数理解析研究所講究録 1206 巻 2001 年 95-10695$G_{2}$-Geometry of Overdetermined Systems of Second Order

By

Keizo YAMAGUCHIIntroduction.
Discovery of E.
$\underline{||}\mathrm{C}^{\rceil}..\rceil$Les$\mathrm{C}\mathrm{a}.\mathrm{r}\mathrm{t}\dot{\mathfrak{c}}\tau.\mathrm{n}$$s.?./\cdot 9t\grave{c,}rr|,e.\backslash \cdot$$.‘\grave,(^{J}c(J7/do\uparrow.d.\cdot re_{\backslash }$Allll.

Overdeterminedinde Pfaff \‘a$c?,\cdot nq.\iota,’ ariables$$\mathrm{E}\mathrm{t}^{\backslash }..\mathrm{N}_{011}.\mathrm{n}\mathrm{a}1\mathrm{e}..27$$(\mathrm{i}_{1’11^{f}\mathrm{C})}1\iota\iota \mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e})\mathrm{L}\mathrm{s}.\mathrm{y}\mathrm{s}\mathrm{t}\cdot \mathrm{e}\mathrm{n}\mathrm{l}$(1910), 10(t) -192:$\frac{\partial_{\sim}^{2}\prime}{\dot{\epsilon}\mathit{1}\tau.\cdot\sim)}.\tilde{.},=‘\frac{1}{3}(’\frac{o^{1}\prime\sim\sim)\prime\vee}{\acute{r})_{ll^{2}}},..\cdot.)^{3}$$(\Lambda-)$et, les \’equations aux deriv\‘ees partielles du.

$. \frac{\acute{r}J^{2_{\wedge}}\prime\vee}{\partial x()_{l/}}.\cdot=\frac{1}{2}(\frac{c^{r})^{2_{Z}}}{\partial y^{2}}.)^{2}$

Single equation of Goursat tyPe:(B)$()_{7}^{2}\mathrm{L}^{\cdot}+12t^{2}(\cdot \mathfrak{l}\cdot t-’.\mathrm{s}^{2}.)+32^{\mathrm{q}^{3}}.-36rst=0$‘$\vee\backslash \cdot \mathrm{v}1\mathrm{l}\mathrm{e}.\mathrm{r}^{\tau}$$r= \frac{\dot{\mathrm{c}}^{-}J^{12_{\gamma}},\prime}{\overline{\epsilon}\dot{\mathit{1}}x^{2}}\ldots|\mathrm{s}=\frac{c7^{2}\acute{\acute{6}}}{\partial x’\partial y}.\prime\prime$

14-dirnerisional Exceptional Shnple Lie Algebra$t= \frac{\partial^{2}z}{\partial^{\mathrm{r}}y^{2}}$$G_{2}$The Plan of This TALK

.

,

.

96
of Jet Spaces (Q1)$*\mathrm{G}\mathrm{r}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{m}\mathrm{a}\mathrm{y}\mathrm{O}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{c}\mathrm{t}\cdot \mathrm{i}\mathrm{o}\mathrm{n}$fD-ma.nifolds (fi4).

$\ovalbox{\tt\small REJECT}$$\bullet$$\mathrm{G}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\cdot \mathrm{r}_{\psi}\mathrm{v}$of Linear Differentia.l Systems( Tana.ka Theory)(\S 2)
Sy
associated with SGLA (Simple Graded Lie Algebras) (\S 3).
-stems$\Rightarrow \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}.\mathrm{n}\mathrm{f}_{1}\mathrm{i}\mathrm{a}.1$$-$$\bullet$

Link
theni
Reduction Theorems for $PD$ uianifolds (\S 4, \S 5)$\mathrm{t}y\mathrm{t}*,\mathrm{t}\mathrm{w}\mathrm{e}\mathrm{e},11$$.\Rightarrow$

Togethcr coxubined to discuss$G_{2^{-}}\mathrm{G}\mathrm{e}.o111\mathrm{e}\mathrm{t}_{1}\mathrm{r}\mathrm{y}$of O.

$\mathrm{v}\mathrm{e}\mathrm{r}(\mathrm{l}\mathrm{e}.,\mathrm{t}\mathrm{e},\mathrm{r}\mathrm{n}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}$Systems of Second Order (\S 6)\S l.Sec0nd Order Contact Manifolds.
$\mathrm{C}_{\mathrm{v}}\mathrm{r}\mathrm{a}s_{\backslash }\mathrm{s}\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{B}\mathrm{u}\mathrm{n}\mathrm{d}1\mathrm{e}$

$M:$

:

a manifokl of dimension $m+n$$\prime J(\mathrm{A}f, n)=\cup J_{\mathrm{J}}x\in hI^{\cdot}$

Canonical System$C\mathrm{o}11J(\Lambda,I, 71\cdot)$’$J_{x}=\mathrm{G}\mathrm{r}(T_{x}(M), n)$.

:$\forall u\in J(\mathrm{A}I, n)$$C,(\tau\iota)=\pi_{*}^{-1}(\tau\iota)\subset \mathit{2}_{u}^{\tau}(J(\mathrm{A}f, n))$Inhomogeneous Grassmann c.oordinate:$U=${$x_{\mathrm{o}}=\pi(u_{o})\in U’;(x_{1}, \cdots, x_{n}, z^{1}, \cdots, z^{m})$$\tau\iota\in\pi^{-1}(U’)|\pi(u)=.\prime \mathrm{t}:\in U’$$\mathrm{C},\mathrm{o}\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{i}11\mathrm{a}\mathrm{t}_{1}\mathrm{e}\mathrm{s}(x_{1\prime}\cdots..’\iota_{\iota},, z^{1}, \cdots, z_{:l^{J_{1}^{1}}}^{\uparrow h}.\cdots\prime p_{r\iota}^{1n})$and$dx_{1}\wedge\cdots\wedge dx_{n}|_{u}\neq 0$.

$\varpi^{r\iota}=dz^{\alpha}-\sum_{i=1}^{n}.p^{\alpha}\dot,dx_{i}$$m=1\Rightarrow.\mathrm{C}\mathrm{o}\mathrm{I}\mathrm{l}\mathrm{t}\mathrm{a}\mathrm{e}\cdot \mathrm{t}$.

$(\mathrm{r}_{1},.\cdots.x_{r\iota.:}z^{1}, \cdots, z^{n}’,p_{1}^{1}, \cdots,p_{r\iota}^{m})$$C=\{\varpi^{1}=\cdots=\varpi^{\mathit{7}\prime\downarrow}$

where

};

are introduced by$dz^{\alpha}|_{\mathfrak{l}l}= \sum_{i=1}^{1\mathrm{l}}\prime p_{i}^{a}(u)d.x_{i},|_{u}$On acanonical coordinate system.

$=0\}$,.

bIanifeyld$\varphi:\mathrm{A}\prime Iarrow \mathrm{A}\hat{\prime}I$:$(1\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}()\mathrm{n}1()\mathrm{r}1^{\mathrm{J}\mathrm{h}\mathrm{i}\S\ln}\Rightarrow\varphi_{*}:(J(\Lambda f, 7l\cdot),$$C)arrow(J(\hat{M}, n),\hat{C})$97
Theorem 2.1 (Bicklund). $M$ , AlA)$\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} 2$$ssurrz,e.\mathrm{r}\mathrm{z}$.

$(\mathrm{I})\ovalbox{\tt\small REJECT}(J(\mathrm{A}\ovalbox{\tt\small REJECT}, \mathrm{r}\mathrm{z}),$that$\ovalbox{\tt\small REJECT} \mathrm{B}_{\mathrm{i}}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} Al\mathrm{e}A,fsu\ovalbox{\tt\small REJECT} h$of dimension$\mathrm{O}\ovalbox{\tt\small REJECT} \mathrm{p}.$$mA\mathit{1}\ovalbox{\tt\small REJECT}$.

; isomorphism$C)\ovalbox{\tt\small REJECT}"\ovalbox{\tt\small REJECT}(J(\ovalbox{\tt\small REJECT} f_{\ovalbox{\tt\small REJECT}}\mathrm{n}), C)$.

Cont.act Ma.n$(J, C)$ :$\mathrm{L}\mathrm{a},\mathrm{g}1^{\cdot}\mathrm{a}.11\mathrm{g}\mathrm{e}- \mathrm{G}\mathrm{r}\mathrm{a}\mathrm{s}.\backslash ^{\backslash }\mathrm{n}1i1\mathrm{n}\mathrm{I}1$$\ovalbox{\tt\small REJECT} m,ani\ovalbox{\tt\small REJECT} olds$$\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{d}\Rightarrow(L(J), E)$

Bumdle$L(.\cdot I\mathrm{I}=\mathrm{U}^{L_{u}arrow J}\prime u\epsilon_{-}J\pi$$L_{u}=${Legendrian stibspaces of(C(u),$d\varpi$)}.

\forall \iota )\in L(力$E(v)=\pi_{*}^{-1}(v)\subset T_{v}(L(’J))arrow T_{u}\iota*(\Gamma J)$

where$(x_{1\backslash }\cdots, 5t_{n}., z,p_{1}, \cdots,p_{7l}.,p11, \cdots,p_{\iota n}.,)$$E$.

and $p_{ij}=pji$$=\{\varpi=\varpi_{1}=\cdots=\varpi_{n}=0\}$

,

where
$\varpi--dz-\sum_{i-1}^{r\iota}--p_{i^{(}}tx_{i}$

,

$\{$

,
$\varpi_{i}=dp_{i}-\sum_{j=1}^{\iota}.p_{ij}dx_{j}$Theorem 2.2. (I) :. $(L\cdot(J), E’)arrow(L(\hat{J}),\hat{E});isomorphism\Rightarrow\exists_{1}$(A : $(J, C)arrow(\hat{J},\hat{C})$such, that (J).
$=\mathrm{i}\rho_{*}$\S 2. Geometry of Linear Diff.erential Systems (Tanaka Theory).
$\mathrm{A}\prime I$

alllallifold of (linlension:$d$of rank$D\acute{\ddot{\mathrm{c}}}T(.\Lambda I):‘ \mathrm{s}\iota \mathrm{l}\mathrm{I}\mathrm{J}\mathrm{l})\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{l}\mathrm{l}\mathrm{e}$$r(.\mathrm{s}+7^{\cdot}=d)$$D=\{\omega_{1}$

$($

11 [.

$D):\mathrm{c}()\dot{1}\mathrm{r}11)1\mathrm{e}\mathrm{t}_{1}‘ 11.\mathrm{v}$$’$illt.eg ra.

.

$\vee--\ldots=\omega_{s}=0\}$.

.

$\mathrm{b}\mathrm{l}\mathrm{e}$$\Leftrightarrow D=\{dx_{1}=\cdots=dx_{s}=\mathrm{t})\}$$\Leftrightarrow d_{\acute{\iota}\iota’}i\equiv 0$(rnoel$\Leftarrow\Rightarrow[D, D]\subset D$$\omega_{1},$$\cdot\cdot$$\omega_{\mathit{8}}$)$(1\leqq\prime i\leqq s1$

where $D=\Gamma(D)$Derived System $\partial D:\partial D=D+[D, D]$ .
Cauchy Characteristic System $Ch(D)$.:$Ch(D)(?^{\backslash })=$

$l_{i-}.\mathrm{t},1_{1}${$X\in D(.x)|X_{\underline{\mathrm{t}}}^{1d_{i\ i}=}..-0$

Derived Svstem$\partial^{k}D$(.lno(l$\omega_{1},$:$\prime J^{k}D=\partial(’\partial^{k-1}D)$A-tl$\mathrm{W}\mathrm{e}\mathrm{a}.\cdot\dot{\mathrm{k}}$Derive.d$\mathrm{S}\mathrm{y}_{\iota}\mathrm{s}$t.elIl$\partial^{(k)}.D$:$\ldots,\omega_{\mathit{8}}.)$

for $i=1,$$\ldots,s$},98(k)D $=\partial^{(k-1)}’.D+[D, \partial^{(k\cdot-1)}D]$ ,Symbol Algebras: regtflar
(S1)
such
$(.\Lambda,f, D)$$\exists l^{\mathit{1}}>0$$\mathrm{t}\cdot 1_{1}\mathrm{a}\mathrm{t},\cdot$

for all$k\geqq l^{\mathit{1}}$,$D^{-k}=\cdots=D^{-\mu\supset}\neq\cdots-\neq T$ $D^{-2}\supset D^{-1}=D\neq$

.

(S2)

$[D_{:}^{\rho}D^{q}]\subset D^{p+q}$$(\Lambda,I, D)$for all$?^{y.j}l<\mathrm{t}\mathfrak{l}$’.

: regular such that, $T(\Lambda f)=D^{-\mu}$ .

$\forall x\in \mathrm{A}I$,$-\mu$$\mathrm{m}(x,)=\oplus$

佳 $p(x)$.

$\mu=-1$

9-1 $(x)=D^{-1}(x),$$\mathfrak{g}_{p}(.\tau.\cdot)=D^{p}(x)/I\mathit{2}^{p+1}(x)$$[X, \mathrm{Y}]---\varpi_{p+q}([\tilde{X},\tilde{\mathrm{Y}}]_{x})$

where

,

$\tilde{X}\in\Gamma(D^{p}),$ $X=\varpi_{p}(\tilde{X}_{\tau}.)\in \mathfrak{g}_{p}(x),\tilde{\mathrm{Y}}’\in\Gamma(D^{q}),$ $\mathrm{Y}=\varpi_{q}(\tilde{\mathrm{Y}}_{x})\in \mathfrak{g}_{q}(x)$$\mathfrak{g}_{p}(x)=[\mathfrak{g}_{p+1}(x), \mathfrak{g}_{-1}(’.r,)]$.

for $p<-1$ .

Conversely given a
Fundamental Graded Lie Algebra :$\mathrm{m}=\oplus\rho=-1-\mu \mathfrak{g}_{p}$Nill)ot Cllt GLA satisfying$\mathrm{i}.\mathrm{t}’...$$\cdot$,$\mathrm{t}_{}.1_{1}\mathrm{e}$$\cdot$generating condition :$\mathfrak{g}_{p}---[\mathfrak{g}_{p+\mathrm{l}}.,\mathfrak{g}_{-1}]$$\Rightarrow$$(\mathrm{A}I(\mathrm{m}), D_{\dot{\mathrm{m}}})$: Standard Differential System of type$\mathrm{m}$$\underline{arrow}$$\mathfrak{g}(\mathrm{m})$: Prolongation of$\mathrm{m}=\oplus_{p<0}\mathfrak{g}_{p}$Our Problem :When does$\mathfrak{g}(.\cdot \mathrm{m})$become finite diniensional arid simple ’?

Symbol Algebra of $(L(J), B)$ :$E=\{\varpi=\varpi_{1}=\cdots=\varpi_{n}=0\}$,

for $p<-1$

99$\mathrm{c}^{2}(n)=\mathrm{c}_{-3}\oplus \mathrm{c}_{-2}\oplus \mathrm{c}_{-1}$wltere$C_{--}.\cdot;=W,$ $\mathrm{c}_{-2}=W\otimes V^{*}$,
沖】 $=V\oplus W\otimes S^{2}(V^{*})$$\mathrm{C}\circ \mathrm{f}\Gamma|c1.\mathrm{n}1\mathrm{C}$,.

Dual frallle::$\{\varpi, \varpi_{i}, dx_{i}, d.p_{?.j}.\}\dot{\prime}$$\{\frac{\partial}{\partial z}, \frac{\partial}{\partial p_{i}}, \frac{d}{dx_{i}}, \frac{\partial}{\partial p_{\mathrm{i}j}}\}$$\mathrm{w}1_{1}\mathrm{e},\mathrm{r}\mathrm{e}$$\frac{d}{dx_{i}}=\frac{d}{\partial’x_{i}}.+p_{i}\frac{?}{\partial^{r}z}.+\sum_{j=1}^{n}(p_{ij}\frac{\partial}{\partial p_{j}}$$[ \frac{\partial’}{\partial p_{ij}},$$\frac{d}{dx_{k^{4}}}.]=\delta_{k}^{i}.\frac{\partial}{\partial p_{j}}+\delta_{k}^{j}\frac{\partial’}{\partial p_{i}}$$[ \frac{\partial}{\partial p_{i}}’\frac{d}{d\prime c_{k}}\ldots]=\delta_{k}^{i}.\frac{\partial}{\partial^{r}z}$$E=\{\varpi=\mathrm{r})\}=\pi_{*}^{-1}C$ ,

\S 3.

,

.
.

$Ch(\partial E)=\mathrm{K}\mathrm{e}\mathrm{r}\pi_{*}$

Systems associated with Simple Graded Lie Algebras.

$\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{l}\dot{\mathrm{e}}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}$Gradation of:Simple Lie Algebra over$\mathfrak{g}$$\mathbb{C}$$\mathfrak{g}$1):$\mathrm{C}\mathrm{a}.\mathrm{r}\mathrm{t}\overline{\mathrm{H}.}.\mathrm{n}$Sllt)alg.

$\triangle=\{(1_{1}’, \cdots, \alpha_{l}\}$stleln$;\Phi\subset \mathfrak{h}^{*}:\mathrm{R}\mathrm{o}\mathrm{o}\mathrm{t}\cdot \mathrm{S}_{J}.\mathrm{v}$:Simple Root. System$\mathfrak{g}=\oplus_{\mathrm{t}\ulcorner}9\alpha\oplus \mathfrak{h}\oplus-.\oplus\alpha\in\Phi\alpha\in\Phi^{+}9-\alpha$$\triangle_{1}’\subseteq\Delta$:Fix,$\Phi^{+}=\bigcup_{p\geqq 0}\Phi_{p}^{+}$

,

,

,

$\Phi_{p}^{+}=\{\alpha=\sum_{i=1}^{\ell}n_{i}\alpha_{i}|.\sum_{\alpha.\in\Delta_{1}}n_{i}=p\}$gp=\oplus \mbox{\boldmath $\alpha$}6。p+g\mbox{\boldmath $\alpha$}’$\{$佳$(p>0)$$0=\oplus_{\alpha\in\Phi_{0}^{+\mathfrak{g}_{\alpha}\oplus \mathfrak{h}\oplus\oplus_{\alpha\in\Phi_{\mathrm{O}}^{+9-\alpha}}}}$,

g-p=\oplus \mbox{\boldmath $\alpha$}\in \Phi \rho +g-\mbox{\boldmath $\alpha$}フThen

Gerleratirlg$\mathrm{C}\mathrm{o}\mathrm{n}\mathrm{e}1\mathrm{i}\mathrm{f}.\mathrm{i}\mathrm{o}11:\mathrm{m}=\oplus_{p<0}\mathfrak{g}_{p}(\star,)$$\triangle_{1}\subset\Delta$

where

for

$[\mathfrak{g}_{p}, \mathfrak{g}_{q}]\subset \mathfrak{g}_{p+q}$$\Rightarrow$$\mathfrak{g}_{p}=[\mathfrak{g}_{p+1}, \mathfrak{g}_{-1}]$$(\swarrow \mathrm{Y}_{\ell}.\triangle_{1})$$l^{\mathit{1}.=\sum_{a_{i}\in\Delta_{1}}n_{i}(\theta),\theta=\sum_{i=1}n_{i}(\theta)\alpha_{\mathrm{i}}}\ell$,:$p,$ $q\in \mathbb{Z}$.

for $p<-1$$\mathfrak{g}=\oplus^{\mu}p=-\mu \mathfrak{g}_{p}$100Theorem 41.
Dynkin$X\ell^{\ovalbox{\tt\small REJECT}}$Classification$\mathfrak{g}\ovalbox{\tt\small REJECT}\oplus_{p\mathrm{e}\mathrm{z}9p^{\ovalbox{\tt\small REJECT}}}Sin\varphi le$$\ovalbox{\tt\small REJECT} iagram$of}Graded Lie$\exists_{1}\Delta_{\mathrm{i}}\mathrm{C}\Delta s.t$$\mathfrak{g}.\ovalbox{\tt\small REJECT}$$A\ovalbox{\tt\small REJECT} ebra$.

$\mathrm{A}f_{\mathrm{m}\prime}$

D。),-space$\mu\geqq 2$$\mathrm{m}---\oplus_{p<09\nu}$Theorem 4.2.
Except for (1), (2.), (3),$9=\oplus_{p\in \mathrm{Z}}\mathfrak{g}_{p}$of Parabolic$9-1\Rightarrow D_{\mathfrak{g}}$

(1)
(2)
(3)

$\mathrm{m}=\oplus_{p<0}\mathfrak{g}_{p}$$\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{a}1\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}\mathrm{s}\mathfrak{g}^{l}=\oplus p\geqq 0\mathfrak{g}_{p}$on$M_{\mathfrak{g}}$.

:Simple $G_{7}aded$ Lie Algebra over$\mathfrak{g}=\oplus p\in \mathrm{Z}\mathfrak{g}_{p}\cong \mathfrak{g}(\mathrm{m})$where$\mathbb{C}$satisfying$(\star)$,.
is of depth 1
is a contact gradation.
is isomorphic, with$(\mathit{1}^{\mathit{1}=}1)$$9=\mathfrak{g}_{-1}\oplus \mathfrak{g}_{0}\oplus\cdot \mathfrak{g}_{1}$.

$\mathrm{Q}=\oplus_{p=-2}^{2}\mathfrak{g}_{p}$$\mathfrak{g}=\oplus_{p\in 7_{l}}\mathfrak{g}_{p}$$(A_{\ell}, \{\alpha_{1}, a_{i}’.\})(1<i<\ell)(C\ell, \{.\alpha_{1}, \alpha_{\ell}\})$

Corresponding R-spaces(1)
Hermitian Symmetric Spaces(2)Cont.act Manifolds$\Rightarrow \mathrm{C}\mathrm{o}\mathrm{m}\mathrm{l})\mathrm{a}\mathrm{c}\mathrm{t}$$\Rightarrow \mathrm{S}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{a}\mathrm{r}\mathrm{d}$$\cdot$$(.\delta.)$$(A_{\ell}, \{\alpha_{1}, \alpha_{i}\})\Rightarrow(J(\mathrm{P}^{\ell}, i-1),$$C)(C\ell, \{\alpha_{1}, \alpha\ell\})\Rightarrow(L(\mathrm{P}^{2\ell-1}), E)$

Q4. Geometry of PD-manifolds.
$R\mathrm{C}L(J)$.

$\mathrm{o}\mathrm{f}\mathfrak{g}=\oplus_{p\in \mathrm{Z}}\mathfrak{g}_{p}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}(\star)$$(X_{\ell}, \Delta_{1})\Rightarrow\Lambda I_{\mathfrak{g}}=G/G’:R$($\mathbb{C}sa\ovalbox{\tt\small REJECT} s\ovalbox{\tt\small REJECT} ing(\star)$$\mathfrak{g}\ovalbox{\tt\small REJECT}\oplus_{p\mathrm{E}\ovalbox{\tt\small REJECT} 1}\mathfrak{g}_{\mathrm{p}}\ovalbox{\tt\small REJECT}(X_{\ell}, \Delta_{1})$$\Leftrightarrow \mathrm{C}\mathrm{l}\mathrm{a}s\mathrm{s}\mathrm{i}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{a}\mathrm{t}\cdot \mathrm{i}\mathrm{o}\mathrm{n}$$(\Lambda I_{\mathfrak{g}}, D_{0})\supset$over:submanifold satisfying(R. )$\mathrm{t}$On $L(J)$ ,On R.,

$)$

$p$

:

$R$. $arrow’/$$C^{1}=\partial\prime E$,$D^{1}=C^{1}.|_{R},$;submersion,$C^{2}=E$$D^{2}=C^{2}|’$.

. satisfies :$(R;D^{1}, D^{2})$: codinl. 1,(R.1)$D^{1}$(R.2)$\partial D^{2}\subset D^{1}$,$D^{2}:$

codinl. $n+1$ ,

.

.

.

101
.

(R.3)

$Ch.(D^{1})\subset D^{2}:\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}$

$n$

,

(R.4)

$Ch.(D^{1})\cap Ch(D^{2})=\{0\}$.

Triplet $(R;D^{1}, D^{2})$ :$\mathrm{P}\mathrm{D}-\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{d}\Leftrightarrow(R.1)\sim(R.4)$$\{()\}=Ch(D^{1})\cap Ch.(D^{2})\subset Ch(D^{1})\subset D^{2}\subset\partial D^{2}\subset D^{1}\subset T(R)$

Realization Theorem for$\mathrm{P}\mathrm{D}$-manifold(i) (R. 1) and $(’R.3)\Rightarrow(J, C)$$J=R/Cl_{l}(D^{1})$ ,wheae$I^{J}$$D^{1}=p_{*}^{-1}(C)$,: $R.-arrow.J=R/Cl_{l}(D^{1})$ .

(ii) (R. 1) and $(R.2)\Rightarrow\iota(v)\cdot=p_{*}(D^{2}(v)(\subset C(u)$ : Legendrian(iii) $(R.4)\Rightarrow\iota:R\vec{.}L(J)$ :immersion

Theorem 5.1.
$.\Phi$: (R.;$\exists_{1}\varphi^{\wedge}:$$D^{1}$.

$D^{2}$)$arrow(\hat{R};\hat{D}^{1},\hat{D}^{2}):ison\iota r)rphis7n\Rightarrow$$(J, C)arrow(.\hat{J}.\hat{C_{r’}})$:contact $diff\rho.os.t.$ ;R.

$\underline{\iota}arrow L(J)$$\Phi\downarrow$$\downarrow\dot{(}\rho_{*}$$\hat{R}arrow’,\wedge L(\hat{J})$.

$\mathrm{C}\mathrm{o}\mathrm{n}1\mathrm{p}\mathrm{a}\mathrm{t}_{1}\mathrm{i}\mathrm{b}\mathrm{i}1\mathrm{i}\mathrm{f}\cdot \mathrm{y}\mathrm{C}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{C}\mathrm{l}\mathrm{i}\mathrm{f}\cdot \mathrm{i}\mathrm{o}\mathrm{n}(C)$$\acute{\backslash _{\backslash }}C)$

wltere$l)(1)$$R^{(\mathrm{J})}$: $R^{(1)}arrow R$ is $or|,to$ .

: the first )$\mathrm{r}o\mathrm{l}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$\mathrm{I}$of.(Rj$D^{1},$ $D^{2}$).

Theorerrt 5.2. $(R:D^{1}, D^{2}):PD$ -rnanifold $satisfyi_{7l}g$ the condition (C).
$\forall\iota;\in Il$:

dinl $D^{1}(8’.)-( \lim$$D^{2}(?.’,1=\mathrm{d}\mathrm{i}_{\mathrm{l}}\mathrm{n}Ch(D^{2})(v)$$Espec.\iota.a_{l}^{7},lyD^{1}=\dot{(}^{-})D^{2}\Leftrightarrow Ch(D^{2})=\{0\}$$\mathrm{I}_{\mathrm{I}1}$

wbere

case rank $Ch(D^{2})>()$ .
Geometry of.

.

$(R;D_{\backslash }^{1}D^{2})\Rightarrow \mathrm{G}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}$$X=R/Ch(D^{2}),\cdot D^{2}---\rho_{*}^{-1}(D),$ $\rho:Rarrow X$.

of $(X, D)$ ,102\S 5. Single Equations of Goursat Type.
$L(J\backslash )\supset R=\{F(x_{i}, z,p_{i},p_{ij})=0\}$:Hypersurface

R. is of (wea.k) parabolic t.ype at. each$\Leftrightarrow(\frac{\dot{c}J\Gamma^{2}}{\partial_{l^{y_{j}}j}}.

(R.

$\Leftrightarrow$(\cdot\iota,’))$$D^{\sim^{)}}.$: rarik 1at. each

) : regular of$\mathrm{t}.\mathrm{J}’\cdot \mathrm{I}$)$(^{\dot{\mathrm{r}}},$$\mathrm{s}.\mathrm{t},$. $p:Rarrow J$ ; submersion,$\mathrm{t}’.\in R$$\iota’\in R$$\epsilon$:$\epsilon=5_{-3}\oplus z_{-2}\oplus s_{-1}$ 、

where$5_{-\backslash }.;=\mathbb{R},$$\triangleleft=r_{-\dot{\grave{A}}}V^{*},$$\epsilon_{-1}=V\oplus \mathrm{f}\mathrm{f}\subset S^{2}(V^{*})$$’$$\Leftrightarrow\exists$

Cofra.nlC,;$(\mathrm{f})^{[perp]}=\langle e^{2}\rangle\subset S^{2}(V),$$e\in V$$\{\varpi_{:}\varpi_{a},\omega_{a}.\varpi_{1\alpha}.\varpi_{\alpha\beta}\}(1\leqq a\leqq n, 2\leqq\alpha\leqq\beta\leqq n)$

.

on

$R$

such

that
$D^{2}=\{\varpi=\varpi_{1}=\cdots=\varpi_{\mathit{7}l}=0\}$,$\{\begin{array}{l}d\varpi\equiv\omega_{\mathrm{l}}\Lambda\varpi_{\mathrm{l}}+\cdots\cdots\cdot+\omega_{n}\wedge\varpi_{n}(\mathrm{m}\mathrm{o}\mathrm{d}\varpi)d\varpi_{1}\equiv\omega_{2}\Lambda\varpi_{12}+\cdots+\omega_{n}\wedge\varpi_{1n}(\mathrm{m}\mathrm{o}\mathrm{d}\varpi,\varpi_{1},\cdots,\varpi_{n})d\varpi_{\alpha}\equiv\omega_{\mathrm{l}}\wedge\varpi_{\alpha \mathrm{l}}+\cdots\cdots\cdot-\vdash\omega_{n}\Lambda\varpi_{\alpha n}(1\mathrm{n}\mathrm{o}\mathrm{d}\varpi,\varpi_{\mathrm{l}},\cdots,\varpi_{n})\end{array}$where$lf–\cdot\leqq n$$\varpi_{\alpha}\rho=\varpi_{\theta\alpha}\varpi_{1a}=\varpi_{\alpha 1}.2\leq_{-}\alpha,$

is aequation of Goursat type
R. ;(weak) parabolic tyPe
is t.he, Monge system ;
where

.

$R$

$\Leftrightarrow$$\mathrm{s}.\mathrm{t}$. $M(E)$ ;completely integrable,$\mathrm{A}I(E)$$M(E)=\{\varpi=\varpi_{1}=\cdots=\varpi_{n}=.\omega_{\alpha}=\varpi_{1\alpha}=0 (2\leqq\alpha\leqq n)\}$.

Tbe First Order Covariaxit System $N(E)$$N=N(E)=\{\varpi=\varpi_{1}=0\}$

By Two)$\mathrm{S}\mathrm{t}_{1}.\mathrm{e}1$

Reductions

Geometry of $(R, D^{2})$ ;Goursat

where

.

$\mathrm{T}\mathrm{y}\mathrm{p}\mathrm{e}\Rightarrow \mathrm{G}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}$$\mathrm{Y}=R/Ch(N),$ $N=\rho_{*}^{-1}(D_{N}),$ $\rho:Rarrow Y$.

of$(\mathrm{Y}, D_{N});$Type$\mathrm{c}^{1}(n-1,2)$

,

103

$\alpha_{\ell-1}\alpha_{\ell}$$\alpha_{2}$$\alpha.\mathrm{l}$$B_{\ell}(\ell\geqq 3)$$A_{\ell}(l\geqq 2)$

0-$–\theta$

つ 2 .......

$2\Leftarrow$】$\alpha_{-\mathrm{l}}’\alpha_{t}$$\alpha_{1}$$C_{\ell}(\ell^{1}\geqq 2)$$F_{4}$$-\theta$$G_{2}$

誌$-.. \prod^{4\prime}\mathrm{a}_{3}$$2() \alpha_{2}.\alpha.\cdot\theta\frac{\overline{\mathfrak{v}}\prime 132}{4\alpha r\alpha\prime 0\alpha,-\alpha_{8}-}$$\ovalbox{\tt\small REJECT}$

Extended Dynkin DiagralIls with the coefficient of\S 6.

$G_{2}$$\mathrm{t}1_{1}\mathrm{e}$highest root

-geometry.

6.1. Standard Contact Manifolds. Lie Algebra over$\mathfrak{g}:\mathrm{S}\mathrm{i}_{111}\mathrm{p}1\mathrm{e}$$\theta$$\mathbb{C}$:Highest. Root$(X_{\ell}, \Delta_{\theta})$$(’J_{\mathfrak{g}\dot,- \mathfrak{g}}c_{})$: Contact. Gra.dation:Standard Contact$\Rightarrow$$\mathfrak{g}=\mathfrak{g}_{-2}\oplus \mathfrak{g}--\iota\oplus 90\oplus-.\mathfrak{g}_{1}\oplus \mathfrak{g}_{2}$$\mathrm{M}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{s}\Leftarrow \mathrm{B}\mathrm{e}$)$\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{b}\mathrm{y}$104[Projectiviation of the (c0-)aj0int orbit through the highest root vector]$\Delta_{\theta}\Leftrightarrow \mathrm{E}\mathrm{x}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{d}$62. Gradation of

Dynkin Diagram$G_{2}$.
$\theta=3\alpha_{1}+2\alpha_{2}$$\alpha_{1}^{\wedge}\Leftarrow\iota \mathrm{u}_{2}(_{-}..\backslash ,$.

$\Delta_{1}\subset\Delta=\{\alpha_{1\prime}\alpha_{2}\}$

(G1)

.

$\Delta_{1}=\{\alpha_{1}\}$$\mu=3$ ,$\mathrm{m}=\mathfrak{g}_{-3}\oplus \mathfrak{g}_{-2}\oplus \mathfrak{g}_{-1}$where cliIn$\mathfrak{g}_{-3}=\mathrm{d}\mathrm{i}\mathrm{n}1\mathfrak{g}_{-1}=2,$(G2)$\Delta_{1}=\{\alpha_{2}.\}$.

$\dim \mathfrak{g}_{-2}=1$$\mu=2$$\mathrm{m}=\mathfrak{g}_{-2}\oplus \mathfrak{g}_{-1}$

(G3)

.

$\Delta_{1}=\{\alpha_{1}, \alpha_{2}\}$.

$/\iota=5$: Contact Gradation,.

$\mathrm{m}=\mathfrak{g}_{-\delta}\ulcorner\oplus\gamma \mathfrak{g}_{-4}\oplus$ $\mathfrak{g}_{-3}\oplus \mathfrak{g}_{-2}\oplus \mathfrak{g}_{-1}$where$\dim \mathfrak{g}_{-1}=2$

Root Systemrankg$\dim \mathfrak{g}_{p}=1$$G_{1\mathit{1}}$:Simple Root System

consists of the following roots$\alpha_{1},$$(\chi_{2}$,$\alpha_{A}"+a_{1\backslash }.\alpha_{2}+2\alpha_{1},$$2\alpha_{2}+3\alpha_{1}$Typefor others.

$=\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{l}\mathfrak{h}=2$$\Delta=\{\alpha_{1},\alpha_{2}\}$$\Phi^{+}$

and$G_{2}$$\alpha_{2}+3\alpha_{1}$,105$(J_{\mathfrak{g}’ \mathrm{g}}C,)$:Standard Contact Manifold$\dim J_{\mathfrak{g}}=5$: Lagrange-Grassmann Bundle$L(J_{\mathfrak{g}})$$\mathrm{d}\mathrm{i}_{\mathrm{l}}\mathrm{n}L(J_{\mathfrak{g}})=8$

Orbits Decoinposition$L(J_{l})=O\cup R_{1}.\cup R_{2}$:Open orbit,

(1)

$O$

(2)

$R_{1}$: Codim 1, the Global Model of (B),(3,)

$R_{2}$:Codim 2, the Global Model of (A).
COlIll)a\iota *.t$R_{\mathit{2}}‘:$$=$$(G_{2}, \{\alpha_{1}, \alpha_{2}.\})$For Exceptiorial Simple Lie Algebras,$X_{\ell},$$\{\alpha_{G}\},1$:$\Delta_{\theta}---\{\alpha_{\theta}\}$$\Rightarrow$$X_{\ell}\not\cong A_{\ell}$($\exists_{1}$$\Leftrightarrow$

佳-2 $=V$$\mathfrak{g}_{-1}=W\otimes V^{*}$$\mathrm{d}\mathrm{i}_{\mathrm{l}}\mathrm{n}\mathfrak{g}_{-3}=2$,: 3next to$(\Lambda I_{\mathfrak{g}}, D_{\mathfrak{g}})$,$NI_{\mathfrak{g}}$,$\alpha c$‘$l^{\iota=\}}\backslash$$\mathfrak{g}_{-3}=\nu V$

i.e. ,(,$D_{\mathfrak{g}}$): regular of type$\Leftrightarrow$$(J_{\mathfrak{g}}, C_{\mathrm{g}}’)$$\mathrm{c}^{1}(.r\cdot, 2)$.

$(X_{\ell}, \{\alpha_{\theta}\})$$\uparrow$$=$$L(J_{\mathfrak{g}}.)\supset R_{\mathit{2}}$$(X_{p}, \{\alpha_{\theta}, \alpha_{G}\})$$\downarrow$(X. $D$ )$L(J_{\mathfrak{g}}\dot{)}\supset R_{1},$$\Leftarrow$$\Leftrightarrow$$(X_{\ell}, \{\alpha_{\mathrm{C}\mathrm{v}},\})$$(\Lambda I_{\mathfrak{g}}, \partial D_{\mathfrak{g}})$.

$\alpha_{\theta}$

106

[C1]
[C2]

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$va\uparrow^{\mathrm{Y}}iables$$\mathrm{E}.\mathrm{C}\mathrm{a}.\mathrm{r}\mathrm{f}\uparrow \mathrm{a}\mathrm{l}\cdot 1$$d$$\iota ux$$ir\iota co7\iota nue$

[Y1]

[Y2]

$\mathrm{K}.\mathrm{Y}\dot{‘}\iota \mathrm{m}\mathrm{a}.\mathrm{g}n\mathrm{c}^{\backslash }.1_{1}\mathrm{i},$$\cdot$$G_{2}$of Science,
Department of Mat.hematics,
Hokkaido Universit.y, Sapporo 060-0810Japan.sci.hokudai.ac.jp
E-nla.il:$\mathrm{F}\mathrm{a}\mathrm{c}\mathrm{u}\mathrm{l}\mathrm{t}\cdot \mathrm{y}$$\mathrm{y}\mathrm{a}.111\mathrm{a}.\mathrm{g}.\iota\iota \mathrm{c}\mathrm{h}_{\sim}’|.\mathrm{C}\dot{0},111\mathrm{a}\mathrm{t}\mathrm{h}$