}, v_{i}, v_{j} ∈ Vec M, v_{i}(q) ∈ Δ_{q}^{i}, v_{j}(q) ∈ Δ_{q}^{j} ∀q, and v_{i}(q_{0}) = 0. Then [v_{i}, v_{j}](q_{0}) ∈ Δ_{q_{0}}^{i+j−1}.

It follows immediately from this lemma that the Lie brackets of vector fields with values in Δ_{q}^{i}, i = 1, 2, …, induce a structure of graded Lie algebra on the space ∑_{i>0} Δ_{q_{0}}^{i}/Δ_{q_{0}}^{i−1}. We denote this graded Lie algebra by Lie_{q_{0}} F. Obviously, Lie_{q_{0}} F is generated by Δ_{q_{0}}^{1}.

Proposition 4.2. Let F be regular and bracket generating at q ∈ M. Then the mapping f ↦ T_{q}^{F}f, f ∈ F, induces a Lie algebra isomorphism of Lie_{q} F and Lie{T_{q}^{F}f : f ∈ F}.

Recall that Lie{T_{q}^{F}f : f ∈ F} is the Lie algebra of a codimension 1 normal subgroup of T_{q}P^{F}, which acts transitively on T_{q}^{F}M (see Proposition 2.1). The quotient of T_{q}P^{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 Lie_{q} F on T_{q}M. Since T_{q}^{F}M is diffeomorphic to ℝ^{n} and has the “origin” (the jet of the constant curve q), we obtain a canonical isomorphism of T_{q}^{F}M with the simply connected Lie group generated by Lie_{q} F.

We now turn to the generic case. Let L_{d} be the free Lie algebra with d generators (all algebras in this paper are over ℝ); in other words, L_{d} is the Lie algebra of commutator polynomials of d variables. We have L_{d} = ⨁_{i=1}^{∞} L_{d}^{i}, where L_{d}^{i} is the space of degree i homogeneous commutator polynomials. We set ℓ_{d}(i) = dim L_{d}^{i}, ℓ_{d}^{(i)} = ∑_{j=1}^{i} ℓ_{d}(i). The classical recursion expression of ℓ_{d}(i) is ℓ_{d}(i) = d^{i} − ∑_{j|i} jℓ_{d}(j).

Below we deal with the space of germs at q ∈ M of d-tuples of smooth vector fields (f_{1}, …, f_{d}) endowed with the standard C^{∞} topology. The following statement is almost obvious.

Proposition 4.3. For an open, everywhere dense set of d-tuples (f_{1}, …, f_{d}), the set F = {f_{1}, …, f_{d}} is regular and bracket generating at q with degree of nonholonomy m_{q} = min{i : ℓ_{d}^{(i)} ≥ n} and dim(Δ_{q}^{i}/Δ_{q}^{i−1}) = ℓ_{d}(i) for i = 1, …, m_{q} − 1.

Take F such that the module F̄ is generated by a generic d-tuple of vector fields. According to Propositions 4.2 and 4.3, the classification of (T_{q}P^{F}, T_{q}^{F}M) for such F is reduced to the classification of generic graded Lie algebras Lie_{q} 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, …, the bidimensions (d, ℓ_{d}^{(i)}) , i = 1, 2, 3, …, are called free; the bidimension (d, d + 1) is called the Darboux bidimension, and the bidimension (d, (d − 1)(d + 2)/2) is called the dual Darboux bidimension.

Theorem 4.4. 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.

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},...,f_{d}}}M. The term comes from singularity theory: a nonholonomic tangent space T_{q}^{{f_{1},...,f_{d}}}M is called simple if its small perturbations T_{q}^{{f′_{1},...,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.

5. Generalization

The construction of the functor T_{q}^{F} is easily generalized to the case of an F with a fixed filtration (when various fields from F have various “weights”). More precisely, let

F_{0} ⊆ F_{1} ⊆ ⋯ ⊆ F_{ν} ⊆ ⋯ = F

and f_{0}(q) = 0 for any f_{0} ∈ F_{0}. Then the (local) diffeomorphism e^{f_{0}} induces a transformation p^{l,0}(f) of C_{q}^{l}. We now set:

• Δ_{q}^{l} = span{[f_{1}, […, [f_{i−1}, f_{i}] ⋯](q) : f_{j} ∈ F_{ν_{j}}, ν_{1} + ⋯ + ν_{i} ≤ l, i > 0};

• P^{l}(F), the group of transformations generated by δ_{α}^{l}, α ∈ ℝ \ 0 and by p^{l,k}(f), f ∈ F_{k}, k ≥ 0;

• P_{∘}^{l}(F), the normal subgroup of P^{l}(F) generated by p^{l,k}(f), f ∈ F_{k−1}, k > 0;

and repeat the whole construction with these modified definitions. The presence of F_{0} brings some new phenomena. In particular, the Lie algebra generated by T_{q}f, f ∈ F, is not, in general, nilpotent. Moreover, the substitution for F of the module F̄ with the induced filtration may enlarge this Lie algebra.

Typical examples are smooth nonlinear control systems with an equilibrium at (q, 0):

(2)  ẋ = f(x, u), x ∈ M, u ∈ ℝ^{r}, f(q, 0) = 0.

We set F_{ν} = \{\frac{∂^{|α|}}{∂u^{α}} f(·, 0) : |α| ≤ ν\}; the induced filtration of the module F̄ 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.

Figure 1. Rigid bidimensions with the indication of the number of isomorphism classes. The free bidimensions lie on the curves.

References

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[8] A. M. Vershik, V. Ya. Gershkovich, Nonholonomic dynamic systems. Geometry of distributions and variational problems. Springer Verlag, EMS 16 (1987), 5–85. MR 89f:58007 [9] A. M. Vershik, V. Ya. Gershkovich, A bundle of nilpotent Lie algebras over a nonholonomic

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[10] A. M. Vershik, V. Ya. Gershkovich, Estimation of the functional dimension of the orbit space of germs of distributions in general position. Mat. Zametki 44 (1988), no. 5, 596–603, 700. English transl., Math. Notes 44 (1988), no. 5-6, 806–810 (1989) MR 90g:58002

Steklov Mathematical Institute, Moscow, Russia Current address: SISSA, Via Beirut 2–4, Trieste, Italy

E-mail address: agrachev@ma.sissa.it

IAC-CNR, Viale Policlinico 136, Roma, Italy

E-mail address: marigo@iac.rm.cnr.it