数理解析研究所講究録
第 1664 巻 2009 年 46-55

46

An equivalence problem for second order PDE and double
fibration as a flat model space名古屋大学大学院多元数理科学研究科 野田尚廣 (Takahiro Noda)
Department of Mathematics, Nagoya University

1. INTRODUCTION
The purpose of the note is to announce my recent result. The detail will be givenin the forthcoming paper [N].
We will study an equivalence problem of second order PDE for one unknown functionof two variables. The equivalence problem for differential equations is simply explained
as follows. We fix classes of differential equations and a group of coordinate transformations. Then, we consider a problem how differential equations change under coordinatetransformations. We can also express this problem in terms of group actions. Let be acoordinate transformation group and $X$ be a set of certain differential equations. Thenthe equivalence problem for differential equations in $X$ is interpreted as the problem ofdetermining the orbit decomposition with respect to the action of on $X$ .
The equivalence problem is studied deeply by Sophus Lie and \’Elie Cartan, and manyother authors. (See [M] for a detailed history of the equivalence problem.)
We consider an equivalence problem of second order PDE for one unknown functionof two variables $y=y(x_{1}, x_{2})$ :$\mathcal{G}$$\mathcal{G}$(1)$\frac{\partial^{2}y}{\partial x_{i}\partial x_{j}}=f_{ij}(x_{1}, x_{2}, y, z_{1}, z_{2})$,where, $f_{ij}(1\leq i, j\leq 2)$ satisfying $f_{ij}=f_{ji}$ are$:=$functions on$\{(x_{1}, x_{2}, y, z_{1}, z_{2})\}$ , and $z_{1}=y_{x}1’ z_{2}=y_{x}2^{\cdot}$ If
all vanish, (1) is called the flat equation.
For this PDE, we take the group ScaleDiffof lifts of scale transformations on
as a transformation group , where scale transformations are defined by:$C^{\infty}$$J^{1}(\mathbb{R}^{2}, \mathbb{R})$$f_{ij}$$(\mathbb{R}^{3})^{cont}$$\mathbb{R}^{3}$$\mathcal{G}$$\phi(x_{1}, x_{2}, y)=(X_{1}(x_{1}), X_{2}(x_{2}), Y(x_{1}, x_{2}, y))$.

Since is the transformation on, we can characterize this one geometricallyas follows. Scale transformations preserve not only fibers on $J^{0}(R^{2}, R)$ , but also the$\phi$$J^{0}(\mathbb{R}^{2},\mathbb{R})$web-structure on the base spaceconstructed by parallel translation of -axis and-axis. For the equivalence problem, we can not apply the Tanaka theory, because
symmetry group is not semi-simple group. Thus, it is necessary to use the Cartan’sclassical method. We will calculate explicitly the curvatures for this equivalence problem
by using Cartan’s equivalence method. We obtain the necessary and sufficient condition
when the second order PDE satisfying integrability condition is equivalent to the flat
equation via a vanishing condition of these curvatures (Theorem 2.3).
We also discuss a duality associated with differential equations via double fibrations.
In particular, we consider a duality between the coordinate space and the solution spaceof the flat equation. Double fibrations play an important role for a study of this duality. Moreover, these fibrations are usually described via some transformation groups
appeared in equivalence problems. For the group ScaleDi, we can not obtaina fibration of compact type, because the group ScaleDi
is too small. Hence,$\mathbb{R}^{2}$$x_{1}$$x_{2}$$ff(\mathbb{R}^{3})^{cont}$$ff(\mathbb{R}^{3})^{cont}$

47

it is natural to consider an existence problem of groups from which double fibration ofcompact-type is obtained as a flat model space. For this problem, we find a non-trivialgroup which gives a fibration of compact-type (Theorem 3.4).
Acknowledgement. The author is supported by Research Fellowships of the Japan
Society for the Promotion of Science for Young Scientists.
2. EQUIVALENCE PROBLEM AND LOCAL INVARIANTIn this section, we introduce an equivalence problem and explain the G-structureassociated with this problem. For this purpose, we prepare some terminology and notation.
For functions of two variables $y=y(x_{1}, x_{2})$ , we consider the second order PDE (1)and diffeomorphisms onof the form$\phi$$\mathbb{R}^{3}$$\phi(x_{1}, x_{2}, y)=(X_{1}(x_{1}), X_{2}(x_{2}), Y(x_{1}, x_{2}, y))$.

of this form is called a scale transformation. A scale transformation
naturally to a contact diffeomorphism ofdefined by:
The map$\phi$$\hat{\phi}$$Z_{1}= \frac{Y_{x_{1}}+Y_{v}z1}{(X_{1})_{x_{1}}},$$Z_{2}= \frac{Y_{x}2+Y_{v^{z}2}}{(X_{2})_{x_{2}}}$$ff(\mathbb{R}^{3})$$:=$$\mathbb{R}^{3}\}$$\{$$ff(\mathbb{R}^{3})^{cont}$$\{$$ff(\mathbb{R}^{3})^{cont}$

$\{$

,

. We introduce the following terminology:,
Scale transformation on$:=$
The lift of Di
to$:=$ScaleDiThe lift of $ScaleDiff(\mathbb{R}^{3})$ to$X:=$ {second order PDE (1)}.
The main problem in the present note is the following.

ScaleDiDi

lifts$J^{1}(\mathbb{R}^{2}, \mathbb{R})$$\hat{\phi}(x_{1}, x_{2}, y, z_{1}, z_{2})=(X_{1}(x_{1}), X_{2}(x_{2}), Y(x_{1}, x_{2}, y), Z_{1}, Z_{2})$where,$\phi$$ff(\mathbb{R}^{3})$$J^{1}(\mathbb{R}^{2}, \mathbb{R})\}$,$J^{1}(\mathbb{R}^{2},\mathbb{R})\}$

Problem 2.1. Examine the orbit decomposition under the action of
on $X$ .

,$ScaleDiff(\mathbb{R}^{3})^{cont}$

In order to resolve the above problem, we use a G-structure associated with theequation (1). First, we replace from data of second order PDE (1) to data of differential
system ([G], [Ol]). We choose the following coframe ofcorresponding to theequation (1),$J^{1}(\mathbb{R}^{2},\mathbb{R})$$\underline{\theta}_{0}:=dy-z_{1}dx_{1}-z_{2}dx_{2}$,$\underline{\theta}_{1}:=dz_{1}-f_{11}dx_{1}-fi_{2}dx_{2}$(2)$\underline{\theta}_{2}:=dz_{2}-f_{21}dx_{1}-f_{22}dx_{2}$,,

望 1 $;=dx_{1}$ ,望 2 $:=dx_{2}$.

We consider the IFlrobenius system(3)$\mathcal{I}:=\{\underline{\theta}_{0},\underline{\theta}_{1},\underline{\theta}_{2}\}$diffwith Se$1\wedge\underline{\omega}_{2}\neq 0$

constructed by this coframe. The correspondence between second order PDE (1) andthe Frobenius system is described as follows. Consider vector fields on $J^{1}(R^{2}, R)$ which
are annihilated by , while are not annihilated by . At any point on, such$\mathcal{I}$$J^{1}(\mathbb{R}^{2}, \mathbb{R})$$\underline{\theta}_{i}$$\underline{\omega}_{i}$48vector fields are generated by two vector fields. The integral surfaces which aretangent to the 2-plane $span\{v_{1},$at any point are the graphs of solutions of thesecond order PDE (1). Then, the parameters
are regarded as a local coordinate
system of this integral surface.
The Frobenius condition (integrability condition) of the Frobenius system is:$v_{1},$$v_{2}$$v_{2}\}$$(x_{1}, x_{2})$$\mathcal{I}$$d\underline{\theta}_{i}\equiv 0$(mod$\theta,\underline{\theta}_{1},$$\lrcorner$)$(i=0,1,2)$ .

$\underline{\theta}_{2})$Then, the above integrability condition is equivalent to $A=B=0$ , where $A$ and $B$ aregiven by
$A=(f_{11})_{x_{2}}-(f_{12})_{x_{1}}+(f_{11})_{y}z_{2}+(f_{11})_{z1}fi_{2}+(fi_{1})_{z_{2}}f_{22}$$-(f_{12})_{y}z_{1}-(f_{12})_{z_{1}}fi_{1}-(f1_{2})_{z}2f_{12}$,$B=(f_{12})_{x}2-(f_{22})_{x_{1}}+(f_{12})_{y}z_{2}+(f_{12})_{z1}f_{12}+(f_{12})_{z2}f_{22}$$-(f_{22})_{y}z_{1}-(f_{22})_{z1}f_{11}-(f_{22})_{z_{2}}f_{12}$.

Remark 2.2. Hereafter, we discuss only the second order PDE (1) with respect tosatisfying $A=B=0$ .

$f_{ij}$A family of integral surfaces of gives a 2-dimensional foliation on. We
describe an infinitesimal automorphism group of the foliation, and consider a principal
bundle overwith this group as a structure group.
The contact lift of the scale transformation preserving satisfies the following$\mathcal{I}$$J^{1}(\mathbb{R}^{2}, \mathbb{R})$$J^{1}(\mathbb{R}^{2}, \mathbb{R})$$\phi$$\mathcal{I}$$\phi$

equations:$\hat{\phi}^{*}\underline{\theta}_{0}=a\underline{\theta}_{0}$$\hat{\phi}^{*}\underline{\theta}_{1}=b\underline{\theta}_{0}+c\underline{\theta}_{1}$(4)$\hat{\phi}^{*}\underline{\theta}_{2}=e\underline{\theta}_{0}+g\underline{\theta}_{2}$$(a\neq 0)$,$(c\neq 0)$,$*\underline{\omega}_{1}=h\underline{\omega}_{1}$$\hat{\phi}^{*}\underline{\omega}_{2}=k\underline{\omega}_{2}$,$(g\neq 0)$

バ$(h\neq 0)$,$(k\neq 0)$.

The equation (4) can be written in the following form:(5)$\{\begin{array}{l}\theta_{0}\theta_{1}\theta_{2}\omega_{l}\omega_{2}\end{array}\}=\{\begin{array}{lllll}a 0 0 0 0b c 0 0 0e 0 g 0 00 0 0 h 00 0 0 0 k\end{array}\}\{\begin{array}{l}\underline{\theta}_{0}\underline{\theta}_{1}\underline{\theta}_{2}\underline{\omega}_{1}\underline{\omega}_{2}\end{array}\}$where,
determined by$a,$ $b,$ $c,$$e_{\wedge}g,$$\phi$$h,$ $k$

are functions. Thus we have linear transformations of coframes. Moreover, the lift$\hat{\phi}$of the scale transformation satisfies:$d\theta_{0}\equiv-\theta_{1}\wedge\omega_{1}-\theta_{2}\wedge\omega_{2}$(6)$(mod \theta_{0})$,$d\theta_{1}\equiv 0$$(mod \theta_{0}, \theta_{1}, \theta_{2})$$d\theta_{2}\equiv 0$$(mod \theta_{0}, \theta_{1}, \theta_{2})$

,

.

49

These relations give conditions $a=ch=gk$ . From these conditions, we get the linear
transformations of coframes of the following form:.

(7)$\{\begin{array}{l}\theta_{0}\theta_{1}\theta_{2}\omega_{l}\omega_{2}\end{array}\}=\{\begin{array}{lllll}ch 0 0 0 0b c 0 0 0e 0 g 0 00 0 0 h 00 0 0 0 k\end{array}\}\{\begin{array}{l}\underline{\theta}_{0}\underline{\theta}_{1}\underline{\theta}_{2}\underline{\omega}_{1}\underline{\omega}_{2}\end{array}\}$Therefore, we obtain the following 5-dimensional Lie group as infinitesimal automorphism group:(8)$G:=\{\{\begin{array}{lllll}ch 0 0 0 0b c 0 0 0e 0 g 0 00 0 0 h 00 0 0 0 k\end{array}\}\in GL(5, \mathbb{R})|ch=gk\}$.

Then, we choose the reduced G-bundleof the coframe bundle
over. This bundle
is called G-structure associated with the second order PDE(1). On the principal bundle, we compute curvatures (local invariant functions) forthe equivalence problem using the Cartan’s equivalence method. ([G], [O2], [St]).
Consequently, we obtain the following curvatures.
$\mathcal{F}_{GL}(\mathbb{R}^{5})$$\mathcal{F}_{G}$$J^{1}(\mathbb{R}^{2}, \mathbb{R})$$\mathcal{F}_{G}$$\mathcal{F}_{G}$$K_{1}=- \frac{c}{gh}(f_{11})_{\underline{\theta}_{2}}$$K_{3}= \frac{1}{hk}\{(f_{12})$,,$+(f_{12})_{\underline{\theta}_{2}}(f_{12})_{\underline{\theta}_{1}}-(f_{12})_{\underline{\theta}_{22_{2}}}\}$

島$K_{4}=- \frac{g}{ck}(f_{22})_{\underline{\theta}_{1}}$$K_{2}=- \frac{1}{gh}(f_{12})_{\underline{\theta}_{2}\underline{\theta}_{2}}$,$K_{5}=- \frac{1}{ck}(fi_{2})_{\underline{\theta}_{1}\underline{\theta}_{1}}$,,$K_{6}= \frac{1}{hk}\{(f_{i2})_{\text{島}}+(f_{12})_{\underline{\theta}_{1}}(f_{12})_{\underline{\theta}_{2}}-(f_{12})_{\underline{\theta}_{1}\underline{\omega}_{1}}\}$,$K_{7}= \frac{1}{ch^{2}}\{(f_{11})_{\underline{\theta}_{2}\underline{\theta}_{1}\underline{\omega}_{2}}+(fi_{1})_{\underline{\theta}_{2}\theta_{2}}(f_{22})_{\underline{\theta}_{1}}+(f_{i1})_{\underline{\theta}_{2}\underline{\theta}_{1}}(f_{22})_{\underline{\theta}_{2}}$$-(f_{12})_{\underline{\theta}_{2}\underline{\theta}_{1}}(fi\iota)_{\underline{\theta}_{2}}-(f_{12})_{\underline{\theta}_{2}\underline{\theta}_{2}}(f_{12})_{\underline{\theta}_{2}}-(f_{12})_{\underline{\theta}_{2}\underline{\theta}_{1}}(fi_{1})_{\underline{\theta}_{1}}$$+2(f_{12})_{\underline{\theta}_{2}\underline{\theta}_{1}}(f_{12})_{\underline{\theta}_{2}}\}$,$K_{8}= \frac{1}{chk}\{(f_{12})_{\underline{\theta}_{2}\underline{\theta}_{1}\underline{\omega}_{2}}-(f_{12})_{\underline{\theta}_{1}\underline{\theta}_{0}}-(f_{12})_{\underline{\theta}_{1}\underline{\theta}_{1}}(f_{12})_{\underline{\theta}_{2}}\}$$K_{9}= \frac{2(f_{12})_{\underline{\theta}_{2}}\rho_{1}-(fi_{1})_{\underline{\theta}_{1}\underline{\theta}_{1}}}{ch}$

,

,

50

$K_{10}= \frac{1}{hk}\{-(f_{12})_{\theta}\lrcorner)$一$(f_{12})_{\underline{\theta}_{1}}(f_{12})_{\underline{\theta}_{2}}+(f_{11})_{\underline{\theta}_{2}}(f_{22})_{\underline{\theta}_{1}}+(f_{12})_{\underline{\theta}_{2^{Q}2}}\}$$K_{11}= \frac{1}{hk}\{(f_{11})_{\underline{\theta}_{14!_{2}}}-2(f_{12})_{\underline{\theta}_{2}}$里2$K_{12}= \frac{1}{ch^{2}}\{(f_{11})_{\underline{\theta}_{1}\underline{\theta}_{0}}-2(f_{12})_{\underline{\theta}_{2}}$

島

,

,

$\}$

$+(f_{11})_{\underline{\theta}_{1}\underline{\theta}_{1}}(f_{12})_{\underline{\theta}_{2}}$$+(fi_{1})_{\theta_{1}\theta_{2}}(f_{12})_{\rho_{1}}-2(f_{12})_{\underline{\theta}_{1}\underline{\theta}_{2}}(fi_{2})_{\underline{\theta}_{2}}-2(f1_{2})_{\underline{\theta}_{2}Q_{2}}(f_{12})_{\underline{\theta}_{1}}\}$$K_{13}= \frac{1}{hk}\{2(f_{12})_{\underline{\theta}_{14_{1}}}-(f_{22})_{\underline{\theta}_{2^{\underline{td}}1}}\}$,,$K_{14}= \frac{1}{chk}\{-2(f_{12})_{\underline{\theta}_{1}\underline{\theta}_{0}}-2(f_{12})_{\underline{\theta}_{1}\underline{\theta}_{1}}(f_{12})_{\underline{\theta}_{2}}-2(f_{12})_{\underline{\theta}_{1}\underline{\theta}_{2}}(f_{12})_{\underline{\theta}_{1}}$$+(f_{22})_{\underline{\theta}_{2}\underline{\theta}_{0}}+(f_{22})_{\underline{\theta}_{1}\underline{\theta}_{2}}(f_{12})_{\underline{\theta}_{2}}+(f_{22})_{\underline{\theta}_{2}\underline{\theta}_{2}}(f_{12})_{\underline{\theta}_{1}}\}$$K_{15}= \frac{2(f_{12})_{\underline{\theta}_{1}\underline{\theta}_{2}}-(f_{22})_{\underline{\theta}_{2}\underline{\theta}_{2}}}{gk}$,.

where, curvature functions are expressed by the dual hame:$\partial_{A}\theta:=\frac{\partial}{\partial y’}$$\partial_{\underline{\theta}_{1}}:=\frac{\partial}{\partial z_{1}}$,$\partial_{\underline{\theta}_{2}}:=\frac{\partial}{\partial z_{2}}$$\partial_{\underline{\omega}_{1}}:=\frac{\partial}{\partial x_{1}}+z_{1}\frac{\partial}{\partial y}+f_{11}\frac{\partial}{\partial z_{1}}+f_{12}\frac{\partial}{\partial z_{2}}$$\partial_{\underline{\omega}_{2}}:=\frac{\partial}{\partial x_{2}}+z_{2}\frac{\partial}{\partial y}+f_{21}\frac{\partial}{\partial z_{1}}+f_{22}\frac{\partial}{\partial z_{2}}$

,

,

.

From the general theory of G-structure, a vanishing condition of curvatures gives thefollowing theorem:
Theorem 2.3 ([N]). Suppose that the second order $PDE$ satisfies the integrability condition $A=B=0$ . Then. the equation (1) is (locally) equivalent to the flat equation
under lifts of scale transformations if and only if the curvatures
vanish.
$K_{i}$We have some results from the theorem:
First, the functions
satisfying $A=B=K_{i}=0$ are written as quadratic polyno. Hence, if there is a polynomial of degree grater than 2 as a polynomial ofmials inamong , then corresponding equation (1) is not equivalent to the flat equation
under lifts of scale transformations.
Next, we give some examples of equation which is equivalent to the flat equation.
$f_{ij}$$z_{1},$$z_{1},$$z_{2}$$z_{2}$$f_{ij}$Corollary 2.4. Suppose that the functionsin (1) are given in the following form:. Then the equation (1) is(locally) equivalent to the flat equation under the lifts of scale transformations if andonly if $P_{y}=Q_{y}=R_{y}=0,$ $P_{x}2=Q_{x_{1}}$ , $Q_{x_{2}}=R_{x1}$ .

$f_{11}=P(x_{1}, x_{2}, y),$$f_{12}=Q(x_{1}, x_{2}, y),$$f_{ij}$$f_{22}=R(x_{1}, x_{2}, y)$Remark 2.5. The conditions $P_{y}=Q_{y}=R_{y}=0,$ $P_{x2}=Q_{x_{1}},$in Corollary
2.4 are obtained by the integrability condition $A=B=0$ . Therefore, the second order
PDE (1) for the functions
given by the above form is equivalent to the flat equation
if and only if it is integrable.
$Q_{x_{2}}=R_{x_{1}}$$f_{ij}$51

3. DUALITY ASSOCIATED WITH DIFFERENTIAL EQUATIONSIn this section, we discuss a duality between the coordinate space and the solution
space associated with the following flat equation;(9)$\frac{\partial^{2}y}{\partial x_{i}\partial x_{j}}=0$$(1\leq i,j\leq 2)$.

For the purpose, we consider the following double fibration.

(10)
where, projections$\pi_{1},$$\pi_{2}$are defined by$\pi_{1}(x_{1}, x_{2}, y, z_{1}, z_{2})=(x_{1}, x_{2}, y)$,$\pi_{2}(x_{1}, x_{2}, y, z_{1}, z_{2})=(z_{1}, z_{2}, y-z_{1}x_{1}-z_{2}x_{2})$.

We call the double fibration (10) the model space of the flat equation or flat model space.
$:=\{(x_{1}, x_{2}, y)\}$ ,In this fibration, we regard the left base space as a coordinate space$:=\{(a, b, c)\}$ . Solutions of (9) are written
and a right base space as a solution spaceas $y=ax_{1}+bx_{2}+c$ for real parameters. Graphs of solutions are planes onorand the 3-parameter family of solutions yields a 2-dimensional foliationon. Then the leaf space of this foliation is interpreted as a solution space of(9). We discuss the compactification of the flat model space. The fibration (10) can beembedded naturally into the following (global) double fibration:$\mathbb{R}^{3}$$\mathbb{R}^{3}$$\mathbb{R}^{3}$$a,$ $b,$ $c$$J^{1}(\mathbb{R}^{2}, \mathbb{R}))$$J^{1}(\mathbb{R}^{2}, \mathbb{R})$(11)where,$V=\mathbb{R}^{4}$, and $Gr(3,4)$ is a Grassmannian manifold and $F_{V}(1,3)$ is a flag variety:$Gr(3,4)=\{E|E$ is a hyperplane of $V$$F_{V}(1,3)=\{(l, E)|l\in \mathbb{R}P^{3},$The, projections$\pi_{1},$$\pi_{2}$$:=\mathbb{R}^{4}\}$,$E\in Gr(3,4)\cong \mathbb{R}P^{3},$$l\subset E\}$.

are defined by$\pi_{2}([u], H):=[f_{H}]$ ,,where,
is a linear functional satisfying $ker(f_{H})=H$ of. (Since is uniquely
defined up to scalar multiplication,is well-defined.) The double fibration (10) do notdepend on coordinate transformation group . So, we introduce the flat model spacedepending on .

$\pi_{1}([u], H):=[u]$$f_{H}$$V^{*}\backslash \{0\}$$\pi_{2}$$\mathcal{G}$$\mathcal{G}$$f_{H}$

52

We fix a coordinate transformation groupsymmetry group ([Ol], [O2]).

Definition 3.1. Let$G$$\mathcal{G}\subset$

Di$ff(\mathbb{R}^{3})$. First, we define the following

be an isotropy subgroup of the flat equation (9) inis called symmetry group of the flat equation for .
$G$$\mathcal{G}$. This group$\mathcal{G}$In the case ofcoordinate spaceFor$\mathcal{G}=Diff(\mathbb{R}^{3})$$\mathbb{R}^{3}$, the symmetry group is $SL(4, \mathbb{R})$ and the action on theis given by:,$(\begin{array}{llll}a_{1} a_{2} a_{3} a_{4}b_{1} b_{2} b_{3} b_{4}c_{1} c_{2} c_{3} c_{4}d_{1} d_{2} d_{3} d_{4}\end{array})\in SL(4, \mathbb{R})$(12)$(x_{1}, x_{2}, y) \mapsto(\frac{a_{1}x_{1}+a_{2}x_{2}+a_{3}y+a_{4}}{d_{1}x_{1}+d_{2}x_{2}+d_{3}y+d_{4}},$

Next, we introduce subgroups of$\frac{b_{1}x_{1}+b_{2}x_{2}+b_{3}y+b_{4}}{d_{1}x_{1}+d_{2}x_{2}+d_{3}y+d_{4}},$$SL(4, \mathbb{R})$$\frac{c_{1}x_{1}+c_{2}x_{2}+c_{3}y+c_{4}}{d_{1}x_{1}+d_{2}x_{2}+d_{3}y+d_{4}})$

as follows:$H_{i}:=\{g\in SL(4, \mathbb{R})|g[e_{i}]=[e_{i}]\}$,$\overline{H}_{i}:=\{g\in SL(4, \mathbb{R})|^{t}g^{-1}[e_{i}]=[e_{i}]\}$,

where, $e_{i}(i=1,\cdots\cdot,4)$ are standard basis of, andare corresponding elements$RP^{3}$in. The subgroups
are isotropy subgroups which preserve lines, and thesubgroups
are isotropy subgroups which preserve hyperplanes spanned by $e_{j}(j\neq i)$
respectively. We used Cartan involution
in the definition of . We consider
the following double fibration.
$\mathbb{R}^{4}$$[e_{i}]$$H_{i}$$[e_{i}]$$\overline{H}_{i}$$\tilde{\theta}(g)={}^{t}g^{-1}$$\overline{H}_{i}$(13)where,
with respect to$H=H_{4}\cap\overline{H}_{1}$.
In the case of. We call this fibration as a model space of the flat equation (9)$\mathcal{G}$, we obtain the following well-known fibration using
Di$G=SL(4,\mathbb{R})$ .
corresponding symmetry group(14)$\mathcal{G}=$$ff(\mathbb{R}^{3})$.

53This fibration equals the fibration (11). Also, this fibration corresponds to the following
picture by Dynkin diagrams:

From the picture, we have the fibration (14) by the following process ([Ya]). Let
be the simple root system andbe the set of positive roots. If wechoose a non-empty subset,
of , then we have the decomposition$k\geq0$
wherefor
are subset defined by$\Phi^{+}$$\triangle=\{\alpha_{1}, \alpha_{2}, \alpha_{3}\}$$\triangle_{1}$$\triangle$$\Phi^{+}=\bigcup_{k\geq 0}\Phi_{k}^{+}$$\Phi_{k}^{+}$$\Phi_{k}^{+}:=\{\alpha=\sum_{i=1}^{3}n_{i}(\alpha)\alpha_{i}\in\Phi^{+}|\sum_{\alpha_{i}\in\triangle_{1}}n_{i}(\alpha)=k\}$Corresponding to the decomposition, we can choose a parabolic subalgebra as follows:$\mathfrak{p}=\mathfrak{g}_{0}\oplus\sum_{\alpha\in\Phi_{0}^{+}}(\mathfrak{g}_{\alpha}\oplus g_{-\alpha})\oplus\sum_{\alpha\in\Phi_{k}^{+},k>0}g_{\alpha}$.

Let $P\subset G=SL(4, \mathbb{R})$ be the parabolic Lie subgroup whose Lie algebra is equal to, and put $M$ $:=G/P$ . This space is called the model space of parabolic geometry
associated with subset
of simple root system . Now, we determine the topologyof the above fibration by Dynkin diagrams using this construction. We choose subsets
of simple root system. Corresponding to the
choice, we have the following parabolic subalgebras:$\mathfrak{p}$$\triangle$$\triangle_{1}$$\triangle_{1}=\{\alpha_{1}\},$$\triangle_{2}=\{\alpha_{3}\},$$\triangle_{12}=\{\alpha_{1}, \alpha_{3}\}$$****$$****1\in \mathcal{B}l(4, \mathbb{R})\}$,$\mathfrak{p}_{12}=\{(000*0***0***$$****1\in\epsilon l(4, \mathbb{R})\}$.

$\mathfrak{p}_{1}=\{(000*$$****$We have corresponding parabolic subgroups$\mathfrak{p}_{2}=\{(**0*0***$$0***$$****1\in\epsilon l(4, \mathbb{R})\}$,and model spaces $G/P_{1}\cong$, whereThus,.
we can identify the above
picture constructed by Dynkin diagrams with the fibration (14). We can use Tanaka
theory for the equivalence problem associated with this case ([Ya]).
In the case of, we calculate the corresponding flat model space.
ScaleDi
From the action (12) ofon , we have the following symmetry group fromrestriction of variables associated with the scale transformation.

$\mathbb{R}P^{3},$$G/P_{2}\cong \mathbb{R}P^{3},$$G/P_{12}\cong F_{V}(1,3)$$\mathcal{G}=$$P_{2},$$P_{12}\subset G$$V=\mathbb{R}^{4}$$ff(\mathbb{R}^{3})$$SL(4, \mathbb{R})$$P_{1},$$\mathbb{R}^{3}$

54$G=\{(0*O*00**000*$$****1\in SL(4, \mathbb{R})\}$.

Then, we have the following.

Proposition 3.2. We obtain the following double fibration as the flat model space associated with:$\mathcal{G}=ScaleDiff(\mathbb{R}^{3})$(15)This fibration is degenerate. Sinoe
ScaleDi $ff(R^{3})$ is very strongly restricted from, this degeneration arises. Hence, we consider the following problem.
Di
Problem 3.3. Find a symmetry group for proper subgroup of, from which
has double fibmtion of compact-type as a flat model space.
For the problem, we consider the following subgroup:$\mathcal{G}=$$ff(\mathbb{R}^{3})$$\mathcal{G}$$Diff(\mathbb{R}^{3})$$G\subset SL(4, \mathbb{R})$(16)$G=\{g\in SL(4, \mathbb{R})|g[e_{3}]=[e_{3}],g[e_{3}]=[e_{3}]\}$ ,(17)$=\{(\begin{array}{ll}* *0** *0*0 0*0* *0*\end{array})\in SL(4, \mathbb{R})\}$.

Note that is a subgroup invariant under Cartan involution . We have the following
fibration of compact-type with respect to the group:
Theorem 3.4 ([N]). A double fibration constructed by the group (16) is the following
fibration of compact-type.
$G$$\tilde{\theta}$(18)

We note that the coordinate transformation group corresponding to this symmetrygroup (16) is constructed by the transformations of the form:$\mathcal{G}$$X_{1}=X_{1}(x_{1}, x_{2}),$$X_{2}=X_{2}(x_{1}, x_{2}),$$Y= \frac{y}{A(x_{1},x_{2})}$.

55

REFERENCES[BG]

R. Bryant, P. Griffiths: Toward a geometry of differential equation, Geometry, Topology, andphysics, International Press (1995), 1-76.
[B]
R. Bryant: \’Elie Cartan and geometric duality, Journees \’Elie Cartan 1998 et 1999, Intitut \’Elie
Cartan 16 (2000), 5-20.
[G]
R. B. Gardner: The method of equivalence and its applications, CBMS-NSF Regional Conf. Ser.
in Appl. math, 58, SIAM. Philadelphia (1989).
[GTW] C. Grissom, G. Thompson, G. Wilkens: Linearization of second order ordinary differential
equations via Cartan’s equivalenoe method, J. Differential Equations, 77, (1989), 1-15.
[HK] Hurtubise, J. C. ,Kamran, $N$ : Projective connections, double fibrations, and formal neighborhoodof lines, Math. Ann. 292, (1992) no.3383-409.
[IL]
Thomas, A. Ivey, J. M. Landsberg: Cartan for Beginners:Differential Geometry via Moving
Flrames and Exterior Differential Systems, American Mathematical Society 2003.
[M]
T. Morimoto: Two great books by \’Elie Cartan, Pleasure of mathematics, no.29, 98-104, NipponHyouronsha, (2002) ( apanese).
[N]
T. Noda: Equivalenoe problem for second order PDE and double fibration as a flat model space,
preprint.
[01] P. Olver: Applications of Lie Groups to Differential Equations (Second Edition), Graduate Texts
in Mathematics, Springer 2000.
[O2] P. Olver: Equivalence, Invariants, and Symmetry, Cambridge University Press, 1995.
[OSS] T. Ozawa, H. Sato, H. Suzuki: Differential Equations and Schwarzian Derivatives, noncommutative geometry and physics 2005, Proceedings of the Intemational Sendai-Beijing Joint Workshop,(2007) 129-149.
[Sa]
H. Sato, Orbit decomposition of space of differential equations, mathematics of 21 century
untrodden peak of geometry $\sim Nippon$ -Hyouronsha, (2004) 267-280 (Japanese).
[St]
Stemberg, Lectures on differential geometry, Chelsea (1983).
[Ta] N. Tanaka: On the equivalence problems associated with simple graded Lie algebras, Hokkaido
Math. J. 8 (1979), no. 1, 23-84.
[Ya] K. Yamaguchi: Differential systems associated with simple graded Lie algebras, Advanced Studies in Pure Math., 22 (1993), 413-494.
[YY] Yoshikawa, Atsuko Yamada: Equivalence problem of third-order ordinary differential equations,Internat. J. Math. 17 (2006), no. 9, 1103-1125.
E-mail address: $m04031$ xOmath. nagoya-u. ac. jp$J$$\sim$