1 Main results↩︎

This paper is devoted to the proof of the existence of a canonical absolute parallelism and to the characterization of maximally symmetric models for bracket generating rank \(2\) distributions that satisfy a natural condition of maximal growth for iterated Lie brackets of length at most \(3\) of their sections with further consequences to rank \(2\) distributions which are not Goursat.

A rank \(l\) vector distribution \(D\) on an \(n\)-dimensional manifold \(M\) or an \((l,n)\)-distribution (where \(l<n\)) is a subbundle of the tangent bundle \(TM\) with \(l\)-dimensional fibers. The weak derived flag at \(p\in M\) of the distribution \(D\) is the flag \(\{D^{i}\}_{i=1}^\infty\) defined by \[\label{week95derived} D^{1}(q) = D(q)\quad \text{and}\quad D^{i}(q) = D^{i-1}(q)+[D,D^{i-1}](q)\;\text{for } i>1.\tag{1}\] The space \(D^i(q)\) is called the \(i\)th power of the distribution \(D\) at point \(q\). In particular, \(D^2(q)\) (respectively, \(D^3(q)\)) is called the square (respectively, the cube) of the distribution \(D\) at \(q\). A distribution \(D\) is called bracket generating if for every \(q\), \(D^k(q)\) coincides with the whole tangent space \(T_q M\) for sufficiently large \(k\). The tuple \((\dim D(q). \dim D^2(q), \ldots \dim D^i(q), \cdots)\) is called the small growth vector of the distribution \(D\) at the point \(q\). The distribution \(D\) is called equiregular at a point \(q_0\) if there exists a neighborhood \(U\) of \(q_0\) such that the small growth vector of \(D\) is the same for all \(q\in U\). If \(D\) is bracket generating, the set of points at which \(D\) is equiregular, is generic.

Elementary counting implies that for a rank \(2\) distribution \(D\), \(\dim D^3(q)\) is at most \(5\). One of the main results of the paper can be formulated as follows:

Note that for \(n>5\) the infinitesimal symmetry algebra of the distribution associated with the Monge equation ?? is isomorphic to the natural semidirect sum of \(\mathfrak{gl}_2(\relax \ifmmode \mathbb{R} \else\) \(\fi)\) with the \((2n-5)\)-dimensional Heisenberg algebra (for details, see [1] or [2]).

É. Cartan proved an analogue of Theorem 1 for the case \(n = 5\) [3]: the dimension of the bundle in item (1) and the corresponding upper bound for the dimension of the symmetry algebra is 14 (equal to the dimension of the exceptional Lie algebra \(G_2\)). The maximally symmetric model in this case is given by equation ?? (the Cartan–Hilbert equation) or, equivalently, by the Pfaffian system ?? with \(n = 5\), and its infinitesimal symmetry algebra is isomorphic to the split real form of \(G_2\).

Theorem 1 strengthens Theorems 1 and 3 of [1] by removing an additional assumption that the distribution \(D\) is of so-called maximal class (see Definitions 3 and 4 in Section 2 for the precise geometric definition.¹ Thus, Theorem 1 is a direct consequence of those theorems from [1] and of the following result, which forms the main technical core of the present paper:

A precise description of the bundle and the canonical frame of item (1) of Theorem 1, using the theory of Jacobi curves [4] and geometry of curves in projective spaces, can be found in [1]. An alternative construction of this bundle and the canonical frame, using Tanaka-Morimoto theory, can be found in section 3 of [2].

Theorem 2 was conjectured in [5] (see also [1]) nearly 20 years ago, but before the present paper, it had been confirmed only in the following very limited specific cases (see, e.g., [6]):

1. \(5\leq n\leq 8\);

2. for distributions with small grow vector \((2,3,5,6,7,\cdots, n)\) at every point; i.e., when \(\dim D^j(q)=j+2\) for \(4\leq j\leq n-2\);

3. in the case of \((2, 14)\)-distributions with “free" small growth vector \((2,3,5,8,14)\);

4. for distributions associated with Monge equations \[z^{(m)} = F\big(x, y, y', \cdots, y^{(n-2-m)}, z,\cdots, z^{(m-1)}(n)\big), \quad \displaystyle{{\partial ^2 F\over \partial (y^{(n-2-m)})^2}\neq 0}.\]

Note that in [7] it was demonstrated that the local model ?? is the most symmetric among Monge distributions. This was done by showing first that Monge distributions with fixed \(1\leq m\leq \lfloor{\frac{n-2}{2}}\rfloor\) have fixed Tanaka symbol. In the sequel, those Tanaka symbols will be called Monge symbols. Then by computing the universal Tanaka prolongation of such symbols, the authors find that the maximally symmetric model corresponds to \(m=1\) and is locally equivalent to the model in ?? . However, for \(n\geq 6\) generic germs of rank 2 distributions are not Monge ²: for \(n\geq 7\) this follows from the fact that generic Tanaka symbols of \((2,n)\)-distributions are not Monge symbols. For \(n=6\) the statement follows from the fact that certain nontrivial invariants vanish for Monge distributions. ³

The previous approaches to proving Theorem 2 relied on attempts to compute the filtration 5 (below) directly in terms of the original distribution \(D\). A key insight in [6] was that it suffices to prove Theorem 2 for flat distributions with prescribed Tanaka symbols ⁴ (for the definition of the Tanaka symbols and flat distributions, see [8], [9] and also the end of section 3, after formula 21 ). However, this method depends on the classification of Tanaka symbols, which becomes infeasible due to the combinatorial explosion of possibilities as the dimension increases. To circumvent this obstacle, a two-stage strategy was proposed in [6]. The first stage involves computing the filtration 5 for free truncated nilpotent Lie algebras with two generators of a given step. The second stage aims to use the result of these computations for all nilpotent graded Lie algebras of the same step. However, the first stage quickly becomes computationally infeasible as the dimension grows, even for computer algebra systems. Moreover, even when the first stage was successfully completed (as in the 5-step case of item (3) above), it remained unclear how to use it to implement the second stage effectively.

In this paper, we adopt a completely different strategy, which enables us to prove Theorem 2 in full generality, without relying on any computer algebra computations. Rather than attempting to compute the filtration 5 directly from the original distribution \(D\) on \(M\), we begin with an abstract filtration of the type 5 on the corresponding submanifold of the projectivized cotangent bundle \(\mathbb{P}T^*M\). Assuming that the filtration enjoys the properties of one associated with a distribution of a given constant class greater than \(1\) ⁵, we show that it corresponds to a bracket-generating distribution only if the class is maximal. A key factor that convinced us of the promise of this method, and motivated us to pursue it, was the realization—based on general reasoning—that if the method were to fail, it would effectively yield a counterexample. Moreover, we recognized that the analysis should not strongly depend on the dimension, but from item (1) above, no counterexamples arise in low-dimensional cases. Finally, this approach is completely independent of the Tanaka symbols of the original distribution.

The next two subsections will expound the implications of Theorem 1 for non-Goursat rank 2 distributions and the implications of Theorem 2 for optimal control problems with constraint given by a rank 2 distribution with 5-dimensional cube, respectively.

1.1 Canonical Frames for Non-Goursat Rank 2 Distributions↩︎

We now explain why the assumption in equation ?? is, in fact, not restrictive. In short, given a distribution which is not Goursat, one can apply iterative Cartan deprolongations (outlined below) at a generic point to reduce to a distribution satisfying ?? . We thereby justify the title of the paper.

First, for \(n = 3\) and \(n = 4\), all generic germs of \((2, n)\)-distributions are locally equivalent to each other, as established by the classical Darboux and Engel theorems. These distributions are modeled by the Cartan (or “contact”) distributions on the jet spaces \(J^1(\mathbb{R}, \mathbb{R})\) and \(J^2(\mathbb{R}, \mathbb{R})\), respectively, where \(J^k(\mathbb{R}, \mathbb{R})\) denotes the space of \(k\)-jets of functions from \(\mathbb{R}\) to itself. For arbitrary \(n \geq 3\), the Cartan distribution on \(J^{n-2}(\mathbb{R}, \mathbb{R})\) provides a canonical model of a \((2, n)\)-distribution, possessing an infinite-dimensional Lie algebra of symmetries given by the group of contact transformations. It is also worth noting that for \(n \geq 4\), all such distributions have a \(4\)-dimensional cube. This shows that without assumption ?? we may get models with infinite-dimensional symmetries.

On the other hand, even if ?? does not hold at generic points, Theorem 1 is applicable (at a generic point) after a certain reduction procedure (see [1]) also called deprolongation in [1]. Indeed, because \(D\) is bracket generating, its cube has dimension at least \(4\) at generic points. Suppose that \(D\) satisfies \(\dim D^3(q)=4\) on an open set \(M^\circ\) of \(M\). Then the rank \(3\) distribution \(D^2\) on \(M^\circ\) has a one-dimensional characteristic sub-distribution lying in \(D\). At any \(q_0\in M^\circ\), we can consider the quotient \(\Pi:U\to \mathop{\mathrm{depr}}^1_{q_0}U\) of a neighborhood \(U\) of \(q_0\) by the corresponding one-dimensional foliation together with a new bracket generating rank \(2\) distribution \(\mathop{\mathrm{depr}}^1_{q_0}D\) on \(\mathop{\mathrm{depr}}^1_{q_0}U\) obtained by the factorization of \(D^2\). In fact, the germ of \(D\) at \(q_0\) can be uniquely reconstructed from \(\mathop{\mathrm{depr}}_{q_0}^1D\), because it is equivalent to its Cartan prolongation (see [2] for details). Therefore \(\mathop{\mathrm{depr}}^1_{q_0}D\) is called the (first) deprolongation of \(D\) at the point \(q_0\).

In the case that \(\mathop{\mathrm{depr}}_{q_0}^1D\) has cube of rank 4 in a neighborhood of \(\Pi(q_0)\), we can repeat this process at \(\Pi(q_0)\) to obtain \(\mathop{\mathrm{depr}}^1_{\Pi(q_0)}\big(\mathop{\mathrm{depr}}^1_{q_0}D\big)\), which we shall denote by \(\mathop{\mathrm{depr}}^2_{q_0}D\) and call the second deprolongation of \(D\) at \(q_0\). The \(s\)th deprolongation is defined inductively on the condition that \(\mathop{\mathrm{depr}}^{s-1}_{q_0}D\) has cube of rank 4 in a neighborhood of image of \(q_0\) under the corresponding composition of quotients. We denote this distribution by \(\mathop{\mathrm{depr}}^s_{q_0}D\).

At a generic point \(q_0\) of \(M\), there exists \(s\) so that after iterating this procedure \(s\) times, one arrives at one of the following two cases:

1. The \((2,n-s)\) distribution \(\mathop{\mathrm{depr}}^s_{q_0}D\) has \(5\)-dimensional cube in a neighborhood of the image of \(q_0\) under the corresponding composition of quotients (so \(s\leq n-5\)).

2. The \((n-4)\)th deprolongation \(\mathop{\mathrm{depr}}^{n-4}_{q_0}D\) is well defined (and in this case is locally equivalent to the Engel distribution).

We call the \(s\) appearing in case (i) the deprolongation degree of \(D\) at \(q_0\). We also set the deprolongation degree for case (ii) equal to \(n-4\). Note that the deprolongation degree is defined at a generic point of \(M\), but in general, this degree is only locally constant. It is not greater than \(n-5\) in case (i).

In case (ii), the original distribution \(D\) is locally equivalent to a Goursat distribution [10] at \(q_0\). Goursat distributions are defined using the strong derived flag \(\{D^{[j]}\}_{j=1}^\infty\), where instead of 1 we have \[\label{strong95derived} D^{[1]}(q) = D(q)\quad \text{and}\quad D^{[i]}(q) = D^{[i-1]}(q)+[D^{[i-1]},D^{[i-1]}](q)\;\text{for } i>1.\tag{2}\] The distribution \(D\) is called Goursat if \(\dim D^{[i]}(q)=i+1\) for every \(i\geq 1\) and every \(q\in M\). It is very well known [10] that any \((2,n)\) Goursat distribution at a generic point is locally equivalent to the Cartan distribution on \(J^{n-2}(\mathbb{R}, \mathbb{R})\) and therefore the germs at such points have an infinite-dimensional infinitesimal symmetry algebra.

In case (ii), the germ of \(D\) at \(q_0\) is Goursat. In case (i) our original distribution is not Goursat near \(q_0\), and we can apply Theorem 1 or Cartan’s result in [3] to \(\mathop{\mathrm{depr}}_{q_0}^sD\) with \(n\) replaced by \(n-s\), namely:

The last theorem together with the classical theory of Goursat distributions covers in principle (i.e., modulo analysis of the invariants coming from the canonical frame in non-Goursat case) the local geometry of all bracket generating rank 2 distributions in a neighborhood of generic points: In the Goursat case, the germs at generic points are all locally equivalent. In the non-Goursat case, after a suitable number of deprolongations, one can construct a canonical structure of absolute parallelism.

We emphasize that our analysis concerns arbitrary bracket generating rank \(2\) distributions, but only in neighborhoods of generic points, where the notion of genericity is explicitly defined by the assumption that the class of the distribution (or its appropriate deprolongation) at a point is maximal (see Definition 4 below).

We do not address the equivalence problem for germs of distributions at points that do not satisfy this genericity condition (which we refer to as singular points). Even in the case of Goursat distributions, the moduli space of all germs is quite wild and consists of all germs appearing in the so-called Monster Tower (see [10])—even though the generic germ is unique up to local equivalence.

In the non-Goursat case, the situation is much more intricate: even generic germs admit functional invariants coming from the canonical frames. Nonetheless, if these function invariants are nontrivial, the canonical frame constructed at generic points may still provide valuable information about the equivalence of germs at singular points, for instance, by studying how the invariants of the distribution behave as one approaches these singularities.

1.2 The Existence of Corank 1 Abnormal Extremals through Generic Points↩︎

Finally, note that Theorem 2 is of independent interest from the perspective of optimal control theory. On the space of Lipschitz curves that are almost everywhere tangent to a given distribution \(D\), called horizontal curves of \(D\), consider any variational problem that assigns a cost to each such curve—for example, the problem of minimizing length with respect to a sub-Riemannian metric. The Pontryagin Maximum Principle [11], [12] characterizes, through the Hamiltonian formalism, a class of curves, known as Pontryagin extremal trajectories, among which the minimizers of such problems (with fixed endpoints) must lie. Extremal trajectories for which the Lagrange multiplier which multiplies the cost vanishes are called abnormal extremal trajectories; they depend only on the distribution \(D\) and not on the specific cost functional, and they are also called singular curves of the distribution \(D\) [13].

While abnormal extremal trajectories can be described purely geometrically using the canonical symplectic form on the cotangent bundle of \(M\) ([13]–[15], see also subsection 2.1 below), for brevity we use here an equivalent description as critical points of the endpoint map: Given a point \(q_0\) and a time \(T\), denote by \(\Omega_{q_0}(T)\) the set of all horizontal curves of \(D\) starting at \(q_0\) defined on \([0,T]\), and by \(F_{q_0, T} : \Omega_{q_0}(T) \to M\) the endpoint map that takes each \(\gamma \in \Omega_{q_0}(T)\) to the endpoint \(\gamma(T)\). Note that if the set \(\Omega_{q_0}(T)\) has the structure of a \((L^\infty([0,T]))^l\)-manifold, where \(l\) is the rank of \(D\).

Obviously, the corank of abnormal extremal trajectory is at least \(1\). Theorem 2 implies the following theorem:

We conjecture that Theorem 4 (and its stronger version, Theorem 9 below) are valid at any point–that is, the phrase “at a generic point" can be omitted—although this remains beyond our current reach. See Remark 10 below for discussion of this issue.

The paper is organized as follows: In Section 2 we recall the definition of the class of a distribution from [1] which is used in Theorem 2. In Section 3 we proof Theorem 2 and therefore Theorems 1 and 3. Finally, in section 4 we prove a stronger version of Theorem 4, which is labeled Theorem 9.

Acknowledgment We would like to thank Boris Doubrov for several valuable comments, which clarified and simplified some arguments.

2 Rank 2 distributions of Maximal class↩︎

The class of a distribution was defined in [1], [5] in the development of the so-called symplectification procedure. For completeness, we define the notion of class here, reproducing the constructions of our previous work [2], which deviates only modestly from the original construction. This is especially important, because in contrast to [1], [2], [5] , which were focused mainly on the case of maximal class, here we need to consider an arbitrary class.

The symplectification procedure utilizes the natural contact structure on the projectivized cotangent bundle \(\mathbb{P}T^*M\) to construct a so-called “even contact structure” on a submanifold \(\mathcal{M}\subseteq \mathbb{P}T^*M\) of codimension 3. The kernel of this even contact structure is a canonical line distribution \(\mathcal{C}\). Lifting \(D\) to \(\mathbb{P}T^*M\) and osculating with \(\mathcal{C}\) yields a flag at each point of \(\mathbb{P}T^*M\). The class of the distribution will then be defined by considering the growth of the dimensions associated with this flag.

2.1 The Characteristic Line Distribution and Regular Abnormal Extremals↩︎

Let \(D\) be a bracket generating rank 2 distribution on a smooth manifold \(M\) of dimension \(n\geq5\). Define the annihilator of \(D^{\ell}\) \[\big(D^{\ell}\big)^\perp = \big\{ (p,q)\in T^*M : p\cdot v = 0\;\forall\;v\in D^{\ell}(q)\big\}.\]

Consider the fiberwise projectivization \(\mathbb{P}T^*M\) of the cotangent bundle. Since each \((D^{\ell})^\perp\) is a linear subbundle of \(T^*M\), we may define a codimension 3 submanifold \[\mathcal{M} = \mathbb{P}\Big((D^{2})^\perp\setminus (D^{3})^\perp\Big)\subseteq \mathbb{P}T^*M.\] Let \(\mathfrak{s}\) be the tautological (Liouville) one-form on \(T^*M\); explicitly, for coordinates \((q^i)\) on \(M\) with conjugate variables \(p_i\), \(\mathfrak{s}=\sum p_i\text{d}q^i\). Recall that \(d\mathfrak{s}\) is the canonical symplectic form on \(T^*M\). The form \(\mathfrak{s}\) passes to a conformal class \(\overline{\mathfrak{s}}\) of 1-forms on \(\mathbb{P}T^*M\) which defines a contact structure.

Since \(\text{rank}(D^{2})=3\), the submanifold \(\mathcal{M}\) has codimension 3 in the contact manifold \(\mathbb{P}T^*M\). Restricting the contact forms \(\overline{\mathfrak{s}}\) to \(\mathcal{M}\) gives a hyperplane distribution \[\label{evencontact95H} H=\ker\left(\overline{\mathfrak{s}}|_{\mathcal{M}}\right)\tag{3}\] with a conformal class of skew-symmetric forms \(\overline{\sigma}=d\overline{\mathfrak{s}}|_{H}\) well-defined on this hyperplane distribution. Since \(H\) is a hyperplane distribution, it has rank \(2n-5\), so the kernel of the form \(\overline{\sigma}\) must have odd rank. In [1], the authors show that \(\ker(\overline{\sigma})\) has the minimal rank of 1, so that \(\mathcal{M}\) is equipped with a so-called even contact structure. We shall write \(\mathcal{C}\) for the line distribution \(\ker(\overline{\sigma})\), called the characteristic line distribution of \(D\). Following ([14]–[16])

The reason why regular abnormal extremal trajectories are indeed abnormal extremal trajectories in the sense of Definition 1 is explained at the beginning of the proof of Theorem 9 in Section 4. In particular, see relation 22 .

2.2 Maximality of Class↩︎

Let \(\pi:\mathcal{M}\to M\) be the canonical projection. The lift of \(D\) to \(\mathcal{M}\) is denoted by: \[\label{J0} \mathcal{J}(\lambda) = \big\{v\in T_\lambda \mathcal{M} : \pi_*(v)\in D\bigl(\pi(\lambda)\bigr)\big\}\tag{4}\] which is a distribution of rank \(n-2\). Osculating with the characteristic line distribution \(\mathcal{C}\), we obtain from \(\mathcal{J}\) a flag at each point of \(\mathcal{M}\). Write \(\mathcal{J}^{(0)} = \mathcal{J}\) and define recursively \[\label{geod95flag} \mathcal{J}^{(i)}(\lambda) = \mathcal{J}^{(i-1)}(\lambda) + [\mathcal{C}, \mathcal{J}^{(i-1)}](\lambda)\quad \text{for}\;i\geq 1\tag{5}\] In Proposition 1 of the paper [1], the authors show that for each \(0\leq i\) and each \(\lambda\in \mathcal{M}\), we have \[\label{jump} \text{dim}\big(\mathcal{J}^{(i+1)}(\lambda)\big) - \text{dim}\big(\mathcal{J}^{(i)}(\lambda)\big) \leq 1.\tag{6}\] so that \[\label{max95dim95i} \dim \mathcal{J}^{(i)}(\lambda) \leq n-2+i.\tag{7}\] Let \(H\) be as in 3 . Since \(\mathcal{C}\) is a Cauchy characteristic of \(H\), and since \(\mathcal{J}\) is contained within this distribution, so too is each \(\mathcal{J}^{(i)}\). Hence \[\label{max95dim} \dim\mathcal{J}^{(i)}(\lambda)\leq \text{rank}(H)= 2n-5.\tag{8}\] Define the integer-valued functions on \(\mathcal{M}\) and \(M\), respectively: \[\begin{gather} \tag{9} \nu(\lambda) = \min\{i\in \relax \ifmmode \mathbb{N} \else \mathbb{N}\fi: \mathcal{J}^{(i+1)}(\lambda) = \mathcal{J}^{(i)}(\lambda)\} \\ \tag{10} m(q) = \max\{\nu(\lambda): \lambda\in \pi^{-1}(q)\} \end{gather}\] One can show that the set \(\{\lambda\in \pi^{-1}(q): \nu(\lambda)=m(q)\}\) is nonempty and Zariski open in the fiber \(\pi^{-1}(q)\) and that \(\nu(\cdot)\) and \(m(\cdot)\) are lower semicontinuous.

The relation between the class at \(\lambda\) of the regular abnormal extremal passing through \(\lambda\) and the corank of the corresponding abnormal extremal trajectory is given in section 4; see 24 . Note that 7 and 8 imply that \(\nu(\lambda)\leq n-3\), and therefore \(m(q)\leq n-3\). The equality \(\nu(\lambda)=n-3\) holds if and only if \(\mathcal{J}^{(n-3)}(\lambda)=H(\lambda)\).

The following lemma is proven as Remark 3.4 in [4] and will be used in the proof of Theorem 2.

Proposition 3.4 of [4] demonstrates that germs of \((2,n)\) distributions of maximal class are generic. Theorem [max95class95conjecture] is a much stronger statement that we want to prove.

Note that the class of the distribution is constant in a neighborhood of a generic point of \(M\). To prove the first sentence of Theorem 2, it suffices to restrict our considerations to such neighborhoods, as we do in Theorem 7 below. Therefore, instead of introducing special notation for these neighborhoods, we will, from now on, assume that the distribution \(D\) has constant class \(m\) on \(M\). Then the set \[\mathcal{R}_{m}=\{\lambda\in \mathcal{M}: \nu(\lambda)=m\}.\] is open and dense in the space \(\mathcal{M}\).

2.3 Involutivity Conditions↩︎

Now let \(D\) be a rank 2 distribution of constant class \(m\) on an \(n\)-dimensional manifold \(M\), \(n\geq5\). Then by 6 for any \(q\in M\) and any \(\lambda\in \mathcal{R}_{m}\), the flag \[\mathcal{J}(\lambda)\subseteq\mathcal{J}^{(1)}(\lambda)\subseteq \cdots\subseteq \mathcal{J}^{(m)}(\lambda) \subseteq H(\lambda)\subseteq T_\lambda\mathcal{R}_{m}\] has the property that \(\text{rank}(\mathcal{J}^{(i+1)})= \text{rank}(\mathcal{J}^{(i)})+1\) for each \(0\leq i \leq m-1\). Further, from the assumption that the class is equal to \(m\) it follows that \[\label{X95stab}[\mathcal{C},\mathcal{J}^{(m)}]\subseteq \mathcal{J}^{(m)}.\tag{11}\] We can use the conformal class of 2-forms \(\overline{\sigma}\) defined in section 2.1 to continue this flag: for each \(i\geq 1\) and each \(\lambda\in \mathcal{R}_{m}\), define \[\mathcal{J}_{(i)}(\lambda) = \big\{v\in T_\lambda \mathcal{R}_{m} : \overline{\sigma}(v,w) = 0\;\forall\;w\in \mathcal{J}^{(i)}(\lambda)\big\},\] the skew complement of \(\mathcal{J}^{(i)}(\lambda)\) with respect to \(\overline{\sigma}\). From the definition of \(\mathcal{M}\), it follows quickly that \(\overline{\sigma}\left(\mathcal{J},\mathcal{J}\right) = 0\); we obtain a flag: \[\label{calJFlag} \mathcal{C}(\lambda)\subseteq \mathcal{J}_{(m)}(\lambda)\subseteq \cdots \subseteq \mathcal{J}_{(1)}(\lambda)\subseteq \mathcal{J}(\lambda)\subseteq \mathcal{J}^{(1)}(\lambda)\subseteq \cdots \subseteq \mathcal{J}^{(m)}(\lambda)\subseteq H(\lambda)\subseteq T_\lambda\mathcal{R}_{m}.\tag{12}\] By 6 , we have for each \(0 < i \leq m\) that \[\label{flag32dims} \dim\big(\mathcal{J}_{(i)}(\lambda)\big) = n-2-i\quad\text{and}\quad \dim\big(\mathcal{J}^{(i)}(\lambda)\big) = n-2+i.\tag{13}\]

At each \(\lambda\in \mathcal{R}_{m}\), one can show that \[\mathcal{J}^{(1)}(\lambda) = \big\{v\in T_\lambda \mathcal{M} : \pi_*(v)\in D^{2}(\lambda)\big\}\] which implies (with some computation) that \(\mathcal{J}_{(1)}(\lambda) = \ker(T_\lambda\pi)\oplus \mathcal{C}(\lambda)\). Define \(V_1(\lambda) = \ker(T_\lambda\pi)\), the vertical subspace over \(\lambda\). For \(i=0\) and for \(2\leq i\leq m,\) define \[\label{Splitting} V_i(\lambda) = \mathcal{J}_{(i)}(\lambda)\cap V_1(\lambda),\tag{14}\] the vertical component of the \((n-2-i)\)-dimensional piece of the flag at \(\lambda\). From 12 it is clear that \[\label{vert95inclusion} V_{i}(\lambda)\subset V_{i-1}(\lambda), \quad i\geq 1.\tag{15}\] Further, observe that \(V_0(\lambda)=V_1(\lambda)\). From 12 , one can also observe that for each \(1\leq i\leq m\) \[\label{J95i326132C324332V95i} \mathcal{J}_{(i)}(\lambda) = V_i(\lambda)\oplus \mathcal{C}(\lambda),\tag{16}\] The \(V_i\) also satisfy involutivity conditions; in the following lemma, which is Lemma 2 in [1], recall that \(V_0=V_1\).

Although the statement is only proven for distributions of maximal class in [1], the proof does not rely on maximality of class. In order to prove the main theorem, we shall need one more fact about the flag 12 .

The following lemma is a small extension of Remark 2 of [1]. In that remark, only one inclusion of the equality [lemma32Alt32J95i] is demonstrated. Later in that paper, the equality is shown assuming maximality of class. We provide a detailed proof here because the lemma is crucial in the proof of Theorem 2.

3 Proof of the Theorem 2↩︎

As was noted at the end of section 2.2, the class of a rank 2 distribution is constant in a neighborhood of a generic point of the base manifold. Therefore to prove the first sentence of Theorem 2, it suffices to prove the following

Now prove the second sentence of Theorem 2. Let \(q_0\) be a point where \(D\) is equiregular and assume that \(\mu\) is the minimal integer such that \(D^\mu(q_0)=T_{q_0}M\). Denote \[\label{Tanaka95comp} \mathfrak{g}_{-1}(q_0):=D(q_0), \quad \mathfrak{g}_{-i}(q_0) : =D^{i}(q_0)/D^{i-1}(q_0), \,\, \forall 1\leq j\leq \mu,\tag{21}\] then the graded space \(\mathfrak{m}(q_0)= \displaystyle{\bigoplus_{j=-\mu}^{-1}\mathfrak{g}_j(q_0)}\), associated with the filtration 1 , is endowed with the structure of the graded nilpotent Lie algebra, called the Tanaka symbol of the distribution \(D\) at the point \(q_0\). The flat distribution \(D_{\mathfrak m(q_0)}\) of constant symbol (or type) \(\mathfrak m(q_0)\) is defined to be the left-invariant distribution corresponding to the \((-1)\)-graded component \(\mathfrak g_{-1}(q_0)\) on the simply connected Lie group with Lie algebra \(\mathfrak{m}(q_0)\). Since the flat distribution \(D_{\mathfrak m(q_0)}\) is left-invariant, its germs at different points are equivalent ⁷, and therefore \(D_{\mathfrak m(q_0)}\) has constant class. Therefore, by Theorem 7 it is of maximal class at every point. Then by [6], the original distribution \(D\) at \(q_0\) is of maximal class.

4 Proof of Theorem 4↩︎

We are going to prove the following stronger theorem:

References↩︎