数理解析研究所講究録 1374 巻 2004 年 114-125

114

On various kinds of the Gray-type Theorem
北海道大・理学研究科, 足立二郎 (Jiro ADACHI)

Department of Mathematics,
Hokkaido University

0

Introduction

The goals of this short article are to introduce various kinds of tangent
distributions on manifolds, for those Gray type theorems hold. Especially
we introduce a recent result by the author ([A1]) on a generalization of
the Gray theorem.
Global stabilities of various kinds of distributions on manifolds areimportant and interesting issues. For example, the well-known Gray thecontact structure, throughorem (see [Gr]) states that a deformation of aacontact
family ofcompact manifold is represented by aafamily
contact structures, on aacompact
global diffeomorphisms. We observe in this article some theorems of thistype for various kinds of distributions. We consider sufficient conditions
under which such distributions are globally stable, that is, the deformations can be represented by families of global diffeomorphisms of theunderlying manifolds.
We observe the following results concerning the stability of tangent
A tangent distribution (or distribution for short) $D$ of rank
distributions. Atangent
a distribution of fc-dimensionalon an -dimensional manifold $M$ is adistribution
stricttangent space at each point $x\in M,$ in aastrict
subspacesof aatangent
subbundle of the tangent bundle.
sense. In other words, it is aasubbundle
In Section 2, we review the original Gray theorem ([Gr]). It is a global
stability theorem where the deformed distributions are contact structures.
$k$

$n$

$D_{x}\subset T_{x}M$

$\ln$

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A contact structure is a distribution of corank one on an odd-dimensional
manifold which is completely non-integrable. The Gray Theorem claims
that deformations of a contact structure on a compact manifolds are
represented by global isotopies (see [Gr]). There is no obstruction for
representing a family of contact structures with a family of global diffeomorphisms. In the following, we introduce some studies of this typefor some distributions on manifolds, which observe obstructions for the
representation by global isotopies.
In Section 3 we introduce results due to R. Montgomery and M. ZhitOin [MZh]. They studied in [MZh] Goursat flags. A Goursat flag is
a sequence of derived distributions rank of each of those is different by one
from the next one (see Section 1 for precise definition). In subsection 3.1,
we observe the case of distributions of corank one. A generalization of
the Gray theorem is obtained here. A notion of Cauchy characteristic
distribution (see Section 1 for definition) plays an important role. It is
proved that a deformation of such a distribution preserving the Cauchy
characteristic distribution is represented by a family of global diffeomorphisms. In subsection 3.2, we observe a Gray type theorem for Goursat
flags. It is proved that a deformation of Goursat flag preserving the
Cauchy characteristic distribution of the derived distribution of corankone is represented by a family of global diffeomorphisms. A Goursat flagof length 2 is, especially, called an Engel structure. In other words, anEngel structure is a distribution of rank 2 on a 4-dimensional manifold
which is maximally non-integrable. Engel structures had been studied by
F. Engel, E. Goursat, E. Cartan, and many other mathematicians for along time. R. Montgomery and A. Golubev proved that Engel structures
are globally stable under some condition about a certain line field (see[Mo], [Go] . A result due to R. Montgomery and M.
can beconsidered as an extension of this result.
In Section 4, we introduce a result in [A1] about global stability of
distributions of higher coranks of derived length one. A subdistribution$K(D)\subset D$ of a distribution $D$ is defined here. It is known that distri$\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{k}\mathrm{i}\dot{1}$$)$$\mathrm{Z}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{k}\mathrm{i}\dot{1}$18butions with integrable $K(D)$ whose derived distribution is the tangent
bundle has unique local normal form (see [KR]). It is proved that a
deformation of such a distribution preserving the subdistribution $K$ isrepresented by a family of global diffeomorphisms.
In Section 5 we introduce results due to B. Jakubczyk and M. Zhitin [JZh3]. They studied global stability of Pfaffian equations.
Pfaffian equations and tangent distributions of corank one have been studstudied their local
ied for a long time. J. Martinet and M.
normal forms (see [Ma], [Zh2]). Recently, B. Jakubczyk and M. ZhitOobtained some results on the classification of Pfaffian equations(see [JZhl], [JZh2], [JZh3]). In the case of Pfaffian equations on odddimensional manifolds, non-contact loci and certain characteristic line
fields on the loci played an important role for the classification. And,in the case of Pfaffian equations on even-dimensional manifolds, certain
characteristic line fields played an important role for the classification.
They obtained a sufficient condition for global stability in terms of above
notions.
In Section 6, we introduce examples of distributions of rank 2 on 4manifolds with non-Engel locus. M. Zhitomirskiiobtained them in [Zhl]
as normal forms. We mention about results, in the forthcoming paper[A2], about global stability of such distributions. Non-Engel loci andcharacteristic line fields play an important role.
$\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{k}\mathrm{i}\dot{1}$$\mathrm{Z}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{k}\mathrm{i}\dot{1}$$\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{k}\mathrm{i}\dot{1}$1$\mathrm{P}\mathrm{r}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{r}\dot{\mathrm{l}}\mathrm{e}\mathrm{s}$In this section, we define some basic notions needed in the following
sections. Some notions needed in one of the sections is defined in each
, $i=1,2$ , . . . , , of a distributionsection. Strong derived distr butions$D$ aredefined pointwise inductively as follows,$D^{i}$$\{\begin{array}{l}D^{1}=DD^{i+1}=D^{i}+[D^{i},D^{i}]\end{array}$

$k$

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Note that it is defined pointwise in terms of sheaves of vector fields which,
are cross-section of the subbundle $D\subset TM.$ Therefore ranks of$p\in M,$ might be different for each point $p\in M.$ We say a distribution $D$
to be Lie square regular if all
are distributions of constant ranks. The
Cauchy characteristic distribution $L(D)$ of a distribution $D$ is defined
pointwise as follows,$D_{p}^{i}$$D^{i}$$L(D)_{p}=$$=${ $X\in D_{p}|[X,$ckv{$Y]\in D_{p}$$X\in D_{p}|\mathrm{X}_{\lrcorner}$, for any $Y\in D_{p}$ },$|_{D_{p}}=0,$

for any$\omega$$\in$$\mathrm{S}(\mathrm{D})$}.

When $L(D)$ is a distribution of constant rank, the distribution $L(D)$is integrable according to the Probenius theorem. Note that $L(D)$ is a
distribution of rank 0 when is a contact structure.
$D$

2

The original Gray theorem

In this section, we introduce the well-known Gray theorem proved in [Gr].
It is one of the most important Theorem for contact topology.

Theorem 2.1 (Gray). Let
be a family of contact structures oncompact orientable manifold M. Then there exists a family: $Marrow M$.
id andof global diffeomorphisms which satisfies$D_{t}$$a$$\Phi_{t}$$\varphi_{0}=$3$(\varphi_{t})_{*}D_{0}=D_{t}$Lie square regular distributions and Goursat flags

In this section, we introduce some results due to R. Montgomery and
M. Ya.
. They studied Goursat flags from the ge0metric view point. The first step of their inductive proof is a generalization of the Gray theorem.
$\mathrm{Z}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{k}\mathrm{i}_{\dot{1}}([\mathrm{M}\mathrm{Z}\mathrm{h}])$3.1

Lie square regular distributions of corank one.

The following is proved in [MZh].

118

. Let

, t $\in[0,$ 1], be
one-parameter family of distributions of corank k $=1$ on a compact
has the Cauchy characteristic
entable manifold M. It is assumed that
distribution $L(D_{t})\equiv L$ of constant rank for any $t\in[0,1]$ . Then, there
exists a family of global diffeomorphisms : $M$ $arrow M_{i}t\in[0,1]$ , which
id andfor any $t\in[0,1]$ .
satisfies

Theorem 3.1$(\mathrm{M}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{g}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{y}- \mathrm{Z}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{k}\mathrm{i}_{\dot{1}})$$D_{t}$$a$$or\dot{\mathrm{v}}-$$D_{t}$$\varphi_{t}$$(\varphi_{t})_{*}D_{0}=D_{t}$$\varphi_{0}=$Distributions appeared in the theorem above is Lie square regular. TheOrem 3.1 is a generalization of the Gray theorem 2.1 since rank $L(D_{t})=0$
if
are contact structures.
$D_{t}$

3.2

Goursat flags.

They also proved a Gray type theorem for Goursat flags. A Goursat flagof length on a manifold $M$ is a sequence
$s$

(F) :

$D_{s}\subset D_{s-1}\subset$. . $\subset D_{1}\subset D_{0}=TM$ , $s\geq 2$of distributions on $M$ which satisfies the following conditions:$\{\begin{array}{l}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}D_{i}=i,i=1,2,\ldots,sD_{i-1}=D_{i}^{2}=D_{i}+[D_{i},D_{i}],i=1,2,\ldots,s\end{array}$

The following is proved in [MZh].
Theorem 3.2. Let (F) :$(\mathrm{M}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{g}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{r}\mathrm{y}- \mathrm{Z}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{k}\mathrm{i}_{\dot{1}})$$D_{s,t}\subset$,.

$\subset$

be a family of Goursat flags of Length on a compact
orientable manifold M. Suppose that $L(D_{1,t})\equiv L(D_{1,0}$ for any $t\in[0,1]$ .
Then there exists a family $\varphi:Marrow M$ of diffeomorphisms which satisfies.
id and$D_{1,t}\subset TM,$ $t\in[0,1]$$\varphi 0=$$s$$(\varphi_{t})_{*}(F_{t})=(F_{0})$A distribution of rank 2 on a 4-manifold which construct a Goursat
ucture. It is known that Engelflag of length 2 is called an Engel
structures have a unique local normal form. A Gray type theorem for$st$Engel structure is studied independently (see [Go], [Mo]).

113

be a family of Engel
Theorem 3.3 (Golubev, Montgomery). Let
structures on a compact orientable 4-manifold M. Suppose thatfor any $t\in[0,1]$ . Then there exists a family $\varphi:Marrow M$ ofdiffeomorphisms which satisfies.
id and$D_{t}$$L(D_{t}^{2})\equiv$$L(D_{0}^{2})$$?\mathit{0}$$(\varphi_{t})_{*}(D_{t})=(D_{0})$$=$Note that Theorem 3.2 can be considered as an extension of Theorem 3.3.

Lie square regular distributions of higher coranks

4

In this section, we introduce a result obtained in [A1], It is a Gray typetheorem for Lie square regular distributions those coranks are greaterthan one.
First of all, we define a certain subdistribution $K(D)$ of a distribution$D$ .
It is defined in terms of the Pfaffian system $S(D)$ . We define a
covariant system associated to a Pfaffian system $S\subset T^{*}M$ according toA. Kumpera and J. L. Rubin (see [KR]), as follows. The bundle map$arrow\wedge^{2}(T^{*}M/S)$(mod )
defined on local sections of as
is called the Martinet structure tensor (see [Ma]). We define the polar
space
of at $p\in M$ as$\delta:S$$S$$\mathrm{P}\mathrm{o}1(S)_{p}$$S$$\delta(\omega)=d\omega$$S$$\mathrm{P}\mathrm{o}1(S)_{p}:=${$w\in T_{p}^{*}M/S_{p}|w\Lambda\delta(\omega)=0,$

for any$\omega$$\in S$}

When the polar space
has a constant rank on $M$ , we define the(Pol(S)), where $q:T^{*}Marrow$covariant system associated to as$T^{*}M/S$ is the quotient map.
For a distribution $D\subset TM,$ let $K(D)$denote the subdistribution of $D$ which is annihilated by the covariant
systemassociated to the Pfaffian system $S(D)$ .
$\mathrm{P}\mathrm{o}1(S)_{p}$$\hat{S}$$S$$\hat{S}:=q^{-1}$$\hat{S}(D)$

Example 4.1. We give an example of the polar space and the covariant
system for the standard distribution on. Let $D_{0}=\{\omega_{1}=$,,with}, where $\omega_{i}:=dx_{2i-1}+x_{2i}$ dt, be a distribution on
the standard coordinates. When $k>1,$ the distribution of$7^{1}(1, k)\cong \mathbb{R}^{2k+1}$$0$$\ldots$$\mathbb{R}^{2k+1}$$\omega_{k}=0$$(x_{1}, \ldots, \mathrm{x}2\mathrm{k}, t)$120polar spaces of $S(D_{0})$ is obtained as follows:$\mathrm{P}\mathrm{o}1(S(D_{0}))=\{w\in$

T*M/S $(D_{0})|$$w\Lambda dx_{2i}\Lambda dt\equiv 0$$=\{dt\}$$(\mathrm{m}\mathrm{o}\mathrm{d} S(D_{0}))$, $i=1,2$ ,$\ldots$

,

$k$

}

.

Then the covariant system is obtained as follows:$\hat{S}(D_{0})=\{\omega_{1}, .

, \omega_{k}, dt\}=\{dx_{1}, dx_{3}, \ldots, dx_{2k-1}, dt\}$.

. They are clearly
Then we haveintegrable. When $k=1$ , $D_{0}=\{dx_{1}-x_{2}dt=0\}$ is the standard con, $5(D_{0})$. Then we havetact structure on$\{dx_{1}, dx_{2}, dt\}$ , and $K(D_{0})=\langle$ 0 .
$K(D_{0})=\langle\partial/\partial x_{2}, \partial/\partial x_{4}, \ldots, \partial/\partial x_{2k}\rangle$$\mathrm{P}\mathrm{o}1(S(D_{0}))=\{dx_{2}, dt\}$$\mathbb{R}^{3}$$=$$\rangle$

The main theorem in [A1] is the following.
$[0, 1]f$ be
one-parameter family of distribuTheorem 4.2. Let ,tions of corank $k\geq 1$ on a compact orientable manifold M. Suppose, for
any $t\in[0,1]$ :$D_{t}$$t\in$$a$(1) the first derived distributions coincide with the tangent bundle of $M$ :$D_{t}^{2}=TM,$(2) there exists a constant integrable subdistribution $K\subset D_{t}$ of corank

one.
Then, there exists a family : $Marrow M$ , $t\in[0,1]f$ of global diffeomorphisms which satisfies
id andfor any $t\in[0,1]$ .
$\varphi_{t}$$\varphi_{0}=$$(\varphi_{t})_{*}D_{0}=D_{t}$Note that Theorem 4.2 is obtained from Theorem 3.1 when $k=1$ since$L(D_{t})=K(D_{t})$ then and they are integrable from the definition of the
Cauchy characteristic distribution.
It is also proved in [KR] that such distributions as in Theorem 4.2 have
have a unique normal form as in Example 4.1.

121

Proposition 4.3 (Kumpera-Rubin). Let $D$ be a distribution of corank$k>1$ on a manifold $M$ whose derived distribution coincides with thetangent bundle: $D^{2}=TM.$ If the distribution $K(D)$ is integrable, thenat each point $p\in M$ the distribution $D$ admits the following local normal$fom$ : $D=\{\omega_{1}=0, \ldots, \omega_{k}=0\}$ ,$\omega_{1}=dx_{1}+X$$i$,w2 $=dx_{3}+$ x4dt,, $\omega_{k}=dx_{2k-1}+x_{2k}$ dt,

where the coordinates Xi, t vanish at p $\in M$

5

Distributions of corank 1 with singularities

In this section, we introduce some results obtained by B. Jakubczyk andM.
in [JZh3]. They studied in [JZh3] global stability of distributions with degeneracy loci. The first typical example is the Martinet
normal form.
$\mathrm{Z}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{k}\mathrm{i}\dot{1}$

Example 5.1 (Martinet). Set $D=\{\alpha=dz-y^{2}dx=0\}$ onwithcoordinates $(x, y, z)$ . Then, we obtain the non-contact locus I as follows:$\mathbb{R}^{3}$

I

$=$

{

$p\in \mathbb{R}^{3}|$(cr $\Lambda d\alpha)_{p}=(2ydx\Lambda dy\Lambda dz)_{p}=0$ } $=\{y=0\}$ .

case $\dim M=$
The results in [JZh3] is mentioned in two cases:$2k$ , and (2) the case $\dim M--2k+1,$
where $M$ is an underlying manifold.
Case(l) Let $P=(\omega)$ be a Pfaffian equation on a manifold $M$ of dimension $n=2k,$ and a generator of . A characteristic vector field $X$
is a volume, whereof $P$ is defined the relation
form. The line field $L(P)$ generated by a characteristic vector field $X$ iscalled the character istic line field of $P$ . Let Sing(L) denote the set of
singular points of $L(P)$ . We introduce an important notion concerning
singular points of $L(P)$ . Let $I_{p}(X)$ be an ideal in the ring of function
germs at $p\in$ Sing(L), generated by the coefficients , . . ’ of a characteristic vector field $X$ , with respect to some coordinate system around. Set. Then the following condition is introduced$(1)\mathrm{t}\mathrm{h}\mathrm{e}$$\omega$$P$$X_{\lrcorner}\omega=\omega\Lambda(d\omega)^{k-1}$$\Omega$$a_{1}$$p$$d_{p}(P):=\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}I_{p}(X)$$a_{n}$

122

in [JZh3] :(A)$d_{p}(P)\geq 3$

for any point $p\in$ Sing(L).

It is proved in [JZh3] that this condition is a genericity condition. The
statement of the result is the following:. Let, $t\in[0,1]$ , be a fam$ily$of Pfaffian equations on $M_{f}^{2k}k\geq 2_{f}$ which satisfies the following
conditions:

Theorem 5.2$(\mathrm{J}\mathrm{a}\mathrm{k}\mathrm{u}\mathrm{b}\mathrm{c}\mathrm{z}\mathrm{y}\mathrm{k}-\mathrm{Z}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{k}\mathrm{i}_{\dot{1}})$$P_{t}$(1) all$P_{t}$

define the common characteristic line field $L=L(Pt)_{f}$(2) all$P_{t}$

satisfy condition (A).

Then, there exists a familyto .

$I)_{t}$: $Marrow M$ of diffeomorphisms sending$P_{t}$$P_{0}$

Case(2) As we observe in Example 5.1, there is a non-contact locus called
the Martinet hypersurface: $S=\{p\in M|(\omega\Lambda(d\omega)^{k})_{p}=0\}$ . In a similar
way to Case(l) above, the characteristic line field $L(P)$ on the Martinet
hypersurface , and the depth $d_{p}(P)$ at singular point $p\in$ Sing(L) aredefined. In this case we need further condition concerning the Martinet, wherehypersurface. Let $H$ be a function defined as $H=$
as itsis a volume form. $H$ determines the Martinet hypersurface
zero level. The ideal (H) is called the Martinet ideal The Martinet ideal(H) is said to have the property of zeros if the ideal generated by thegermof $H$ at $p\in S=\{H=0\}$ in the ring of all function germs
at coincides with the ideal consisting of function germs vanishing on
the germ at of in the same ring, for any $p\in S.$ We need one morebe the Frech\’e space of smooth functions on $M$ ,condition. Let
and $C^{\infty}(M, 5)$ its closed subspace consists of functions which vanish on .
Set $C^{\infty}(S)=C^{\infty}(M)/C^{\infty}(M, \mathrm{S})$ . The Martinet hypersurface is said
to have the extension property if there exists a continuous linear operator$\lambda:C^{\infty}(S)arrow C^{\infty}(M)$ which satisfy $\lambda(f)|s=f$ for all $f\in C^{\infty}(S)$ .
The
statement of the result is the following:$S$$\mathrm{i}$$\Lambda(d\omega)^{k}/D$$S$$\Omega$$H_{p}$

$p$

$p$

$S$

$C^{\infty}(M)$

$S$

$S$

123

$[0, 1]$ , be a family
Theorem 5.3 (Jakubczyk-Zhitomirskii). Let ,of Pfaffian equations on a compact orientable manifold $M^{2k+1}$ , $k\geq 1,$
which satisfies the following conditions:$P_{t}$$t\in$(1) all 6 have the common Martinet hypersurface , which has the extension property, and the Martinet ideals have the property of zeros,$S$(2) all$P_{t}$

define the common characteristic line field $L=L(P_{t})$ ,(3) all$P_{t}$

satisfy condition (A).

Then, there exists a familyto .

$I)_{t}$: $Marrow M$ of diffeomorphisms sending$P_{t}$$P_{0}$

6

Distributions of corank 2 on 4-manifold with nonEngel locus

In this section, we consider distributions of rank 2 on 4-manifold with
non-Engel loci. First, we define the notion of non-Engel loci. Let $D$ be
an Engel structure. Recall that the derived distributions ,may not
be distributions in a strict sense. In fact, they may have a point where
the rank of distribution degenerates. We set$D^{2}$$\Sigma_{1}(D):=${ $\in M|$ rank$p$

$D_{p}^{2}<3$},$\Sigma_{2}(D):=$$D^{3}${$p\in M|$ rank

$D_{p}^{3}<4$},

and call them the first and the second non-Engel loci of $D$ respectively.
We call the union
S2(D) $=:\Sigma(D)$ just the non-Engel locus of $D$ .
$\Sigma_{1}(D)\cup$

Example 6.1. We observe normal forms obtained by M.
in[Zhl]. We regard them distributions onwith coordinates $(x, y, z, w)$ .
(1) $D=\{\omega_{1}=dx+z^{2}dwz-0, \omega_{2}=dy+zwdw =0\}$$\mathrm{Z}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{k}\mathrm{i}\dot{1}$$\mathbb{R}^{4}$$=\langle\partial/\partial w-z^{2}\partial/\partial x-zw\partial/\partial y, \partial/\partial z\rangle$

In this case,$D^{2}= \langle\frac{\partial}{\partial w}-z^{2}\frac{\partial}{\partial x}-zw\frac{\partial}{\partial y},\frac{\partial}{\partial z},2z\frac{\partial}{\partial x}+w\frac{\partial}{\partial y}\rangle)$$D^{3}= \langle\frac{\partial}{\partial w}-z^{2}\frac{\partial}{\partial x}-zw\frac{\partial}{\partial y},\frac{\partial}{\partial z},2z\frac{\partial}{\partial x}+w\frac{\partial}{\partial y},\frac{\partial}{\partial x},\frac{\partial}{\partial y}\rangle$

124

Therefore,$\Sigma_{1}=\{\dim D_{p}^{2}<3\}=\{z=0, w=0\}$,$\Sigma_{2}=\{\dim D_{p}^{3}<4\}=\emptyset$.

(2) $D=\{\omega_{1}=dx+zdw=0, \omega_{2}=dy+z^{2}wdw=0\}$$=\langle\partial/\partial w-z\partial/\partial x-z^{2}w\partial/\partial y, \partial/\partial z\rangle$

In this case,$D^{2}= \langle\frac{\partial}{\partial w}-z\frac{\partial}{\partial x}-z^{2}w\frac{\partial}{\partial y}$,$\frac{\partial}{\partial z})\frac{\partial}{\partial x}+2zw\frac{\partial}{\partial y}\rangle$$D^{3}=w}-z\frac{\partial}{\partialx}-z^{2}w\frac{\partial}{\partialy}$y}$$D^{3}= \langle\frac{\partial}{\partial\langle\frac{\partial}{\partialw}-z\frac{\partial}{\partialx}-h\frac{\partial}{\partial,$\frac{\partial}{\partial z}$,$\frac{\partial}{\partial x}$$\frac{\partial}{\partial x}+2zw\frac{\partial}{\partial y}$,,$2w \frac{\partial}{\partial y})2z\frac{\partial}{\partial y}\rangle$

Therefore,$\Sigma_{1}=\{\dim D_{p}^{2}<3\}=\emptyset$,$\Sigma_{2}=\{\dim D_{p}^{3}<4\}=\{w=0, z=0\}$.

In a similar way to Section 5, a characteristic line field for the derived
can be defined. Non-Engel loci and the characteristic line
distribution
fields play an important role in the arguments on Gray type theorem for
distributions with non-Engel loci, in the forthcoming paper [A2].
$D^{2}$

References[A1]

J. Adachi, Global stability of distributions of higher corank of derived length one,
. Not. 2003 no. 49, 2621-2638.
Int. Math.
${\rm Res}$[A2]

J. Adachi, Deformation of tangent distributions of rank 2 on 4-dimensional rnanifolds, (preprint).

[Go]-plane fields
A. Golubev, On the global stability of maximally nonholonomic
11,523-529.
. Notices 1997 no.
in four dimensions, Internat. Math.
$rwo$${\rm Res}$[Gr]

J. W. Gray, Some global properties of contact structures, Ann. of Math. (2) 69(1959) 421-450.

[JZhl] B. Jakubczyk and M. Zhitomirskii, Odd-dimensional Pfaffian equations: reduction to the hypersurface of singular points, C. R. Acad. Sci. Paris Sr. I Math. 325(1997), no. 4, 423-428.

125[JZh2] B. Jakubczyk and M. Zhitomirskii, Local reduction theorems and invariants for
singular contact structures, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 1, 237-

295.
[JZh3] B. Jakubczyk and M. Zhitomirskii, Distributions of corank 7 and their charactenstic vector fields, Hans. Amer. Math. Soc. 355 (2003), 2857-2883.

[KMS] M. Kazarian, R. Montgomery and B. Shapiro, Characteristic classes for thedegenerations of twO-plane fields in four dimensions, Pacific J. Math. 179 (1997),
no. 2, 355-370.

[KR]

A. Kumpera, and J. L. Rubin, Multi-flag systems and ordinary differential equations, Nagoya Math. J. 166 (2002), 1-27.

[Ma]

J. Martinet, Sur les singularitis des formes diffirentielles, Ann. Inst. Fourier(Grenoble) 20 (1970), 95-178.

[Mo]

R. Montgomery, Engel defomations and contact structures, Northern California
Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2, 196, 103-117,
Amer. Math. Soc, Providence, RI, 1999.

[MZh] R. Montgomery and M. Ya. Zhitomirskii, Geometric approach to Goursat flags,Ann. Inst. H. Poincare’ Anal. Non Lin\’eaire 18 (2001), no. 4, 459-493.

[Zhl] M. Ya.
, Normal forms of germs of-dimensional distributions on, Funct. Anal. Appl. 24 (1990), no. 2, 150-152.
$\mathrm{Z}\mathrm{h}\mathrm{i}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{k}\mathrm{i}_{\dot{1}}$$rwo$$\mathbb{R}^{4}$[Zh2] M. Ya. Zhitomirskii, Typical singularities of differential 1-forms and Pfaffian
equations, Transl. Math. Monogr., 113 Amer. Math. Soc, Providence RI, 1992.
Department of Mathematics,
Hokkaido University,
Sapporo 060-0810 Japan.
$\mathrm{e}$-mail : j-adachi@math.sci.hokudai.ac.jp