数理解析研究所講究録
第 725 巻 1990 年 15-46

Maslov class of an isotropic map-germarising from one dimensional symplectic reduction
Goo ISHIKAWA

0. Introduction
Let$(M^{2\tau\iota}, \omega)$be a symplectic manifold of dimension $2n$ and $N^{n}\subset M^{2n}$ be a Lagrangiansubmanifold with singularities. For each regular point$x$of $N,$ $T_{x}N$ is a Lagrangian sub-space of the sympletic vector space $T_{x}M$ .
To investigate the local structure of $N$ near a singular point$x_{0}$

of $N$ , it is natural to

study the behavior of the distribution { $T_{x}N|x$ is a regular point of $N$ } near$x_{0}$. Then we

can grasp an invariant of the singularity, which is called the Maslov class in this paper.

Originary, the notion of Maslov class (or Keller-Maslov-Arnol’d class) is a global one,and it appears in context of asymptotic method of linear partial differential equation andsymplectic topology ([Al],[GS],[Gr],[H\"o],[Ma],[V],[W]).

Maslov classes represent obstruction for transversality of two Lagrangian subbundles.

Applying this, we define the Maslov class of isotopic mapping as an obstruction for extendability of a partially defined Lagrangian subbundle.
Related to variational problems, Arnol’d introduced a Lagrangian variety, so called

an open swallowtail, and invesigated it ([A3],[J]).

On the other hand, related to the problem of Lagrangian immersion of surfaces to four
dimensional symplectic manifolds, Givental’ introduced a Lagrangian variety, so called an
open Whitney umbrella, and investigated some local and grobal problems. Especially hecalculated the ‘’Maslov index” of an open Whitney umbrella.

Our results generalize this result of Givental’.
The purpose of this paper is to develop a general theory on objects containing La-

grangian varieties mentioned above.
Singular Lagrangian varieties appear typically in the process of symplectic reduction.

1

16

Let $M^{2(n+k)}$ be a sympectic manifold of dimension $2(n+k)$ , and $K^{2n+k}$ a coisotropic
submanifold of $M$ of codimention $K$ . Then naturally we have a symplectic manifold$M^{\prime 2}$“of dimension $2n$ , at least locally, and we call $M’$ a k-dimensional reduction of $M$ . Let$L^{n+h}$

be a Lagrangian submanifold (without singularities) of $M$ . Suppose $N=L\cap K$

is an n-dimensional submanifold of $K$ . If

$L$intersects transversely to $K$ , then we havean immersed Lagrangian manifold $L’$ in $M’$ . Otherwise, we have a singular Lagrangianvariety by reduction. (Open swallowtails are obtained in this situation; [A3],[G1],[J1]).
More generally, let $N$ “ be an isotropic submanifold of $M$ contained in $K$ . If a charac-

teristic direction of $K$ is tangent to $N$ , then, by reduction, we have a singular Lagrnagianvariety.
Notice that singular Lagrangian varieties obtained by reduction are parametrized byisotropic mappings.
We consider, in this paper, Maslov classes of isotropic mappings obtained by one

dimensional reduction process.

The first result is on vanishing of Maslov classes: The Maslov class of an isotropic

map-germ obtained by l-dimensional reduction of a Lagrangian manifold is zero (Theorem
6.1).

Thus, for a singularity of l-dimensional reduction of an isotropic manifold, the Maslov

class has a meaning of obstruction for representability as an intersection of a Lagrangiansubmanifold and a hypersurface.
In general, Maslov classes do not vanish. We give local model of singularities ofisotropic mappings generically obtained by l-dimensional reduction of isotropic submanifolds, up to local symplectic diffeomorphisms of the reduced symplectic manifold (Theorem9.1). These models are open whitney umbrellas of arbitrary dimension and their suspen-

sions, and their Maslov classes do not vanish (Theorem 10.3). Therefore we see that ageneric isotropic submanifold in a hypersurface of a symplectic manifold is not an intersection of a Lagrnagian submanifold and the hypersurface, $1o_{-}caUy$ at each point, where thecharacteristic direction tangent to the hypersurface (Corollary 10.4).

2

Contents: 1.Classical Maslov class $/2.Maslov$ class pf an isotropic map-ping /3.Symplectic equivalence $/4.Maslov$ class of an isotropic map-germ /5.Symplectic reduction /6.Reduction of a Lagrnagian manifold and Maslov
class $/7.Re$ duction of symplectic vector bundles $/8.Proof$ of Theorem 6.1 /9. Local models for generic l-dimnensional reductions $/10.Maslov$ class of anopen Whitney umbrella /11.Generating functions / 12. $A$ differential analy-

sis /13.Generic condition for isotropic submanifolds /14.Variety of singularisotropic jets / 15.Transversality / 16. $Proof$ of Thoerem 9.1 / 17.Universal
Maslov class /18.Calculation of an universal Masl $ov$ class $/19.Proof$ of Tho-erem 10.3

Throughout this paper, all manifolds and maps are of class$C^{\infty}$.

The author would like to thank to Professor Haruo Suzuki for valuable comments andto Keiko Ishikawa for nice typing of $AMS- \mathfrak{M}$ .

1. Classical Maslov class

Let
and

$\pi$

$(M^{2n}, \omega)$be a symplectic manifold, $N^{n}\subset M$ a Lagrnagian submanifold $(\omega|N=0)$ ,: $Marrow B$ a Lagrangian fibration. Then the symplectic vector bundle $E=TM|N$

has two Lagrangian subbundles $L=Ker\pi_{*}|N$ and $L’=TN$ .

In general, for a symplectic vector bundle

bundles$L$and $L’$ of

$E$

$E$

of rank $2n$ over $N$ and Lagrnagian sub-, the Maslov class $m(E;L, L’)\in H^{1}(N, Z)$ is defined as follows.

Consider the bundle A(E) over $N$ of Lagrangian subspaces of fibers of E. The La-

grangian subbundle $L’$ define a section $s(L’)$ : $Narrow\Lambda(E)$ by $s(L’)(x)=L_{l}^{t}\in A(Eae)\subset$$\Lambda(E),$$(x\in N)$

Let$\Omega$.

denote the symplectic form of E. Then there exist a $c6mplex$ structurea Hermitian form$G$

. Denote by

$g$

$G$

on $E,$ unique up to homotopy, such thatthe real part of

$G$

.
3

$\Omega$$J$

andis the imaginary part of

1

$S$

Let

be an orthonormal frame of$\{e_{1}, \ldots , e_{n}\}$respect to . Then$\{e_{1}, \ldots , e_{n}\}$

$g$

$L$

over an open subset$U$

of $N$ , withturns to be a unitary frame of the Hermitian vector bundle$(E;J, G)$ over $U$ . Then we have an isomorphism $E|U\cong C^{n}\cross U$ as Hermitian vector bundle,

which maps $L|U$ to $R^{n}\cross U$ .

Since $U(n)$ acts on the space A(C’) of Lagrangian subspac es oftransitively,$C^{n}$$\Lambda(C^{n})$is identified with the homogeneous space $A(n)=U(n)/O(n)$ , (see [A]).
Thus we have $\Lambda(E)|Uarrow\Lambda(n)$ , which is glued to a$A(n)$$C^{\infty}$

mapping $\Phi(L)$ : $\Lambda(E)arrow$.
Set $\Phi=\Phi(L)os(L’)$ : $Narrow\Lambda(n)=U(n)/O(n)$ . The homotopy type of$\Phi$

isindependent of the choice of $(J, G)$ .
If$\{e_{1}, \ldots, e_{n}’\}$

is an othonormal frame of $L’$ over$U$

, then, at $x\in U,$$e_{j}’= \sum_{i=1}^{n}a_{ij}e_{i}$,for some $A=(a_{ij})\in U(n)$ . Then $\Phi(x)=[A]\in A(n)$ . Remark that $G(e_{j}^{l}, e_{i})=a_{ij}$ .

Define $m(E;L, L’)$ to be the image of the generator of $H^{1}(S^{1}, Z)\cong Z$ , relatively tocounterclockwise orientation, under$(\det^{2}o\Phi)^{*}:$$H^{1}(S^{1}, Z)arrow H^{1}(N, Z)$,

where $\det 2$ : $U(n)/O(n)arrow S^{1}$ is defined by $\det^{2}([A])=(\det A)^{2},$ $A\in U(n)$ .
LEMMA 1.1. The Maslov class sa tisfi es followin $gpr$ operties:(0) $m(E;L, L)=0$ .

(1) $m(E;L, L’)+m(E;L’L”)=m(E;L, L”)$ .

(2)

$L$

and

$L$ ‘

(3) If th ere$L_{1},$$L_{1}’$

to$L_{2},$$L_{2}’$ar transverse in

$is$

$e$

$E$

, th en $m(E;L, L’)=0$ .

an isom orphism between symplectic vector bun dles

$E_{1}$

an$dE_{2}$

mappin$g$, then $m(E_{1}; L_{1}, L_{1})=m(E_{2} ; L_{2}, L_{2}’)$ .

(4) $m(h” E;h^{*}L, h^{*}L^{t})=h^{*}m(E;L, L’)\in H^{1}(P, Z)$ .
Here $L,$ $L,$

$L$ “

are agrangian subbundles of
$L$

$E$

and

$h$

: $Parrow N$ .

Remark 1.2. Vaisman [V] defines the Maslov classes $\mu_{h}(E;L, L’)\in H^{4h-3}(N, R),$ $(h=$$1,2,3,$$\ldots$), for two Lagrangian subbundles $L,$

such that4$L^{t}$

of a symplectic vector bundle$E$

over $N$ ,19(i) $\mu_{1}(E;L, L’)=(1/2)m(E;L, L’)\in H^{1}(N, R)$(ii)$\mu_{h}$satisfies the properties of Lemma 1.1.

Though Lemma 1.1 is well known (see[V]), we give here an elementary proof of thisfundamental lemma in this paper.
PROOF OF LEMMA 1.1: Since $\Phi=\Phi(L)os(L)$ is constant, we have (0).
To show (1), we will show the followings:(5) For Langrangian subbundles

dles$E_{1}$

and$E_{2}$$m(E_{2}, L_{2}, L_{2}’)$$L_{1},$

of$L_{1}^{t}$and$E_{1}$$L_{2},$$L_{2}’$

of$E_{2}$

of symplectic bun-over $N$ respectively, $m(E_{1}\oplus E_{2}; L_{1}\oplus L_{2}, L_{1}’\oplus L_{2}’)=m(E_{1}, L_{1}, L_{1}^{t})+$.

(6) For Lagrangian subbundles$L_{2})=m(E;L, L_{2}\oplus L_{1})$$L_{1},$$L_{2}$

of

$E$

and

$L$

of $E\oplus E$ respectively, $m(E;L,$.

If we have (5) and (6), then$m(E;L, L’)+m(E;L’, L^{tt})=m(E\oplus E^{t}, L\oplus L, L’\oplus L’’),$ $(by(5))$$=m(E\oplus E’, L\oplus L’, L^{t\prime}\oplus L’),$$(by(6))$$=m(E;L, L”)+m(E;L’, L’),$ $(by(5))$$=m(E;L, L^{t/}),$ $(by(O))$.

Thus we have (1).
To show (5), compare three maps$\Phi_{1}=\Phi(L_{1})os(L_{1}’)$: $Narrow A(n)$ ,$\Phi_{2}=\Phi(L_{2})os(L_{2}^{t})$: $Narrow\Lambda(m)$ ,and$\Phi_{3}=\Phi(L_{1}\oplus L_{2})os(L_{1}’\oplus L_{2}^{t})$where we set rank $E_{1}=2n$ and rank $E_{2}=2m$ .

5: $Narrow\Lambda(n+m)$ ,$L_{1}\oplus$

20

Define$\Delta$: $A(n)\cross\Lambda(m)arrow A(n+m)$by$\Delta([A], [B])=[(\begin{array}{ll}A OO B\end{array})]$,where we denote by [A] the class of $A\in U(n)$ in $\Lambda(n)=U(n)/O(n)$ .
Then$\Phi_{3}=\Lambda o(\Phi_{1}, \Phi_{2})$$(\det^{2}o\Phi_{1})\cdot(\det^{2}o\Phi_{2})$and $\det^{2}o\Lambda([A], [B])=\det^{2}([A])\cdot\det^{2}([B])$ . Thus $\det^{2}o\Phi_{3}=$: $Narrow S^{1}$ .

For the generator $1\in H^{1}(S^{1}, Z)$ ,$m(E_{1}\oplus E_{2}; L_{1}\oplus L_{2}, L_{1}^{t}\oplus L_{2}’)=(\det^{2}o\Phi_{3})^{*}1$$=((\det^{2}o\Phi_{1})\cdot(\det^{2}o\Phi_{2}))^{*}1$$=(\det^{2}o\Phi_{1})^{*}1+(\det^{2}o\Phi_{2})^{*}1$$=m(E_{1}; L_{1}, L_{1}^{t})+m(E_{2};L_{2}, L_{2}^{t})$Therefore we have (5).
To see (6), compare two maps$\Phi=\Phi(L)\circ s(L_{1}\oplus L_{2}):Narrow A(2n)$$\Phi’=\Phi(L)os(L_{2}\oplus L_{1})$Define$\kappa$,: $Narrow A(2n)$ ,: $\Lambda(2n)arrow\Lambda(2n)$ by$\kappa([(\begin{array}{ll}A BC D\end{array})])=[(\begin{array}{ll}B AD C\end{array})]$for$(\begin{array}{ll}A BC D\end{array})\in U(2n)$with $A,$ $B,$ $C,$ $D\in M_{n}(C)$ . Then $\kappa 0\Phi’=\Phi$ and,$\det^{2}o\kappa=\det^{2}$$m(\text{\‘{E}}; L, L_{1}\oplus L_{2})=(\det^{2}o\Phi)1$$=(\det^{2}o\kappa 0\Phi’)1$$=(\det^{2}o\Phi’)^{*}1$$=m(E;L, L_{2}\oplus L_{1})$6.

. Thus.

21

Therefore we have(6).
To see (2), remark that, if

$L$

and

$L^{t}$

are transverse, then we can choose $(J, G)$ suchthat $JL=L’$ . Then $\Phi=\Phi(L)os(L^{t})$ is constant, and $m(E;L, L^{t})=0$ .

For (3), suppose

tian structure$\tilde{\alpha}$$\alpha$: $(E_{1}; L_{1}, L_{1}’)arrow(E_{2}; L_{2}, L_{2}^{l})$ is an isomorphism. Choose a Hermi-$(\alpha oJo\alpha^{-1}, Go(\alpha^{-1}, \alpha^{-1}))$: $\Lambda(E_{1})arrow\Lambda(E_{2})$ . Then we have$\Phi(L_{1})os(L_{1}’)=\Phi(L_{2})os(L_{2}’)$

Let

$h$

: $Parrow N$ be a$\Lambda(h^{*}E)arrow\Lambda(n)$on$E_{2}$. Furthermore$s(L_{2}^{t})=\tilde{\alpha}os(L_{1}’)$and$\alpha$induces a diffeomorphism$\Phi(L_{2})0\tilde{\alpha}=\Phi(L_{1})$. Therefore, and $m(E_{1}; L_{1}, L_{1}’)=m(E_{2} ; L_{2}, L_{2}’)$ .

$C^{\infty}$

map. Consider $s(h^{*}L‘)$ : $Parrow\Lambda(h^{*}E)$ and $\Phi(h^{*}L)$ :. Then we have $\Phi(h^{*}L)os(h^{*}L’)=\Phi(L)os(L’)oh$ . Thus$m(h^{*}E;h\cdot L, h^{*}L’)=(\det^{2}o\Phi(h^{*}L)os(h^{*}L’))^{*}1$$=(\det^{2}o\Phi(L)os(L’)oh)^{*}1$$=h^{*}m(E;L, L^{t})$.

Thus we have (4). Q.E.D.

COROLLARY 1.3. $m(E;L, L^{t})=-m(E;L^{t}, L)$
PROOF: By Lemmal.1.(1),(2),$m(E;L, L’)+m(E, L^{t}, L)=m(E;L, L)=0$ .

Returning to the first situation, we define the Maslov class $m(N)$ by$m(N)=m(TM|N, Ker\pi_{*}|N, TN)\in H^{1}(N, Z)$

7

.

22

2. Maslov class of an isotropic mappingLet$(M^{2n}, \omega)$dimension

A$C^{\infty}$$n$

be a symplectic manifold of dimension $2n$ , and$N^{n}$

be a$C^{\infty}$

manifold of.

mapping$F$: $Narrow M$ is called an isotropic mapping if, for each $x\in N$ , theimage of $Taef$ : $T_{x}Narrow TaeM$ is an isotropic subspace of the symplectic vector space $T_{x}M$ ,that is, if $f^{*}\omega=0$ .
For an isotropic mapping

$f$

, set

$\Sigma=\Sigma(f)=${ $x\in N|T_{x}f$ is not injective}.

Then the restriction $f|N-\Sigma$ : $N-\Sigmaarrow M$ is a Lagrangian immersion.

Set $\Lambda(M)=\Lambda(TM)$ , and denote bysymplectic vector bundle $\pi^{*}TM$ over$\pi$

: $\Lambda(M)arrow M$ the canonical projection. In the$\Lambda(M)$, define the tautological bundle$\mathcal{L}_{(y,\lambda)}=\lambda\subset T_{y}M$$(y, \lambda)\in\Lambda(M)$$\mathcal{L}$

by

,

.

Associated to , define$f$$\varphi(f)$: $N-\Sigmaarrow A(M)$ ,by $\varphi(f)(x)=(f(x), {\rm Im}(T_{x}f))$ . Then $\pi 0\varphi(f)=f$ .

Set$N-\Sigma$$L_{f}=\varphi(f)^{*}\mathcal{L}$. Then$L_{f}$

is a Lagrangian subbundle of $f^{*}TM=\varphi(f)^{*}\pi^{*}TM$ over.

Definition 2.1. Assume $f^{*}TM$ has a Lagrangian subbundle (over $N$ ). Then define theMaslov class of$f$by$m(f)=\delta(m(f^{*}TM;L, L_{f}))\in H^{2}(N, N-\Sigma;Z)$

where$\Sigma;Z)$$L$,

is a Lagrangian subbundle of $f^{*}TM$ over $N$ , and 6: $H^{1}(N-\Sigma;Z)arrow H^{2}(N,$ $N-$is the coboundary map.

8

23Remark that $m(f^{*}TM, L, L_{f})\in H^{1}(N-\Sigma;Z)$ . For an another Lagrangian subbundle$L^{t}$

of $f^{*}TM$ over $N$ , we have$m(f^{*}TM;L’, L_{f})=m(f^{*}TM;L’, L)+m(f^{*}TM;L, L_{f})$ ,in $H^{1}(N-\Sigma;Z)$ , by Lemma 1.1.(2).

Since $m(f^{*}TM;L^{t}, L)$ comes from an elem ent of $H^{1}(N;Z)$ , we have$\delta(m(f^{*}TM;L^{t}, L_{f}))=\delta(m(f^{*}TM;L, L_{f}))$.

Therefore $m(f)$ is independent of the choise of L.
Remark 2.2. By Remark 1.2 and the argument as above, we can define a class $\mu_{h}(f)\in$$H^{4h-2}(N, N-\Sigma;Z)$ for an isotropic

mapping$f$: $Narrow M$ by$\mu_{h}(f)=\delta(\mu_{h}(f^{*}TM, L, L_{f}))$,$(h=1,2, \ldots)$ , provided $f^{*}TM$ has a Lagrangian subbundle L.

3. Symplectic equivalance
Let

$f$

: $N^{n}arrow(M^{2n}, \omega)$ andDefnition 3.1. A pairdiffeomorphism$\tau$$f^{l}$: $N^{tn}arrow(M^{t2n}, \omega^{t})$ be isotropic mappings.

$(\sigma, \tau)$: $Marrow M’,$of a diffeomorphism$(\tau^{*}\omega’=\omega)$

and $f’$ if $\tau of=f’o\sigma$ . Then we callIf$\sigma^{*}$$(\sigma, \tau)$$f$: $Narrow N^{t}$ and a symplectic, is called a sympectic equivalance between

is symplectically equivalent to , and write$f$is a symplectic equivalance between$f$and $f’$ , then

: $H^{2}(N’, N’-\Sigma’; Z)arrow H^{2}(N, N-\Sigma;Z))$ where$f^{*}TM$$\sigma$$\sigma$$\Sigma’=\Sigma(f^{t})$$\tau$$\tau’$$\tau^{\prime t}$: $\sigma^{*}f^{\prime*}TM’=f^{*}\tau^{*}TM^{t}arrow f^{*}TM$9.

, and $\sigma^{*}m(f’)=m(f)$ , if

induces isomorphisims: $\tau^{*}TM^{t}arrow TM$ and$f\sim f^{t}$incluces an isomorphism

has a Lagrangian subbundle L.

In fact,$f$

24of symplectic vector bundles over $M$ and $N$ respectively, and$\tau$“

maps$\sigma^{*}L_{f’}$to$L_{f}$. Thus$\sigma^{*}m(f^{l*}TM^{t}, \tau^{\prime\prime-1}L, L_{f’})=m(\sigma f^{t*}TM’; \sigma^{*}\tau^{\prime\prime-1}L, \sigma^{*}L_{f’})$$=m(f^{*}TM, L_{1}, L_{f})$for some Lagrangian subbundle$L_{1}$.

of $f^{*}TM$ , by Lemmal.1.(3) and (4). Therefore$\sigma^{*}m(f’)=\sigma^{*}6m(f^{t*}TM^{l}, \tau^{\prime t-1}L, L_{f’})$$=\delta(m(f^{*}TM;L_{1}, L_{f}))$$=m(f)$

4. Maslov class of an isotropic map-germ
Let

:

$f$

$N^{n},$ $xarrow(M^{2n},\omega)$such that$(f, U)$$f$

be a germ of an isotropic mapping. For each representative: $Uarrow M$ is isotropic and$U$

is a contractible neighborhood of , we$z$have $m(f, U)\in H^{2}(U, U-\Sigma;Z)$ , since $f^{*}TM$ is trivial overneighborhood of , with $V\subset U$ and$x$

restriction, then“$\iota$$\iota^{*}$

$U$

$U$

$V$

is a contractible: $H^{2}(U, U-\Sigma;Z)arrow H^{2}(V, V-\Sigma;Z)$ is the$(m(f, U))=m(f, V)$ by Lemmal.1.(3). Set$H^{2}(N, N- \Sigma, Z)_{x}=\lim_{arrow}H^{2}(U, U-\Sigma;Z)$

where

. If

runs over contractible neighborhoods of$x$,. Then we have an element$m(f)\in H^{2}(N, N-\Sigma;Z)_{x}$.

We call it the Maslov class of the isotropic map-germ$f$.

We can define the notin of symplectic equivalance between two isotropic map-germs
similarly as \S 3.
If

then

$(\sigma, \tau)$$\sigma^{*}$

is a symplectic equivalance between$f$and another$f^{t}$: $N’,$ $x^{t}arrow(M’, \omega’)$ ,: $H^{2}(N^{t}, N’-\Sigma^{t}, Z)arrow H^{2}(N, N-\Sigma;Z)$ maps $m(f’)$ to $m(f)$ .

10

5. Symplectic reduction

Let $(M^{2(n+k)}, \omega)$ be a symplectic manifold of dimension $2(n+k)$ , and $K^{2n+k}\subset M$ bea coisotropic submanifold of codimension$k$. We denote by$(TK)^{\perp}$

the skewothogonal

complement to $TK$ in $TM|K$ .

Remark that the rank of
$TK$

and

$(TK)^{\perp}$$(TK)^{\perp}$

is equal tois integrable ([AM]). We call$k$

. Since$(TK)^{\perp}$$K$

is coisotropic,$(TK)^{\perp}\subset$(resp. induced foliation on $K$ ) the

characteristic distribution (resp. foliation) relatively to $K$ .
Let $x\in K$ . Then, in an open neighborhood
induced, where $M$ ‘ is the leaf space. Thento symplectic diffeomorphisms of$M^{t}$$M^{t}$

such that$U$of in $K$ , a submersion
$x$

$\pi$

: $Uarrow M^{\prime 2n}$

has the unique symplectic structure$\pi^{*}\omega’=\omega|K$$\omega^{t}$up([AM]).

(1) By this reduction procesure, Lagrangian submanifolds of $M$ also reduced to ‘’La-

grangian varieties”.
Now, let $L^{n+k}\subset M$ be a Lagrangian submanifold and $x\in L$ . If $N=L\cap K$ is ann-dimensional submanifold of $K$ in a neighborhood of , then$x$$f=\pi|N$ : $N,$ $xarrow M’$

is an isotropic map-germ. In fact
$f^{*}\omega’=\pi^{*}\omega’|N=\omega|N=0$.

if and only if $T_{x}L\cap(T_{x}K)^{\perp}=0$ .

Remark that$f$is an immersion at

Especially, if$L$is transverse to $K$ , then we have an immersed Lagrangian submanifoldin the reduced symplectic manifold$M^{t}$$z$.

In fact, in this case, $T_{x}L\cap(T_{x}K)^{\perp}=T_{x}L\cap(T_{x}L+T_{x}K)^{\perp}=T_{x}L\cap(T_{x}M)^{\perp}=$$T_{x}L\cap 0=0$.

In general,$f$is not an immersion and has a singularity.

Defition 5.1. Let$f$be as in above. Then$f$

is called an isotropic map-germ arisingfrom a k-dimensional reduction of a Lagrangian manifold.

11

26

(2) More in general, let$K^{2n+k}$

be an isotropic submanifold of $M^{2(n+k)}$ contained in$N^{n}$and containing .
$x$Then $f=\pi|N$ : $N,$ $zarrow M’$ is isotropic and$(T_{x}K)^{\perp}=0$$f$

is immersive if and only if $T_{x}N\cap$.

Defnition 5.2. Such germ$f$is simply call $ed$ an isotropic map-germ arising from k-dimensional reduction.
In what follows, we concentrate to the case $k=1$ .

6. Reduction of a Lagrangian manifold and Maslov class
In \S 4, we have defined the Maslov class $m(f)\in H^{2}(N, N-\Sigma, Z)_{x}$ , for an isotropic map

germ

$f$

:

$N^{n},$$xarrow M^{2n}$THEOREM 6.1. Let
$eq$

$f$

, where$\Sigma$is the singular set of .
$f$

: $N,$ $xarrow M$ be an isot ropic map-germ. If$f$is symplecticallyuivalent to an isotropic map-germ arisin from a l-dim ensional reduction of a Lagrangian$g$

man ifold, then $m(f)=0\in H^{2}(N, N-\Sigma, Z)_{x}$ .

Precisely, for an open neighborh $oodU$ of , an for any respersentative
$x$

$y$

$d$

$f$

: $Uarrow M$

of , there exist a contractible open neighborh $oodV$ such that $x\in V\subset U$ and $m(f|V)=0$$f$

in $H^{2}(V, V-\Sigma;Z)$ .

12

27

7. Reduction of symplectic vector bundles(1) Let$E$be a symplectic vector bundle over a manifold $X$ , and $K$ be a coisotropic

subbundles. Then the bundle$K/K^{\perp}$

has the induced symplectic structure, where$K^{\perp}$

isthe skew-orthogonal complement of $K$ in $E,(see[AM],[W])$ .
Let

$L$

be a Lagrangian subbundle of E. lf $LCK$ , then $K^{\perp}\subset L^{\perp}=L\subset K$ , and$L/K^{\perp}\subset K/K^{\perp}$

is a Lagrangian subbundle.

LEMMA 7.1. Let
$E$

, and

$L$

(resp.

$E$$L$

be a symplectic vector bun $dle$ over $X,$ $K$ a coisotropic subbundle of‘) be a Lagrangian subbundle contained in K. Then$m(K/K^{\perp} ; L/K^{\perp}, L’/K^{\perp})=m(E;L, L^{t})$in $H^{1}(X;Z)$ , (cf. \S 1).
PROOF: Set rankE $=2(n+k)$ and rankK $=2n+k$ . Then $rankK^{\perp}=k$ .

Compare$\Phi_{1}=\Phi(L)os(L’)$: $Xarrow\Lambda(n+k)$ ,and$\Phi_{2}=\Phi(L/K^{\perp})os(L^{t}/K^{\perp})$

Set $\Lambda(n+k, k)=$ { $\lambda\in A(C^{n+k})|\lambda C$ C’
structure on$E$$\cross R^{k}$such that $\Phi_{1}(X)C\Lambda(n+k, k)$ and

C’ is the projection and$(\lambda\in\Lambda(n+k, k))$$\tilde{\pi}$: $Xarrow\Lambda(n)$ .

}. Then we can choose a Hermitian$\tilde{\pi}0\Phi_{1}=\Phi_{2}$: $\Lambda(n+k, k)arrow A(n)$ is defined by. Remark that.

$\det^{2}o\tilde{\pi}=\det^{2}$

Thus we have required result.

$t^{2}o$the situation of \S 5,(1).

13

$\pi$

: $C^{n}\cross C^{k}arrow$$\tilde{\pi}(\lambda)=\pi(\lambda)\subset C^{n}$,: $A(n+k, k)arrow S^{1}$ . Then $\det^{2}o\Phi_{2}=$$\det^{2}\circ\tilde{\pi}\circ\Phi_{1}=\det^{2}\circ\Phi_{1}$(2) We apply Lemma 7.1, where28

Shrinking $K$ , aroundcomes from a submisssion

$z$

$\pi$

if necessary, we assume that the characteristic foliation of $K$: $Karrow M’,$ $N=L\cap K$ is an n-dimential submanifold in $K$ ,and that $K$ and $N$ are contractible.

Set $E=TM|N-\Sigma,$ $K=TK|N-\Sigma,$ $K’=(TN)^{\perp}|N-\Sigma,$ $L=TL|N-\Sigma$ , and $L’=$$TN+(TK)^{\perp}|N-\Sigma$.

Notice that $TN+(TK)^{\perp}$ is a direct sum in $TM$ over $N-\Sigma$ . therefore $L’$ is a subbundleof$E$of rank $n+k$ . }furthermore $L^{t\perp}=K\cap K\supset L’$ . Therefore

Thus we have a symplectic vector bundle

$E$

$L$ ‘

is Lagrangian.

, coisotropic subbundles $K$ and$K^{t}$of rank$2n+k$ and $n+2k$ respectively, and Lagrnagian subbundles $L$ and $L$ ‘ with $L\subset K’,$ $L’CK$

and $L’\subset K’$ .

Since $M’$ is a symplectic reduction of $M$ relatively to $K$ , we have anisomorphism$\alpha$: $TK/(TK)^{\perp}arrow\pi^{*}TM^{l}$ ,which induces an isomorphism$\beta$: $TK/(TK)^{\perp}|Narrow f^{*}TM^{\iota}$ .

For each $y\in N-\Sigma,$ $\beta(L_{y}’/K_{y}^{\perp})=T_{y}f(T_{y}N)=(L_{f})_{y}$ in the fiber $(f\cdot TM^{t})$ over .
$y$

By restriction,$\beta$induces an isomorphism$\gamma$: $K/K^{\perp}arrow f^{*}TM^{t}|N-\Sigma$ ,such that $\beta(L^{t}/K^{\perp})=L_{f}$ .

Therefore, for a Lagrangian subbundle$L_{1}$

of $f^{*}TM’$ over $N$ , we have$m(f^{*}TM^{t};L_{1}, L_{f})=m(K/K^{\perp};\beta^{-1}(L_{1}), L’/K^{\perp})$,in $H^{1}(N-\Sigma;Z)$ , by Lemma 1.1.(3).
Take the Lagrangian subbundle$\beta^{-1}(L_{1})\subset(TK/TK^{\perp})|N$$L_{2}$

of $TM|N$ contained in $TK|N$ which projects to. Then, by Lemma 7.1,$m(K/K^{\perp};\beta(L_{1}), L’/K^{\perp})=m(E;L_{2}, L^{t})$

14

.

29

By Lemma 1.1.(1),$m(E;L_{2}, L’)=m(E;L_{2}, L)+m(E;L, L’)$ .

\S ince $L=TL|N-\Sigma$ is a restriction of the Lagrangian subbundle $TL|N$ over $N$ ,$m(E;L_{2}, L)$

is the restriction of an element in $H^{1}(N, Z)$ .

For these arguments are valid over any contractible neighborhood $V$ of$x$

in $N$ , wehave Theorem 6.1 if $m(E;L, L’)=0$ in $H^{1}(N-\Sigma, Z)ae$ .

Furthermore, by Lemma 7.1 again,$m(E;L, L’)=m(K’/K^{\prime\perp}; L/K^{i\perp}, L^{t}/K^{\mathfrak{l}\perp})$.

In the next section, we $wiU$ show the right hand side is equal to zero in $H^{1}(N-\Sigma, Z)_{x}$ ,at least if $k=1$ .

8. Proof of Theorem 6.1
It is sufficient to show $m(K’/K^{\perp}; L/K^{\perp}, L^{t}/K^{l\perp})=0$ in $H^{1}(V-\Sigma, Z)$ for any sufficiently

small contractible neighborhood
Let $h=0$ be a loc

$a1$

$V$

of , under the notations in \S 7,(2).
$x$equation of $K$ in $M$ , where $h\in C^{\infty}(M)$ . By the sign of

$h,$

$L-N$

is devided into two part: $L-N=L_{+}\cup L_{-},$ $L\pm=\{y\in N|\pm h(y)>0\}$ .
Take a vector field$v$

tangent to$L$tawarding to $L+overN$ . Then $dh(v)\geq 0$ .

Let $W$ be the Hamiltonian vector field with Hamilton$G(w, v)$ is equal to $\Omega(w, v)=(w\rfloor\Omega)(v)=-dh(v)\leq 0$

Remark that normalized$L^{t}/K^{t\perp})$, Therefore, for$\Phi$$v$$h$. Then the imaginary part of.

(resp. w) turns into an orthomormal frame of$L/K^{\perp}$(resp.

: $V-\Sigmaarrow\Lambda(1)=U(1)fO(1)$ in the definition of Maslov class,

we see $\det o\Phi$ : $V-\Sigmaarrow S^{1}CC$ has non-positive imaginary part. Therefore $\det o\Phi$ ishomotopically zero, and so is $\det^{2}o\Phi.$ Thus
$m(K’/K^{\perp} ; L/K^{\prime\perp}, L’/K^{\prime\perp})=(det^{2}o\Phi)^{*}1=0$

in $H^{1}(V-\Sigma;Z)$ for any sufficiently small contractible neighborhood

15

$V$

of

$x$

in $N$ . Q.E.D.

30

9. Local models for generic l-dimensional reductions

We consider a generic local classification by symplectic equivalances of isotropic mapping

arising from a symplectic reduction relatively to hypersurface (i.e. l-dimentioanl reduction),(see Definition 5.2).
Let $(M^{2n+2}, \omega)$ be a symplectic manifold, $K^{2n+1}$ be a hypersurface if $M$ , and$N^{n}$

bean n-dimensional manifold.
Denote by$\mathcal{I}$with the Whitney

the set of isotropic embeddings$C^{\infty}$$i$: $Narrow M$ with $i(N)\subset K$ , endowed

topology.

Next we prepare special isotropic map-germs$f_{n,k}$of isotropic mappings. Consider the cotangent bundle$q_{1},$$\ldots$space,$q_{n}$$R^{n}$;$p_{1}\ldots,p_{n}$as local modes for singularities$T$ “ $R^{n}$with canonical coordinates

and with the symplectic form $\omega=\sum_{i}dp:\wedge dq;$ . Besides, consider thewith coordinates$x_{1},$$f_{n,k}$$\ldots,$:$x_{n}$$R^{n},$. Then$0arrow T^{*}R^{n},$ $(0\leq k\leq[n/2])$

is defined by$q:^{of_{n,h}=x_{i},(1}\leq i\leq n-1)$,$u=q_{n} of_{n,k}=\frac{x_{n}^{h+1}}{(k+1)!}+\sum_{:=1}^{h-1}x;\frac{x_{n}^{k-i}}{(k-i)!}$$v=p_{n} of_{n,k}=\sum_{:=0}^{k-1}x_{k+i}\frac{x_{n}^{k-i}}{(k-i)!}$

,

,

and
$p_{j} of_{n,k}=\int_{0}^{n_{n}}(\frac{\partial v}{\partial x_{j}}\frac{\partial u}{\partial x_{n}}-\frac{\partial v}{\partial x_{n}}\frac{\partial u}{\partial x_{j}})dx_{n},$$(l\leq j\leq n-1)$ ,that is,$p_{j}=\{\begin{array}{l}-\sum_{i=0}^{k-1}x_{h+i^{\frac{W^{2k-\cdot-j}}{(h-i-1)!(k-j)’.(2k-i-j)}}},(1\leq j\leq k-1)\frac{x^{3k-j+1}}{h!(2h-j),.(Sh-j+1)}+\sum_{i=1}^{h-1}l_{i^{\frac{W^{3k-\cdot-j}}{(k-\cdot-1)!(2h-j)!(3k-\cdot-j)}}}..,(k\leq j\leq 2k-1)0,(2k\leq j\leq n)\end{array}$Remark that each$f_{n,k}$is a polynomial mapping of kernel rank one and of very simple

form.

16

31

THEOREM 9.1. There exists an open dense subset
for each $x\in N$ , the isotropic$m$

ap-germ$f$:$N^{n},$$\mathcal{G}$in$\mathcal{I}$such that, for each$xarrow M^{\prime 2n}$

red uction relatively to $K$ is symplectically equivalent to somean$i\in \mathcal{G}$$d$arising from l-dimensional$f_{n,k},$$(0\leq k\leq[n/2])$.

10. Maslov class of an open Whitney umbrella

Let us study properties of local models $hk$ :For $k=0,$

is just the zero-section$f_{n,0}$

For $k\neq 0$ , easily we verify$R^{n},$$\zeta_{n}$:$0arrow T^{*}R^{n},$ $(0\leq k\leq[n/2])$$R^{n},$ $0arrow T^{*}R^{n}$, in \S 9,

and is an immersion.

$\Sigma=\Sigma(f_{n,k})=\{\partial u/\partial x_{n}=\partial v/\partial x_{n}=0\}$, is a submanifoldof codimension 2 in $R$“. Thus we have$(*)$

: $H^{2}(R^{n}, R^{n}-\Sigma, Z)_{0}\cong H^{1}(R^{n}-\Sigma, Z)_{0}\cong Z$ .

By defnition, we can write $f_{n,k}=f_{2k,h}\cross\zeta_{n-2k}$ . Then

Definition 10.1.
Remark 10.2.

$f_{2n,n}$$f_{4,2}$$f_{n,k}$is a “suspension” of$f_{2k,k}$.

is called the n-dimensional open Whitney umbrella.

is just the (2-dimensional) open Whitney umbrella inroduced by

Givental’[G].
For Maslov classes, we have

THEOREM 10.3. Under th ident ification$e$$(^{*})$,$m(f_{n,k})=\{\begin{array}{l}0,k=0\pm 2,0<k\leq[\frac{n}{2}]\end{array}$

COROLLARY 10.4. For a generic, th at is belongin to
$g$

$m$

$at$

anifold$i$$\mathcal{G}$in Theorem 9.1, $iso$ tropic $su$ b-: $N^{n}arrow K^{2n+1}\subset M^{2n+2}$ , if $T_{x}N$ contains the$cA$aracteristic direction of $K$

a point $x\in N$ , then $N$ is ne $ver$ representable as an intersection of any Lagrangiansubmanifold and $K$ , as germ at$x$

PROOF: If $N$ is an intersection of a Lagrangian submanifold and $K$ , then the Maslov

17

32

class of isotropic map-germ arising from reduction relatively to $K$ necessarily vanishes, byTheorem 6.1.
By Theorem 9.1, that map-germ is symplectically equivalent to some$f_{n,k},$ $(k\neq

0)$.

By Theorem 10.3, $m(f_{n,k}),$ $(k\neq 0)$ , does not vanish. Combined with the argument in \S 4,
this leads a contradiction.

Q.E.D.

11. Generating functions

Let $g=$$(g_{1}, \ldots , g_{n})$:$R^{n},$$0arrow R^{n},$ $0$

an isotropic map-germ covering

projection. Set$\theta=\sum_{:}^{n_{=1}}p_{i}dq:$$g$

be a map-germ, and, that is, $\pi of=g$ , where, (Louiville form on

Pioncar\’e lemma, $de=f^{*}\theta$ , for some $e\in E_{n}$ , where

germs of$R^{n},$ $0$

function of$f$$T^{*}R^{n}$

:

$f$

$\pi$

). Then$R^{n},$$0arrow T^{*}R^{n}$be: $T^{*}R^{n}arrow R^{n}$ is the$f^{*}\theta$

is closed. Thus, bydenotes the R-algebra of function-$E_{\tau\iota}$. The function-germ (unique up to additive constant)$e$

is called a generating.

Denote by
LEMMA 11.1.

$H_{g}$

the set of generating functions of isotropic map-germs covering .
$g$$H_{g}=\{e\in E_{n}|de\in\langle dg_{1}, \ldots , dg_{n}\rangle E_{n}\}$.

PROOF: Let $e\in H_{g}$ . Then $de=f^{*}\theta$ , for some isotropic$de= \sum_{i=1}^{n}p_{i}ofdg_{i}$$f$: $R^{n}arrow T^{*}R^{n}$ . Then.

Conversely, suppose $de= \sum_{i=1}^{n}a:dg_{i}$ , for some $a:\in E_{n}$ . Then define
and $q_{i}of=g_{i},$ $(1\leq i\leq n)$ . then$f$is isotropic and covers

Definition 11.2. Two isotropic map-germs

$f$

$g$

$f$

by $p_{i}of=a$:.

and $f’$ :Lagrangian equivalent if there exist a diffeomorphism-germ$R^{n},$$\sigma$:$0arrow T^{*}R^{n}$$R^{n},$$0arrow R^{n},$ $0$-preserving symplectic diffeomorphism-germ $T:T^{\cdot}R^{n},$ $f(0)arrow T^{*}R^{n},$$\pi- fiber$that $f’o\sigma=Tof$ .

18

are called
and a$f^{t}(0)$

such

33

Definition 11.3. Let
$R$

$g$

and

$g’$

:

$R^{n},$$0arrow R$ “,

be map-germs and$0$

be function-germs. Then two pairs $(e, g)$ and $(e‘, g’)$ are called

exist diffeomorphism-germs$\sigma$and$\tau$:$R^{n},$ $0arrow R^{n},$

$0$

and

$e$

$R^{+}$$e’$:$R^{n},$$0arrow$-equivalent if there

are function-germ$\alpha$:$R^{n},$$0arrow R$

such that$e=e’o\sigma+\alpha og$, and $g^{t}o\sigma=\tau og$ .

(See [AVG].)PROPOSITION 11.4. Let

$f$

and

with generating functions$g^{t}$:$R^{n},$$0arrow R^{n},$ $0$

an $d(e^{t},g^{t})$ are$Fu$$R^{+}$

and$f^{l}$:-equivalent, then$R^{n},$$f$$0arrow T^{*}R^{n}$$e$be isotropic map-germs covering

an $de$ ‘ respecti vely. Then, if $(e, g)$

and $f’$ re agrangian equivalent.
$a$

$L$

rthermore, assume the critical point set-germ of$g’$is owhere dense. Then, if

an $df’$ are Lagrangian equivalent, then $(e, g)$ an $d(e’, g’)$ are

PROOF: First suppose

alence. Write

$T$

$f$

$R^{+}$$f$-equi valent.

$(\sigma,T)$

realizes this equiv-in the form

Then $\sum_{i}P_{*}\cdot d\tau_{i}-\sum_{i}p:dq_{i}=dA(p, q)$ , for some$(1\leq i\leq n)$$n$

and $f’$ are Lagrangian equivalent, and$(p, q)\mapsto(P(p,q),$ $\tau(q))$

$0,$

$g$

, we have$A=\pi^{*}\tilde{\alpha}$

for some:$\tilde{\alpha}$$R^{n},$

$A$

:

.

$T^{*}R^{n},$$0arrow R$$f(O)arrow$ R.

. ThusSince$\partial A/\partial p_{i}=$$T^{*}\theta=\theta+\pi^{*}d\tilde{\alpha}$.

Since $f’o\sigma=Tof$ we see)$d(\sigma^{*}e^{t})=\sigma^{*}f^{i*}\theta=f^{*}T^{*}\theta=f^{*}(\theta+\pi^{*}d\tilde{\alpha})$$=d(e+g^{*}\tilde{\alpha})$,Thus $e=eo\sigma-\tilde{\alpha}og+c$ for some $c\in R$ . then it suffices to set$(e, g)$and $(e^{t},g’)$ are$R^{+}$$\alpha=-\tilde{\alpha}+c$

to see-equivalent.

Next suppose $(e,g)$ and $(e’, g’)$ are$R^{+}$-equivalent, and$(\sigma, \tau, \alpha)$

alence. Define a fiber-preserving symplectic diffeomorphism-germ

19

$T$

realizes this equiv:$T^{*}R^{n},$ $T_{0^{*}}R^{n}arrow$34$T^{*}R^{n},$ $T_{0}^{*}R^{n}$covering$\tau$

by$T^{*}\theta-\theta=-d\pi^{*}\alpha$$(Tofo\sigma^{-1})^{*}\theta=\sigma^{-}$. Then,$f^{*}(\theta-d\pi^{*}\alpha)=\sigma^{-}$(de–dg’ )$\alpha$$=\sigma^{-1*}(d\sigma^{*}e’)=de’$$=f^{J*}\theta$

Both $Tofo\sigma^{-1}$ and$f^{t}$

cover$g^{t}$.

. Since the critical point set of

as easily verfied, each map-germ covering

Hence $Tofo\sigma^{-1}=f’$ .
Now, assume
Let

$f$

:

$g$

is nowhere dense,

is uniquely determined by the pull-back of .
$\theta$

Q.E.D.

is of form $g=(x’, u(x’, x_{n}))$ , where $x’=(x_{1}, \ldots, x_{n-1})$ .

$0arrow T^{*}R^{n}$$R^{n},$$g^{t}$$g’$

be an isotropic lifting of

LEMMA 11.5. For a genera ting function

$e$

$g$

. Set $v=p_{n}of$ . Then we haveof , there exists a function-germ $\alpha=\alpha(x’)$$f$such that$e(x’, x_{n})= \int_{0}^{x_{\mathfrak{n}}}v(x^{t},t)\frac{\partial u}{\partial t}(x^{t}, t)dt+\alpha(x’)$

PROOF: Since $de= \sum_{i=1}^{n-1}p:^{ofdx_{i}}+vdu$ , we have.

$\partial e/\partial x_{n}=v(\partial u/\partial x_{n})$. This leads therequired result.

Given $v\in E_{n}$ , with $v(x’, 0)=0$ , define$f_{v}$:$R^{n},$$0arrow T^{*}R^{n}$by $\pi of_{v}=g,p_{n}of_{v}=v$

and,$p_{j} of=\int_{0}^{x_{n}}\{\frac{\partial v}{\partial x_{j}}(x’,t)\frac{\partial u}{\partial t}(x^{l},t)-\frac{\partial v}{\partial t}(x’,t)\frac{\partial u}{\partial x_{j}}(x’,t)\}dt$$(1\leq i\leq n-1)$.

LEMMA 11.6. Let $v\in E_{n}$ with $v(x’, 0)=0$ . Then
more, any isotropic liftin$gf$ of
PROOF: Set$e_{v}= \int_{0}^{x}$“$g$$f_{v}$

is an isotropic lifting of . Further$g$with $p_{n}of=v$ is Lagrangian equi valen to$t$$v(x’, t)(\partial u/\partial t)(x’,t)dt$. Then we see$f_{v}$.

$\partial e_{v}/\partial x_{j}=p_{j}of_{v}+v(\partial u/\partial x_{j})$.

Thus we have $de_{v}=f^{*}\theta$ . Hence we have the first half. By Proposition 11.4 and Lemma

11.5, we have the second half.

20

35

Example 11.7. Set $g=(x_{1}, \ldots, x_{n-1}, u(x’, x_{\pi})),$ $u=x_{n}^{k+1}/(k+1)!+ \sum_{i=1^{X}}^{h-1}:x^{k-i}/(k-$$i)!$. Then $f_{n,k}=f_{v}$ , where$v= \sum^{h_{=^{-}0}1}x_{k+i}x_{n}^{k-*}/(k-i)!$. (See \S 9.)

12. A differential analysis

Consider the space$R^{n}\cross R$

with coordinates$R^{n}|y_{\tau+1}=\cdots=y_{n}=0\},$ $(0\leq r\leq n)$$y_{1},$$y_{\pi}$$\ldots,$; . Write $R‘=\{(y_{1}, \ldots, y_{n})\in$. Denote by

$t$

$j$

, : $R^{r-1}\cross Rarrow R$ ‘$\cross R$

the inclusion defined by $j,(y_{1}, \ldots, y,-1;t)=(y_{1}, \ldots, y_{\tau-1},0;t),$ $(1\leq 7\leq n)$ . Then, set$\phi_{\tau}=j_{n}o\cdots\circ j_{r+1}$: $R^{\tau}\cross Rarrow R^{\pi}\cross R,$ $(0\leq r\leq n-1)$ , andDenote by $E_{r+1}$ the R-algebra ofhas unique maximal ideal$m_{\tau+1}$$C^{\infty}$$\phi_{n}=id_{R^{n}\cross R}$.

function germs at $(0,0)$ in $R’\cross R$ . Then $E_{r+1}$

generated by$y_{1},$$\ldots$,and$y_{r}$$t$.

Now fix an element of $E.,+1$ ,$P=t^{k}/k!+a_{1}(y)t^{k-1}/(k-1)!+\cdots+a_{h}(y)$ ,which is a monic polynomial with respect to , and assume each$t$$a_{i}$

is an analytic map-germwith $a_{i}(0)=0,$ $(1\leq i\leq k)$ .
Then, set
$B_{\tau}=${$h\in E_{+1}|\partial h/\partial t$

is a multiple of$\phi:P$

in $E_{+1}$ }, $(0\leq r\leq n)$ ,$H_{l}(y,t)= \int_{0}^{t}\frac{s^{l}}{t!}P(y, s)ds\in E_{n+1},$ $(0\leq t\leq k)$,

and$\Gamma_{\tau}=$$(y_{1}, \ldots , y, , \phi:H_{0})$: $R^{\tau}\cross Rarrow R’\cross R,$ $(0\leq\gamma\leq n)$ .

PROPOSITION 12.1. We have$(a_{r})1,$ $\phi:H_{1},$$\ldots,$$\phi:H_{k}$genera $teB_{r}$ over$(b_{\tau})Kerj_{\tau+1}^{*}\subset\Gamma:_{+1}m_{r+1+1}B_{r+1}$$(c_{\tau})j_{\tau+1}^{*}$$E_{r+1}$, where$j_{\tau^{*}+1}$via$\Gamma$::$E_{\tau+1}arrow E_{r+1},$ $(0\leq r\leq n)$: $B.+1arrow B,,$ $(0\leq r\leq n-1)$ .

induces an isomorphism$j$: $B_{\tau+1}/r;_{+1}m_{\tau+1+1}B_{+1}\cong B_{\tau}/\Gamma:m_{\tau+1}B_{r}$ ,

21

.

36
of R-vec $tor$ spac es, $(0\leq r\leq n-1)$ .
PROOF: Step 1:$(a_{0})$.

It is easy to see that
$B_{0}=$

is generated by 1,
generated by 1,$(a_{0})$$t^{k+2},$$\phi_{0}^{*}H_{1},$,$\ldots$$\ldots,${$h(t)|\partial h/\partial t$

is a multiple ofover $E_{0+1}$ via$t^{2k+1}$

over $E_{0+1}$ via$\phi_{\dot{0}}H_{k}$$(t^{k+1})^{*}$$\Gamma_{0}^{*}$$t^{h}$}: $E_{0+1}arrow E_{0+1}$ . Thus

$B_{0}$

is: $E_{0+1}arrow E_{0+1}$ . Therefore we have.
Step 2:$(0\leq\gamma\leq n-1)$$(b_{\tau}),$.

Let $h(y_{1}, \ldots, y_{r}, 0;t)=0$ . Then $h=y_{r+1}g(y_{1}, \ldots, y_{r+1}’ t)$ for some $g\in E_{\tau+1+1}$ and. On the other hand,for some $w\in E_{r+1+1}$ . Set$y_{\tau+1}=0$ . Then $w(y_{1}, \ldots, y,,, 0, t)\phi^{*}P=0$ . Since
is not flat, we see. Thus. Therefore,
for some. Hence,. This implies $g\in B_{r+1}$ and $h\in r;_{+1}m_{\tau+1+1}B_{\tau+1}$ . We have$(0\leq r\leq n-1)$ .
$(0\leq r\leq n-1)$ .
Step 3:,For this, first we will show
. By
is surjective. In fact, let$\partial h/\partial t=y_{\tau+1}(\partial g/\partial t)$$\partial h/\partial t=w\phi:_{+}{}_{1}P$$w(y_{1}, \ldots, y_{\tau}, 0, t)=$$\phi:P$$0$$w=y_{\tau+1}\tilde{w}$$y_{r+1}(\partial g/\partial t-\tilde{w}\phi:_{+1}P)=0$$\tilde{e}\in E_{r+1+1}$$(b_{r})$$\partial g/\partial t=\tilde{w}\phi:_{+1}P$$(a_{\tau})+(b,)\Rightarrow(c_{r}),$$h= r;w_{0}+\sum_{\iota=1}^{k}\Gamma:_{+\iota^{w\phi:_{+1}H_{\ell}}}$for some$w_{0},$ $w_{1},$$\ldots,$$w_{k}\in m_{r+1}$

,

. Set

$\tilde{h}=\Gamma:_{+1}\tilde{w}_{0}+\sum_{l=1}^{k}\Gamma:_{+1}\tilde{w}.\phi:_{+1}H.$

where
$h$

$\tilde{w},$$\in m_{r+1+1}$. Therefore is$j$$surjectiv^{\rho}$. .

Then $j:_{+1}h= \sum$ $r;w$$j_{r+1}^{*}\tilde{u}$

.

$=u.$

,

is defined by $\tilde{w}.(y_{1}, \ldots, y_{r}, y,+r;t)=w.(y_{1}, \ldots , y_{r} ; t)$ . Then

Next we will show

that$(a_{r})$$h\in B_{\tau}$$j_{\tau+1}^{*}$

. Set

$j$

$j_{r+1}^{*}\tilde{h}=$.

is injective. For this, let $h\in B,+1$ with$\cdot u.,$

$(u$

.

$\in B_{r}, w$$\tilde{h}=\sum_{\ell’+1}\Gamma^{*}\tilde{w}.\tau^{\sim}\iota$.

.

. By $(a,)$ , there exists. Then$\in\Gamma_{+1}^{*}m,B,$$\in m_{\tau+1})$.

$j_{+1}^{*}h\in\Gamma:m_{\tau+1}B_{\tau}$$\tilde{u}$$\in B,+1$$j_{+1}^{*}\tilde{h}=j_{+1}^{*}h$

22

.

such

. Thus37$j_{\tau+1}^{*}(h-\tilde{h})=0$, and$h\in\Gamma:_{+1+1+1+1}m,B,$

Step 4:$h-\tilde{h}\in$$Kerj_{r+1}^{*}$. By$(b_{r}),$$h-\tilde{h}\in r:_{+\iota^{m}’+1+1}B_{+1}$. Hence.
$(0\leq r\leq n-1)$$(a_{\tau^{t}})+(c_{r})\Rightarrow(a,+1),$.

Remark that.

$j_{r+1}^{*}\phi:_{+1}H$

By$(a_{\tau}),$$1,$ $\phi:H_{1},$$\ldots,$$=(\phi_{\tau+1}oj_{r+1})^{*}H_{*}=\phi:$

generate$\phi:H_{k}$$B_{\tau}/\Gamma_{\tau}m_{\tau+1}B_{\tau}$

H., $(1\leq s\leq k)$ .

over R. Thus 1,$\phi:_{+1}H_{1},$$\ldots$,$\phi^{*}{}_{+1}H_{k}$generate $B_{+1}/r;_{+1}m_{r+1+1}B_{r+1}$ over $R$ , by $(c,)$ .
By [I], we see $B_{r+1}$ is a differentiable algebra in the sense of Malgrange. Then, wehave$(a_{\tau+1})$

by Malgrange’s preparation theorem ([M1]).

13. Generic condition for isotropic submanifolds
Let$\mathcal{I}$Definebe as in \S 9.
$\mathcal{G}_{1}\subset \mathcal{I}$exist a chart (V;$x_{1},$

by the following condition:$\ldots$, $x_{n-1},t$ ) around$x$$i\in \mathcal{G}_{1}$if and only if, for any $x\in N$ , there

and a symplectic chart $(U;p_{0}, \ldots , p_{n}, q_{0}, \ldots, q_{n})$around $i(x)$ such that (\dagger ) $i(V)\subset U,$ $K\cap U=\{p_{0}=0\},$$qjoi=x_{j}$ ,$1\leq j\leq n-1$$\omega|U=\sum_{j=0}^{n}dp_{j}\wedge dq_{j}$,$q_{\pi} oi=u(x’,t)=t^{k+1}/(k+1)!+\sum_{j=1}^{k-1}x_{j}t^{h-j}/(k-j)$$p_{\pi}oi=v(x^{t},t)$$k,$$(0\leq k\leq n)$, where $x‘=$.

It is easy to verify that$\mathcal{G}_{1}$Let $i\in g_{1}$ and $x\in N$ . Set$i^{*}\theta$!,,with $(\partial v/\partial t)(O, 0)=\cdots=(\partial^{h}v/\partial t)(0,0)=0$ , for some$(x_{1}, \ldots, x_{n-1})$

andis a closed l-form on$V$is open dense in .
$\mathcal{I}$$\theta=\sum_{j=1}^{n}p_{j}dq_{j}$

on

$U$

. Then $d\theta|K\cap U=\omega|K\cap U$ . Thus

. Therefore, for some function-germ

additive constant),$de= \sum_{i=1}^{n-1}p_{j}oidx_{i}+vdu$

23

.

$e$

: $N,$ $xarrow R$ (up to

38

Then

$\partial e/\partial t=v(\partial u/\partial t)$$(x_{1}, \ldots , x_{n-1}, u)$By$a_{0},$ $a_{1},$$(a_{n})$$\ldots$,$a_{k}$

Definepoint$e$belongs to$\dot{B}_{n+1}^{}$

introduced in \S 12, setting $P=$

, with respect to the coordinate $(tt)$ . Remark that, in this situation, $H_{0}=u$ and$\partial u/\partial t$$\Gamma_{\pi}=$. Thus$x$

is identified with $g=(q_{1}oi, \ldots , q_{n}oi)$ : $N,$ $xarrow R^{n},$ .
$0$of Proposition 12.1, $e=a_{0} og+\sum_{j=1}^{k}a_{j}ogH_{j}$ , for some function-germ:$R^{n},$$\mathcal{G}\subset \mathcal{I}$$0arrow R$.

by the following condition:, there exist a chart $(U;x^{l}, t)$ around$x$$i\in \mathcal{G}$if and only if$i\in \mathcal{G}_{1}$and, for eachand a symplectic chart $(U,p, q)$ around $i(x)$satisfying (\dagger ) and (\dag er \dag er ):,$dq_{1}\wedge\cdots\wedge dq_{k-1}\wedge da_{1}\wedge\cdots\wedge da_{k}\Lambda dq_{\pi}\neq 0$

at

$0$

in

$R^{n}$.

Remark 13.1. (1) The condition (\dag er \dag er ) is independent of the choice(2) The condition (\dag er \dag er ) is equivalent to that $a=(a_{1}, \ldots , a_{k})$ :$a_{0},$$R^{n},$$a_{1},$$\ldots,$$a_{k}$.

$0arrow R^{k},$ $0$

ia a

submersion and that $g(\Sigma^{1_{k},0}(g))=\{q_{1}=\cdots=q_{k-1}=q_{n}=0\}$ is transverse to $a^{-1}(0)$ ,1, . . ., 1,0
where$\Sigma^{1_{k},0}=\Sigma^{\bigvee_{k}}$is the Thom-Boardman symbol.

LEMMA 13.2. For any $zEN$ and for any isotropic embeddin $gi$ : $Narrow K$ belonging

to$\mathcal{G}$, the isotropic map-germ

symplec tically equivalen to
$t$

$f$

$f_{n,h}$at$x$arising from symplec $tic$ reduction relatively to $K$ isfor some

PROOF: Choose charts satisfying (\dagger ) and$k,$ $(0\leq$(\uparrow\dagger)$k\leq[n/2])$. Then define.
$\sigma$: $N,$ $xarrow N,$

$x$

by

$x_{i}o\sigma=\{\begin{array}{l}x_{i},(1\leq i\leq k-1,2k\leq i\leq n)a_{2h-i}og,(k\leq i\leq 2k-1)\end{array}$

Then $H_{j}o\sigma=H_{j}\circ’(1\leq j\leq k)$ , and $go\sigma=\tau og$ , for some diffeomorphism-germ$\tau$:$R^{n},$$0arrow R^{n},$ $0$.

Furthermore,,$e=a_{0} \circ g+\sum_{j=1}^{k}a_{j}\circ gH_{i}=a_{0}og+e^{t}o\sigma$

24

39

where,$e’= \sum_{j=1}^{k}x_{2k-};H_{j}=\int_{0}^{\ell}\{\sum_{j=1}^{k}x_{2k-};s^{j}/j!\}\partial u/\partial s(x’, s)ds$

is a generating function of$f_{n,h}$. Thus $(e,g)$ and $(e‘, g)$ are

grangian, hence symplectically, equivalent to$f_{n,h}$$R^{+}$-equivalent, and$f$is La-by Proposition 11.4.

14. Variety of singular isotropic jets
Let $N$ be a manifold of dimension$n$and $M$ be a symplectic manifold of dimension $2n$ . In

the k-jet bundle $J^{k}(N, M)$ , we set$J_{I}^{k}(N, M)=$

{ $j^{k}f(x)\in J^{k}(N,$ $M)|f$ :$N,$ $xarrow M$

isotropic},and$\overline{\Sigma}=${$j^{1}f(x)\in J_{I}^{1}(N,$$M)|f$ : $N,$ $xarrow M$ is not an immersion}.

Further set$\overline{\Sigma}^{i}=\{j^{1}f(x)\in J_{I}^{1}(N, M)|\dim KerT_{x}f=i\}$.

Then we have$\overline{\Sigma}=\bigcup_{i=1}^{n}\Sigma^{\sim}:$.

Set$\Sigma^{j}^{-}=\bigcup_{i=j}^{n}\Sigma^{i}^{\sim}$

PROPOSITION 14.1. The set$J_{I}^{1}(N, M)-\Sigma^{2}^{-}$a submanifold of $J^{1}(N, M)$ . Furth er,$\tilde{\Sigma}^{1}$.

of isotropic l-jets with kernel $dim$ ension

is a subman ifold of$J_{I}(N, M)-\Sigma^{2}^{-}$$\leq 2$isof codimension

2.
Set $V=Hom_{R}(R^{n}, C^{n})\cong M_{n}(C)$ .
Let

ture

$[\cdot, \cdot]’$$\langle\cdot, \cdot\rangle$

ondenote the standard Hermitian structure on$C^{n}$

by$[u, v]={\rm Im}\langle u, v\rangle,$ $u,$ $v\in C^{\tau\iota}$25$C^{n}$. Define the symplectic struc-. Let X $CV$ be the set of isotropic linear40

maps $R^{n}arrow C^{n}$ , anddimension

$i$

. Set

be the set of isotropic linear maps $R^{n}arrow C^{\pi}$ with kernel$\Sigma^{i}\subset X$$S^{j}= \bigcup_{:}^{n_{=j}}\Sigma^{i}$.

‘To prove Proposition 14.1, it is sufficient to show
LEMMA 14.2. $X$ is$a$real algebraic variety in$V$, with Sing(X)FurtI] er,$\subset S^{2}.$$\Sigma^{1}$is asubmanifold of codimension 2 in $X-S^{2}$ .
PROOF: Denote by $Alt(n)$ the set of skewsymmetric bilinear forms$R^{\pi}$and by $Sp(n)$ the group of symplectic linear isomorphisms on$a$: $R^{n}\cross R^{n}arrow R$ on$(C^{\pi}$, [, ] .
$)$Set $G=GL(n, R)\cross Sp(n)$ . Define G-actions on $V=Hom_{R}(R^{n}, C^{n})$ and $Alt(n)$ by,$(\sigma, \tau)t=\tau oto\sigma^{-1}$$(\sigma, \tau)a=ao(\sigma^{-1}\cross\sigma^{-1})$$((\sigma, \tau)\in G,$ $\ell\in V,$ $a\inConsider the map
Then$\rho$

Alt(n))$$\rho$,, respectively.

: $Varrow Alt(n)$ defined by $\rho(t)(u, v)=[tu, tv],$ $(t\in V, u, v\in R^{n})$ .

is a G-equivariant polynomial map and $X=\rho^{-1}(O)$ . Especially, $X$ is a real

analytic variety.

Let $t\in X$ . Then rank $(t)=i,$ $(0\leq i\leq n)$ if and only if there exists $g\in G$ ,.

$g\cdot t=(\begin{array}{ll}E.\cdot OO O\end{array})$

In fact, if rank(/) $=i$ , then there exists $\tau\in U(n)\subset Sp(n)$ , such that $\tau(imaget)=$$R^{i}\cross 0\subset C^{n}$$R^{i}\cross 0\subset C^{n}$. Thus, for some $\sigma\in GL(n, R),$
. The converse is clear.

$\tau oto\sigma^{-1}$

Remark that the matrix representation of$\rho$

is$A+\sqrt{-1}B\mapsto\rangle tBA-{}^{t}AB\in Alt(n)$

Let

$t$

be isotropic. Then$\rho$

is submersion at$t$,$A,$ $B\in

M_{n}(R)$.

if and only if kernel rank $(t)\leq 1$ .

To see this, we may assume$t=(\begin{array}{ll}E_{i} OO O\end{array})$26: $R^{n}arrow$ C’ is the projection to,41
without loss of generarity. The tangent map of$\rho$

at ,$t$$T_{l}(\rho):T_{1}Varrow T_{\rho(l)}Alt(n)$is described by$A’+\sqrt{-1}B’\vdash\star\ell B’t-\ell tB’=(\begin{array}{ll}{}^{t}B_{11}-B_{11} -B_{12}t_{B_{12}} O\end{array})$

,

where
$B’=(\begin{array}{ll}B_{11} B_{12}B_{21} B_{22}\end{array})$$B_{11},$ $B_{12},$ $B_{21}$and$B_{22}$,are real matrices of type $(i, i),$ $(i, n-i),$ $(n-i, i)$ and $(n-i, n-i)$

respectively.

Therefore

is surjective if and only if $i=n$ or $i=n-1$ .

$T_{l}(\rho)$

Q.E.D.

15. TransversalityLet$i\in \mathcal{G}_{1}$in the situation of \S 13 and $x\in N$ . Take a chart (V, ”, t) around

$x$

and a

symplectic chart $(U,p, q)$ around $i(x)$ satisfying (\dagger ). Define$f’$: $Varrow R^{\pi+1}$

by $f^{t}=$ $(q_{1}oi, \ldots , q_{n}oi;p_{\pi}oi)$ . Then we have

PROPOSITION 15.1. Let

extension$j^{\tau}f^{t}$$i\in \mathcal{G}_{1}$and $\prime r>[n/2]+1$ . Then

: $Varrow J^{r}(N, R^{n+1})$ of $f’$ is trans verse to$i\in \mathcal{G}$$\Sigma^{1_{l},0},$if and only if th r-jet$e$$(0\leq t<r)$ , for all $x\in N$.

(See $[B],[MatI].$ )PROOF: By assumption, $j’f’(x)\in\Sigma^{1_{h},0}$ for some . Under the notion of \S 13, $e=a_{0}og+$$k$$\sum_{j=1}^{k}a_{j}ogH_{j}$, and$\partial e/\partial t=v\partial u/\partial t$. Therefore, we have$p_{\pi} oi=v=(\partial a_{0}/\partial y_{n})og+\sum_{j=1}^{\pi}(\partial a_{j}/\partial y_{\pi})ogH_{j}+\sum_{j=1}^{h}a_{j}ogt^{j}/j!$

27

.

42

Since $j^{k+1}H_{j}(x)=0$ , whereto$\Sigma^{1_{k_{1}}0}$

at

$x$

$x$

is the origine of coordinates, we see

if and only if $k\leq[n/2]$ and$j^{k+1}f^{t}$is transverse$dq_{1}\wedge\cdots\Lambda dq_{k-1}\wedge da_{1}\wedge\cdots\wedge da_{h}\wedge dq_{n}\neq 0$$[B],[Matl])$ , and this is equivalent to that $j^{\tau+1}f’$ is transverse to$\Sigma^{1_{l},0},$$(0\leq t\leq r)$, (see, nearZ.

This is valid for all $x\in V$ . Thus we have required result.

Q.E.D.

16. Proof of Theorem 9.1
By Proposition 15.1,$\mathcal{G}$defined in \S 13 is open dense in

topology. Futher, by Lemma 13.2, we see$\mathcal{G}$$\mathcal{I}$endowed with the Whitney$C^{\infty}$

Q.E.D.

satisfies the required property.

17. Universal Maslov class
The calculation of Maslov classes of isotropic map-germ can be reduced to that in jet$s$spaces.

Define$\Psi$: $J_{I}^{1}(N, M)-\tilde{\Sigma}arrow A(M)$ by,$\Psi(j^{1}f(x))=T_{l}f(T_{x}N)\subset T_{f(x)}M$$(j^{1}f(x)\in J_{I}^{1}(N, M)-\overline{\Sigma})$.

Remark that $\Psi oj^{1}f=\varphi(f)$ , (see \S 2).
Definition 17.1. The universal Maslov class of an isotropic l-jet $z=j^{1}f(x)$ is definedby$m(z)=\delta(m(\Psi^{*}\pi^{*}TM;L, \Psi^{*}\mathcal{L}))\in H^{2}(J_{I}^{1}(N, M),$ $J_{I}^{1}(N, M)-\tilde{\Sigma},$ $Z)_{z}$

where

$L$

,

is a Lagrnagian subbundle of $\Psi^{*}\pi^{*}TM|U$ over a contractible neighborhood ofin $J_{I}^{1}(N, M)$ and$\mathcal{L}\subset\pi^{*}TM$LEMMA 17.2. Let$J_{I}^{1}(N, M)$$f$is the tautological Lagrangian subbundle over $A(M)$ .

: $N,$ $xarrow M$ be an is tropic map-germ.
$d$

Then

$j^{1}f$;$N,$ $xarrow$induces$(j^{1}f)^{*}$$z$: $H^{2}(J_{I}^{1}(N, M),$

$J_{I}^{1}(N, M)-\tilde{\Sigma},$28$Z)_{j^{1}f(x)}arrow H^{2}(N, N-\Sigma, Z)_{x}$,43which maps $m(j^{1}f(x))$ to $m(f)$ .
PROOF: We have$(j^{1}f)^{*}m(j^{1}f(x))=(j^{1}f)^{*}\delta m(L, \Psi^{*}\mathcal{L})$$=\delta m((j^{1}f)^{*}L, (\Psi\circ j^{1}f)^{*}\mathcal{L})$$=\delta m((j^{1}f)^{*}L, (\varphi(f))^{*}\mathcal{L})$$=\delta m((j^{1}f)^{*}L, L_{f})$$=m(f)$ .

18. Calculation of an universal Maslov classPROPOSITION 18.1. Let$z\in\tilde{\Sigma}^{1}$. Then$H^{2}(J_{I}^{1}(N, M),$ $J_{I}^{1}(N, M)-\overline{\Sigma};Z)_{z}\cong Z$,

and $m(z)=\pm 2$ .

PROOF: The first half is clear from Proposition 14.1.

Without loss of generarity, we may assume $z=j^{1}f(0)\in J_{I}^{1}(R^{\pi},T^{*}R^{n})$ with $q_{i}of=$$x_{i},$$(1\leq i\leq n-1),$ $q_{n}of=0,p:^{of}=0,$ $(1\leq i\leq n)$. Define$c:R^{2}arrow J_{I}^{1}(R^{n},T^{*}R^{\pi})$$c(t, \epsilon)=j^{1}(x_{1}, \ldots, x_{n-1}, tx_{n};0, \ldots, 0, sx_{n})(0)$$(t, s)\in R^{2}$. Then $c(O)=z$ and is transverse to$c$Take a $smaU$ loop$\ell_{\epsilon}$tois a generator of$\ell_{e}$$t_{e}$$\Psi 0\ell_{e}$,.

: $S^{1}arrow J_{I}^{1}(R^{n}, T" R^{n})$ , where$H_{1}(J_{I}^{1}(R^{n}, T^{*}R^{\pi})-\tilde{\Sigma},$ $Z)_{z}$. Remark that$\overline{\Sigma}^{1}$by$t_{e}(e^{i\theta})=c(e\cos\theta, \epsilon\sin\theta)$. Then. Thus $|m(z)|$ is determined by the evaluationis represented by$e^{i\theta}rightarrow(\begin{array}{ll}E_{n-1} OO e^{i\theta}\end{array})\in U(n)$Thus $\det^{2}o\Psi ot_{e}$ : $S^{1}arrow S^{1}$ is of degree 2. Therefore, $|m(z)|=2$ .

29

44

19. Proof of Theorem 10.3

LEMMA 19.1. Let$\overline{\Sigma}^{1}$$f$: $N,$ $xarrow M$ be isotropi . If$c$$j^{1}f(x)\in\tilde{\Sigma}^{1}$and$j^{1}f$is transverse toin $J_{I}^{1}(N, M)$ . Then
$H^{2}(N, N-\Sigma;Z)_{x}\cong Z$

and $m(f)=\pm 2$ , where
PROOF: Since$j^{1}f$$\Sigma=\Sigma(f)=(j^{1}f)^{-1}(\overline{\Sigma})$

is transverse to$\tilde{\Sigma}^{1}$

, and

,

.

$\Sigma=(j^{1}f)^{-1}(\overline{\Sigma}^{1})$, we see$\Sigma$

is a submanifold ofcodimension 2 in $N$ near , and$x$$(j^{1}f)^{*}$: $H^{2}(N, N-\Sigma;Z)\cong H^{2}(J_{I}^{1}(N, M),$$\cong Z$$J_{I}^{1}(N, M)-\tilde{\Sigma};Z)_{j^{1}f(x)}$.

Since $m(j^{1}f(x))=\pm 2$ by Proposition 18.1, we have$m(f)=(j^{1}f)^{*}m(j^{1}f(z))=\pm 2$,

relatively to the above isomorphism.
LEMMA 19.2.

$j^{1}f_{n,h},$ $(k\neq

0)$, is transverse to$\tilde{\Sigma}^{1}$in$J_{I}^{1}(R^{n}, T^{*}R^{n})$at $j^{1}f_{n,k}(0)$ .

PROOF: It is clear, by checking 2-jets.
Now, Theorem 10.3 is clear from Lemmata 19.1 and 2.

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Department of Mathematics,

Faculty of Science,
Hokkaido University,

32