数理解析研究所講究録 1328 巻 2003 年 126-143

126

Tangentially degenerate fronts andtheir singularities: Asurvey.
GO-o ISHIKAWA
Department of Mathematics, Hokkaido University,
Sapporo 060-0810, JAPAN.
石川剛郎 (いしかわ・ごうお), 北海道大学理学研究科*

1

Introduction.

Tangentially degenerate submanifolds in projective spaces are studied from various
aspects; differetial geometry, algebraic geometry, singularity theory and so on. In
particular, P. Griffith and J. Harris [18] and A. Akivis and. Goldberg [2] [3] gave
the description of tangentially degenerate submanifolds in detail.
Looking at unit normal vectors or tangent planes to space surfaces is the most
. Gauss [16]. He, infundamental method in differential geometry initiated byparticular, considered the class of tangentially degenerate surfaces by means of his(Gauss) mappings.
Naturally we are led to consider tangentially submanifolds in Euclidean spaces, ormore naturally in projective spaces by means of Gauss mappings. One of importantclasses of tangentially degenerate submanifolds, then, consists of submanifolds withdegenerate Gauss mappings. Another important class consists of hypersurfaces withdegenerate projective dual. The tangential degeneracy of ahypersurface can be described by the deneneracy of its projective dual; the variety, in the dual projective
space, consisting of tangent hyperplanes to the hypersurface. Moreover we notice
that, also for submanifolds of codimension greater than two, the tangential degeneracy can be described by means of projective duality. This means that the Gauss$\mathrm{V}.\mathrm{V}$$\mathrm{C}.\mathrm{F}$Key words: -degenerate Legendre submanifold, projective duality, Ferus inequality, cuspidaledge, open folded umbrella, open swallowtail, frontal mapping, Grassmann duality.
2000 Mathernatics Subject Classification: Primary;Secondly,$\mathrm{b}\mathrm{i}$$58\mathrm{K}40$$55\mathrm{Q}25,57\mathrm{R}25,57\mathrm{R}45$$53\mathrm{C}15,53\mathrm{C}30,53\mathrm{D}10$.

’Partially supported by Grants-in-Aid for Scientific Research, No. 14340020.

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mapping is degenerate, then the projective dual is necessarily degenerate [18]. Thus
among tangentially degenerate submanifolds, we study, in this paper, submanifolds
with degenerate projective duals, possibly with singularities.
The notions of projective duality and of incidence relation play the central rolein projective geometry. In this survey article, on particular, we re–formulate thestudy on submanifolds with degenerate Gauss mappings using the incidence relationin projective geometry via contact geometry. Also we introduce the notion of “frontal
mappings” and discuss the relations with “poly-symplectic geometry”. Moreover the
projective duality is generalized to “Grassmann duality”in very natural way.

2Degenerate andifolds.

$\mathrm{b}\mathrm{i}$-degenerate Legendre subman-

We denote by
the -dimensional projective space and by
the -dimensional dual projective space. Heremeans the dual
vector space to.
Any submanifoldof thelifts to aLegendre submanifoldof contact elements (tangent hyperplanes) of. Actually $M$ isdefined to be the projective conormal bundleof $M$ . Hereis the conormal bundle of $M$ in. Note that, independently of $m$ , thedimension of the Legendre lifting
is equal to . In general the image of aLegendre
submanifold by the projection :is called awave front or
simply afront. Therefore any submanifold of
can be regarded as afront. This
is not the case just only for: any submanifold $M$ of any manifold $X$ lifts to a
Legendre submanifold $P(T_{M}^{*}X)$ of $P(T^{*}X)$ .
The special feature ofis
has natural double Legendre fibration:$\mathrm{R}P^{n+1}=P(\mathrm{R}^{n+2})$$P((\mathrm{R}^{n+2})^{*})$$\mathrm{R}P^{n+1*}=$

$n$

$n$

$(\mathrm{R}^{n+2})^{*}$$\mathrm{R}^{n+2}$$\overline{M}$$M^{m}\subseteq \mathrm{R}P^{n+1}$$\mathrm{m}\mathrm{a}\mathrm{n}\underline{\mathrm{i}\mathrm{f}\mathrm{o}}\mathrm{l}\mathrm{d}$$P(T^{*}\mathrm{R}P^{n+1})$$\mathrm{R}P^{n+1}$$P(T_{M}^{*}\mathrm{R}P^{n+1})$$T^{*}\mathrm{R}P^{n+1}$$T_{M}^{*}\mathrm{R}P^{n+1}\subseteq$$\mathrm{R}P^{n+1}$$\overline{M}$

$n$

$\pi$

$P(T^{*}\mathrm{R}P^{n+1})arrow \mathrm{R}P^{n+1}$$\mathrm{R}P^{n+1}$$\mathrm{R}P^{n+1}$$\mathrm{R}P^{n+1}$$P(T^{*}\mathrm{R}P^{n+1})$$\mathrm{R}P^{n+1}arrow P(T^{*}\mathrm{R}P^{n+1})arrow \mathrm{R}P^{n+1*}$,

and to the dual projective space.
Inversing the process, first we can consider Legendre submanifolds in the manifoldof cotact elements, the projective cotangent bundle, then second study.
their projections by ::and byThe above constructions can be described in term of projective duality. Set

to$\mathrm{R}P^{n+1}$$\mathrm{R}P^{n+1*}$$P(T^{*}\mathrm{R}P^{n+1})$$\pi$$P(T^{*}\mathrm{R}P^{n+1})arrow \mathrm{R}P^{n+1}$$\pi^{*}$$P(T^{*}\mathrm{R}P^{n+1})arrow \mathrm{R}P^{n+1*}$$\tilde{Q}:=\{(x, y)\in \mathrm{R}^{n+2}\cross(\mathrm{R}^{n+2})^{*}|x\cdot y=0\}$

where$x\cdot y$denotes the canonical pairing of elements$x\in \mathrm{R}^{n+2}$,

and$y\in(\mathrm{R}^{\tau\iota+2})*$

.

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On

$\tilde{Q}$, we have $0=d(x\cdot y)=dx\cdot y+x\cdot dy$ . Moreover we set$Q:=\{([x], [y])\in \mathrm{R}P^{n+1}\cross \mathrm{R}P^{n+1*}|x\cdot y=0\}$

the manifold of incident pairs or the incidence relation. Thenand has the contact structure$Q$,

is of dimension $2n+1$$Q$$D:=\{dx\cdot y=0\}=\{x\cdot dy=0\}\subset TQ$.

Namely, atangent vector $(u, v)\in T([x],[y])Q$ belongs to the contact distribution $D$ if
and, if and only if $x\cdot v=0$ .
axid only if: $Qarrow \mathrm{R}P^{n+1*}$ ) indentify , as
: $Qarrow \mathrm{R}P^{n+1}$ (resp.
The projection(resp.
ofcontact manifolds, with the fiberwise projectivization).
ofAsubmanifold $L\subset Q$ is called a Legendre submanifold if is an integral submanifold of the contact distribution $D$ of dimension . The integrality condition means$u\cdot y=\mathrm{O}$$Q$$\pi^{*}$$\pi$$T^{*}\mathrm{R}P^{n+1}$$P(\Gamma \mathrm{R}P^{n+1})$$T^{*}\mathrm{R}P^{n+1*}$$P(T^{*}\mathrm{R}P^{n+1*})$

$L$

$n$

.
Now, to any submanifold $M$ of
the Legendre submanifold in :that

$TL\subset D|_{L}$$\mathrm{R}P^{n+1}$

of any codimension $m$ , there corresponds$Q$$\overline{M}:=\{([x], [y])\in Q[x]\in M, |(T_{x}\overline{\mathrm{J}/I})\cdot y=0\}$

which is called the Legendre lifting of $M$ . Here.
$(m+1)$ -dimensional submanifold to$N$ ofof any codimension
Also to any submanifold
the Legendre submanifold in :,is the corresponding$\overline{M}\subseteq \mathrm{R}^{n+2}\backslash \{0\}$$M\subseteq \mathrm{R}P^{n+1}$$\mathrm{R}P^{n+1*}$$m^{*}$, there corresponds$Q$$\tilde{N}:=\{([x], [y])\in Q[y]\in N, |(x\cdot T_{y}\hat{N})=0\}$,is the correwhich is also called the Legendre lifting of $N$ . Here$(m^{*}+1)$.
-dimensional submanifold tosponding) is, by definition, the image of by(resp. in
Afront of in$\hat{N}\subseteq \mathrm{R}^{n+2*}\backslash \{0\}$$N\subseteq \mathrm{R}P^{n+1*}$

$L$

$L$

$\mathrm{R}P^{n+1*}$$\mathrm{R}P^{n+1}$

$\pi$

).
(resp.
) can be regarded as afront in(resp.
Thus any submanifold of) of aLegendre submanifold of . However affont may have
(resp. insingularities, which also we are interested in.
$\pi^{*}$$\mathrm{R}P^{n+1}$

Let$\mathrm{R}P^{n+1*}$$\mathrm{R}P^{n+1*}$$\mathrm{R}P^{n+1}$$L\subset Q$$Q$

be aLegendre submanifold. Set$m:= \sup\{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}_{q}(d(\pi|_{L}) :T_{q}Larrow T_{\pi(q)}\mathrm{R}P^{n+1})|q\in L\}$.

Moreover set$m^{*}:= \sup\{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}_{q}(d(\pi^{*}|_{L}) :T_{q}Larrow T_{\pi^{\mathrm{r}}(q)}\mathrm{R}P^{n+1*})|q\in L\}$

.

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We call$L$$m^{*}<n$ .

degenerate if $m^{*}<n$ . Moreover we call $Lbi$ -degenerate if $m<n$ and

Now we call afont in(resp. in) tangentially degenerate or briefly
degenerate if $m^{*}<n$ (resp. $m<n$ ). Moreover we call afront in(resp. in$m^{*}<n$tangentially -degenerate or briefly -degenerate if bothand $m<n$ .
$\mathrm{R}P^{n+1}$$\mathrm{R}P^{n+1*}$$\mathrm{R}P^{n+1}$$bi$$\mathrm{R}P^{n+1*})$$bi$

Example 2.1 Let $n,$ $m$ be integers with $0\leq m\leq n$ Let $M=\mathrm{R}7"\subset \mathrm{R}P^{n+1}$ be a
projective subspace of dimension $m$ . We denote by
the projective dual
to $M;M^{\vee}$ consists of hyperplanes containing $M$ , andis aprojective subspace ofof dimension $n-m$ . Set $L:=M\cross M^{\vee}\subset Q$ . Then is the Legendre lifting$M$of . Then is degenerate if and only if $0<m\leq n$ . Moreover is -degenerate if
and only if $0<m<n$ .
$M^{\sqrt}\subset \mathrm{R}P^{n+1*}$$NI^{\vee}$$\mathrm{R}P^{n+1*}$

$L$

$L$

$L$

$\mathrm{b}\mathrm{i}$Example 2.2 Let $M^{m}\subset \mathrm{R}P^{n+1},0\leq m\leq n$ , be asubmanifold with degenerate
Gauss mapping. Recall that the Gauss mapping :is defined$([x]\inM)$
by. Then the required condition is that ranky $<m$ . Thus
we are assuming $0<m\leq n$ . Lots of examples have been found of submanifolds
with degenerate Gauss mappings [27][29]. Let be the Legendre lifting of $M$ . We
have $M=\pi(L)$ and $\pi^{*}(L)=:M^{\vee}\subset \mathrm{R}P^{n+1*}$ is the projective dual of $M$ . Thenis degenerate. Moreover is -degenerate if $m<n$ . In other words, asubmanifoldwith degenerate Gauss mapping is adegenerate ffont. Moreover if it is of codimension, then it is a -degenerate front.
$\gamma$$Marrow \mathrm{G}\mathrm{r}(m+1, \mathrm{R}^{n+2})$$\gamma([x]):=T_{x}\overline{M},$

$L$

$L$

$L$

$\geq 2$$\mathrm{b}\mathrm{i}$$\mathrm{b}\mathrm{i}$Example 2.3 Let.
be acomplex submanifold of complex dimension
Consider the Hopf fibration :. Set $M:=h^{-1}N\subset \mathrm{R}P^{2n+1}$ . Then
$M$ is a
submanifold of real dimension $21+$ 1with degenerate Gauss mapping.
$L:=M\subset
Q\subset\mathrm{R}P^{2n+1}\cross \mathrm{R}P^{2n+1*}$ beLet
the Legendre lifting of $M$ . Thenis -degenerate. In fact
, for the complex projective dual
and the Hopf fibration. Now suppose $W$ is anon-singularcomplex quadric hypersurface in. Thenis anon-singular complex quadrichypersurface in. Then bothandare of constant rank $2n-1$ . In thisexample $m=2n-1=m^{*}$ and $m+m^{*}-2n=2n-2$ . If $n=2$ , then $m=3=m^{*}$ ,$\dim L=4$ and $m+m^{*}-4=2$ .
$W\subset \mathrm{C}P^{n}$$h$$\ell\leq n$$\mathrm{R}P^{2n+1}arrow \mathrm{C}P^{n}$$\mathrm{r}\mathrm{e}\mathrm{a}\underline{1}$$L$$\pi^{*}(L)=\overline{h^{*}}W^{\vee}1$$\mathrm{b}\mathrm{i}$$W^{\vee}\subset$$\mathrm{R}P^{2n+1*}arrow \mathrm{C}P^{n*}$$\mathrm{C}P^{n*}$$\mathrm{C}P^{n}$$\mathrm{C}P^{n*}$$\pi|_{L}$$W^{\vee}$$\pi^{*}|_{L}$In the last exaanple, we have observed the Legendre submanifold has the constantrank projections
andso that $\pi(L)$ andare both non-singular degenerate$\pi|_{L}$$\pi^{*}|_{L}$$\pi^{*}(L)$130

3Symmetric Ferus inequalities for degenerate Legendre submanifolds.
In this section, we give aformulation of Ferus inequality [14] [15] in projective andsymmetric form.
First we recall the Ferus inequality for submanifolds in asphere or in aprojective
space with degenerate Gauss mappings [14][15]. See also $[7][27]$ .
be asubmanifold with degenerate Gauss mapping. See Example
Let, the rank of Gauss mapping of $M$ .
2.2. Set
ffom algebraic topology. The
First recall the Adams number $A(k)$ for
number $A(k)$ is by definition the maximal number of independent vector fields over
. For example, since Euler number of is not equal to zero, there
the spheredoes not exist nowhere vanishing vector field over , so we have $A(3)=0$ . Sinceandare trivial, we have $A(2)=1$ andandare parallelizable, namely,$A(4)=3$ . One of great results in algebraic topology (or homotopy theory), is thefollowing surprisingly simple formula due to Adams:$M^{m}\subset \mathrm{R}P^{n+1}$$r=\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}7$$\gamma$$k\in \mathrm{N}$$S^{k-1}$$S^{2}$$S^{1}$$S^{2}$$TS^{1}$$S^{3}$$TS^{3}$$A((2b+1)2^{c+4d})=2^{c}+8d-1,$ $(b, c, d\in \mathrm{N}\cup\{0\}, 0\leq c\leq 3)$ .

In particular $A(k)$ depends only on the exponent to 2in the primary decompositionof .
byThen define the Ferus number for$k$$m\in \mathrm{N}$.

$F(m):=\mathrm{l}\mathrm{n}\mathrm{i}\mathrm{n}\{k\in \mathrm{N}|A(k)+k\geq m\}$Then Ferus showed, in the framework of Riemannian geometry, the following crucial
result:with $r=$
be a closed and immersed submanifold of
Theorem 3.1 Letrank(7) $<m$ . Then $r<F(m)$ implies $r=0$ . In particular, if $M$ is a closed andand $M$ is not a projective subspace, then $F(m)\leq r$ .
connected submanifold of$\mathrm{R}P^{n+1}$$NI^{m}$$\mathrm{R}P^{n+1}$

We write down $F(m)$ , for smaller $m$ :$F(1)=1,$ $F(2)=2,$ $F(3)=2,$ $F(4)=4,$ $F(5)=4,$ $F(6)=4,$ $F(7)=4,$ $F(8)=8$ ,$F(9)=8,$ $F(10)=8,$ $F(11)=8,$ $F(12)=8,$ $F(13)=8,$ $F(14)=8,$ $F(15)=8$ ,$F(m)=16,$ $(16\leq m\leq 24),$ $F(m)=24,$ $(25\leq m\leq 31),$ $F(m)=32,$ $(32\leq m\leq 41)$ ,$F(m)=40,$ $(42\leq m\leq 47),$ $F(m)=48,$ $(48\leq m\leq 56),$ $F(m)=56,$ $(57\leq m\leq 63)$ ,$F(m)=64,$ $(64\leq m\leq 75),$ $F(m)=72,$ $(76\leq m\leq 79),$ $F(m)=80,$ $(80\leq m\leq 88)$ ,$F(m)=88,$ $(89\leq m\leq 95),$ $F(m)=96,$ $(96\leq m\leq 105)$ and so on. Moreover we have$F(m)=m$ if $m$ is apower of 2.

131

In this paper we call the inequality $F(m)\leq r$ Ferus inequality. Many examples
satisfying in fact Ferus equality $F(m)=r$ have been found related to isoparametricsubmanifold, homogeneous submanifolds, austere subamnifolds and so on ([27][29]).
However we may feel something missing, by the fact that, in Ferus inequality or
Ferus equality, there appear just $m$ and , but, there does not appear the number ,or the dimension of the ambient space.
Now we are going to formulate Ferus type inequality in term of Legendre sugmanifolds and in more symmetric form.
$n$

$r$

$\mathrm{R}P^{n+1}$

Theorem 3.2 Let be a closed (cornpact without boundary) immersed Legendre sub. Supposeandaremanifold of the incidence relation
constant rank $m$ andrespectively, and is not the Legendre lifting of a projective
subspace. Then we have$L$

$Q\subset \mathrm{R}P^{n+1}\cross \mathrm{R}P^{n+1*}$$\pi|_{L}$$\pi^{*}|_{L}$$L$$m^{*}$$F(m)\leq m+m^{*}-n$ ,$F(m^{*})\leq m^{*}+m-n$.

Note that $n\leq m+m^{*}$ . Moreover we see, if $m$ f-rn’ $=n$ in the situation of Theorem
3.2, then is the Legendre lifting of aprojective subspace (Example 2.1).
$L$

Proof of Theorem 3.2: Set $M=\pi(L)$ . Thenin$\mathrm{R}P^{n+1}$. It is easy to see that$M$

is aclosed and immersed submanifold

rank(7) $\leq m+m^{*}-n$ .

Thus we have $F(m)\leq m+m^{*}-n$ if $M$ is not aprojective subspace. By the symmetry,
we also have $F(m^{*})\leq m^{*}+m-n$ . Thus we have Theorem 3.2.
Now we are led to the following fundamental question:$\square$Question: For any positive integers $n,$ $m,$
$F(m)=m+m^{*}-n$ ,$m^{*}$

satisfying$F(m^{*})=m^{*}+m-n$ ,the symmetric Ferus equalities, find example8 of closed Legendre submanifolds
is of constantsuch thatis of constant rank $m$ and.
rank$L^{n}\subset$$Q^{2n+1}\subset \mathrm{R}P^{n+1}\cross \mathrm{R}P^{n+1*}$$\pi|_{L}$$\pi^{*}|_{L}$$m^{*}$

If the symmetric Ferus equalities are satisfied, then we have$F(m)=F(m^{*})$

and $n=m+m^{*}-F(m)(=m^{*}+m-F(m^{*}))$ .

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Since m $\geq F(m)$ and $m^{*}\geq F(m^{*})$ , the inequalities mfulfilled. Thus the question can be -written as follows:$\leq n,$ $m^{*}\leq n$

are necessarily$\mathrm{r}\mathrm{e}$satisfying $F(m)=F(m^{*})$ , find examples
Question’: For any positive integers $m,$$n=n=m+m^{*}-$of closed Legendre submanifolds$F(m)(=m^{*}+m-F(m^{*}))$ , such thatis of constant rank $m$ andis of constant$m^{*}$$L^{n}\subset Q^{2n+1}\subset \mathrm{R}P^{n+1}\cross \mathrm{R}P^{n+1*},$$\pi|_{L}$rank$m^{*}$$\pi^{*}|_{L}$.

We give here some of known examples:
Example 3.3 By Example 2.3, we have examples for $(m, m^{*})=(3,3),$ $(5,5),$ $(9,9)$ ,$(17, 17)$ , $(25, 25)$ , $(33, 33)$ , $(49, 49)$ , $(57, 57)$ , $(65, 65)$ , $(81, 81)$ , $(89, 89)$ , $(97, 97)$ , and so on.
.
Moreover, we have examples for the sequence : $(\not\simeq+1,2^{\ell}+1),$ $P=1,2,3,$$\ldots$

Example 3.4 (Cartan hypersurfaces.)(1) $(m, m^{*})=(3,2)$ . Let
be the Cartan hypersurface. Then $n=m=$$m^{*}=2$ . Note that $F(3)=2=F(2)$ . Thus we see the symmetric Ferus equalities
hold.
(2) $(m, m^{*})=(6,4)$ . Let
be the Cartan hypersurface. Then $n=m=$$m^{*}=4$ . Note that $F(6)=4=F(4)$ . Thus we see the symmetric Ferus equalities
hold.
(3) $(m, m^{*})=(12,8)$ . Let
be the Cartan hypersurface. Then $n=$$m^{*}=8$. Note that $F(12)=8=F(8)$ . Thus we see the symmetric Ferus$m=12,$
equalities hold.
(4) $(m, m^{*})=(24,16)$ . Let
be the Cartan hypersurface. Then$n=m=24,$ $m^{*}=16$ . Note that $F(24)=16=F(16)$ . Thus we see the symmetric
Ferus equalities hold.
$M^{3}\in \mathrm{R}P^{4}$$3,$$M^{6}\in \mathrm{R}P^{7}$$6,$$M^{12}\in \mathrm{R}\mathit{7}$$13$$M^{24}\in \mathrm{R}P^{25}$Moreover by Kimura’s constructions([27]), we have examples, for instance, for$(m, m^{*})=(6,5),$ $(11,10),$ $(21,20)$.

4Bi-degenerate fronts in four dimensional spaces.
Now we turn to singularities. We study Legendre submanifolds in the incidence relawithtion
and rank \pi $|_{L})=m^{*}$ , not assuming
may have singularities.
andare of constant rank. Then $\pi(L)$ andIn the case $(m, m^{*})=(n, 1)$ , the difleomorphism classification of the singularitiesof degenerate fronts are studied in detail in [20][22][23]. Note that, if $n\geq 2,$ $F(n)>$$Q\subset \mathrm{R}P^{n+1}\cross \mathrm{R}P^{n+1*}$$\pi|_{L}$$\pi^{*}|_{L}$$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(\pi|_{L})=m$$($$\pi^{*}(L)$133$1=n+1-n$ , so

is never of constant rank. For example, in the case $n=2$ , thetypical singularities of degenerate fronts of dimension 2inare acuspidal edge,
afolded umbrella and aswallowtail. These are singularities of tangent developables ofspace curves of types (1, 2, 3), (1, 2, 4) and (2, 3, 4), respectively.
For the classification by aweaker equivalence relation, namely by the homeomorphism classification is given in [25].
In this section, we give the classification of singularities of -degenerate Legendre
submanifold in case $n=3,$ $m=2,$ $m^{*}=2$ . Note that, in this case, $F(2)=2>1=$$2+2-3$ , so thatandare never of constant rank.
Consider the flag manifold$\pi|_{L}$$\mathrm{R}P^{3}$$\mathrm{b}\mathrm{i}$$\pi|_{L}$$\pi^{*}|_{L}$$\mathcal{F}:=\{V : \{0\}\subset V_{1}\subset V_{2}\subset V_{3}\subset V_{4}\subset \mathrm{R}^{5}\}$

Then we see $\dim \mathcal{F}=10$ . On
following: acurve$V(t)$

on $F$ is tangent to:$\mathcal{F}$.

, we define the canonical distrebution$D\subset T\mathcal{F}$by teh$\{0\}\subset V_{1}(t)\subset V_{2}(t)\subset V_{3}(t)\subset V_{4}(t)\subset \mathrm{R}^{5}$

at $t=t_{0}$ if the infinitesimal defomation ofatbe, the infinitesimal defomation of I4(t) at belongs tolongs to, and theinfinitesimal defomation of. Then we see rankD $=4$ .
at belongs to
We define the projection) by $\pi_{1}(V)=V_{1}$(resp.
::$(\pi_{4}(V)=V_{4})$ .
by:
Also we define the projection$\pi_{1,4}(V)=(V_{1}, V_{4})$ . Then we have.
andTypical singularites appearing in -degenerate fronts in this situation are cones
and l-developables.
$D$$V_{1}(t)$$V_{2}(t_{0})$$V_{3}(t_{0})$$t_{0}$$V_{3}(t)$$\pi_{1}$$t_{0}$$V_{4}(t_{0})$$t_{0}$$Farrow \mathrm{R}P^{4}$$\mathcal{F}arrow \mathrm{R}P^{4*}$$\pi_{4}$$\pi_{1,4}$$\pi_{1}=\pi\circ\pi_{1,4}$$\mathcal{F}arrow Q\subset \mathrm{R}P^{4}\cross \mathrm{R}P^{4*}$$\pi_{4}=\pi^{*}0\pi_{1,4}$$\mathrm{b}\mathrm{i}$Let$c:\mathrm{R}arrow \mathrm{R}P^{4}$,$c(t)=[x(t)]=[x_{0}(t), x_{1}(t),x_{2}(t), x_{3}(t), x_{4}(t))]$be asmooth curve. Consider the surface ruled by tangent (projective) lines to thecurve. We call it 1-developable of the curve. Then the tangent planes to regular pointsof the 1-developable are constant along each ruling. In fact the tangent plane to the
1-developable at apoint on atangent line coincides with the osculating 2-plane at thetangent point of the tangent line to the curve.
Let , a2,
be integers with $1\leq a_{1}<a_{2}<a_{3}<a_{4}$ . The curve is called oftyPe ( , a2,) at
if there exist asmooth coordinate of centered atand an affine coordinate
such that $c(t)$ is represented near in the form$c$$a_{3},$ $a_{4}$$a_{1}$$a_{1}$$t$$t_{0}\in \mathrm{R}$$a_{3},$ $a_{4}$$\mathrm{R}$$x_{1},$ $x_{2},$ $x_{3},$ $x_{4}$$x_{1}(t)=t^{a_{1}}+o(t^{a_{1}}),$$x_{2}(t)=t^{a_{2}}+o(t^{a\mathrm{z}}),$$t_{0}$$t_{0}$$x_{3}(t)=t^{a_{3}}+o(t^{a_{3}}),$$x_{4}(t)=t^{a_{4}}+o(t^{a_{4}})$.

The curve is of finite type at if there exist such integers
so thatis of type ( , a2,). The curve itself is called of finite type if it is of finite type$c$$a_{1}$$t_{0}$$a_{3},$ $a_{4}$$a_{1},$$a_{\underline{9}},$$a_{3},$$a_{4}$$c$134

of finite type lifts to unique D-integral
at every point. Any curve :, by using osculating subspaces of dimension 1(the tangent line), of
curve
dimension 2, of dimension 3and of dimension 4. Moreover ?
, then) at
is of type
is of finite type. If the original is of type ( , a2,. We call the dual curve to $c([40])$ .
( -a3, $a_{4}-a_{2},$ $a_{4}-a_{1},$ ) at
Then we have the following fundamental result:$\mathrm{R}arrow \mathrm{R}P^{4}$$c$$\tilde{c}:\mathrm{R}arrow \mathcal{F}$$:=\pi_{4}\circ\tilde{c}:\mathrm{R}arrow \mathrm{R}P^{4*}$$c$$a_{4}$$c^{*}$$c^{*}$$t_{0}\in \mathrm{R}$$a_{4}$$a_{4}$$a_{3},$$a_{1}$$t_{0}\in \mathrm{R}$) is a biTheorem 4.1 The 1-develpable of a curve inof type ( , a2,$m^{*}=2$1-developable.
$m=2,$
dual
theprojectiveItsisdegenerate front wethof thedual curve of type ( $a_{4}-a_{3},$ -a2, $a_{4}-a_{1},$ ).
$\mathrm{R}P^{4}$$c$$c^{*}$$a_{1}$$a_{3},$$a_{4}$$a_{4}$$a_{4}$

we must define, at least, alocal
To classify singularities of subsets inof amanifold $N$ at apoint $p_{0}\in N$ and asubset
equivalence relation: asubset
of amanifold $N’$ at apoint $p_{0}’\in N’$ are called diffeornorphic if there exists
adiffeomorphism : $Uarrow U’$ of an open neighbourhood of in $N$ and an open
neighbourhood $U’$ of in $N’$ which maps $A\cap U$ to $A’\cap U’$ .
Since an open dense part of $\pi(L)$ is asubmanifold of dimension $m$ , it is natural toconsider aparametrization by an $m$ dimensional manifold. Then smooth mappings: $Marrow N$ at apoint $t_{0}\in M$ and $f’$ : $M’arrow N’$ at apoint $t_{0}’\in NI’$ are called
diffeomorphic if there exist adiffeomorphism : $Varrow V’$ of of an open neighbourhood
and adiffeomorphism : $Uarrow$of in $M$ and an open neighbourhood $V’$ of in$U’$
of of an open neighbourhood of $720=f(t_{0})$ in $M$ and an open neighbourhood $V’$on .
of $p_{0}’=f’(t’)$ in $M’$ such that$\mathrm{R}P^{n+1}$$A\subseteq N$$A’\subseteq N’$$U$$\varphi$$p_{0}$$p_{0}’$$f$$\psi$$V$$\mathrm{J}/I’$$t_{0}’$$t_{0}$$\varphi$

$U$

$U$

$\varphi\circ f=f’\circ\psi$

Theorem 4.2 (cf. [22]) Let
at is of one of following tyPes:: $(1, 2, 3, 3+r),$ $r=1,2,$: (2, 3, 4, 5),: (1, 3, 4, 5),: (1, 2, 4, 5),(III) : (3, 4, 5, 6).

$c:\mathrm{R}arrow \mathrm{R}P^{4}$

be a smooth curve and$t_{0}\in \mathrm{R}$. Suppose$c$$t_{0}$$(\mathrm{I})_{r}$$\ldots$,$(\mathrm{I}\mathrm{I})_{0}$$(\mathrm{I}\mathrm{I})_{1}$$(\mathrm{I}\mathrm{I})_{2}$Then the diffeomorphism class inof the 1-developable of the curve at the point
by
only
type.
In
its
other words, if two curves have the same type,is determined
then their 1-developables are locally diffeomorphic.
$\mathrm{R}P^{4}$$c$$c(t_{0})$, only points of types:: (1, 2, 3, 4) and
For ageneric curve in(1, 2, 3, 5) appear. Moreover, for the dual curve of ageneric curve, only points of: (2, 3, 4, 5)
types (I)1 : $(1_{7}2,3,4)$ andear.
We call the 1-developable surface cuspidal edge in the case of type (1, 2, 3, 4), andopen swallowtail in the case of type (2, 3, 4, 5).
$\mathrm{R}P^{4}$$(\mathrm{I})_{1}$$(\mathrm{I}\mathrm{I})_{0}$$\mathrm{a}\mathrm{p}\mathrm{p}$$(\mathrm{I})_{2}$135

Example 4.3 (Cuspidal edge.) The 1-developable surface of acurve of type (1, 2, 3, 4)
has the normal form under the diffeomorphisms:$(x, t) \mapsto(x, 3t^{2}+2xt, 2t^{3}+xt^{2}, \frac{3}{4}t^{4}+\frac{1}{3}xt^{3})$.

Moreover it is diffeomorphic to$(x, t)\mapsto(x, t^{2}, t^{3},0)$.

Example 4.4 The 1-developable surface of acurve of type (1, 2, 3, 5) has the normal
form under the diffeomorphisms:$(x, t) \mapsto(x, 3t^{2}+2xt, 2t^{3}+xt^{2}, \frac{2}{5}t^{5}+\frac{1}{6}xt^{4})$.

However it is actually deffieomorphic to$(x, t)\mapsto(x, t^{2}, t^{3},0)$,naanely, diffeomorphic to the cuspidal edge.
Actualy we can prove the following:

Theorem 4.5 The 1developable of a curve of typeis diffeomorphic to the cuspidal edge.

$(\mathrm{I})_{r}$: $(1, 2, 3, 3+r),$ $(r=1,2,3, \ldots)$

Also we observe that the dual of 1-developable of acurve of type (1, 2, 3, 4) andthe dual of 1-developable of acurve of type (1, 2, 3, 5) are not diffeomorphic:
Example 4.6 (Open swallowtail.) The 1-developable surface of acurve of type(2, 3, 4, 5) has the normal form under the diffeomorphisms:$(x, t) \mapsto(x, 3t^{3}+2xt, \frac{9}{4}t^{4}+xt^{2}, \frac{9}{10}t^{5}+\frac{1}{3}xt^{3})$This is not diffeomorphic to the cuspidal edge.

.

136

5Frontal mappings.
In this section, we introduce the notion of frontal mappings and show an attempt
to generalize Legendre singularity theory, clarifying their applications to the study of
singularities appearing in the Grassmannian duality, or more generally in the Flag
duality, and to poly-symplectic geometry.
mapping. Assume is immersive
Let : $M^{m}arrow N^{n+1},$ $m<n+1$ , be a
$M$ . Thenis called afrontal mapping if,ofsubsetdense
outside of anowhere$x\inM$, there exists aunique limit
for any$C^{\infty}$$f$$, \lim_{xarrow x}f_{*}(T_{x’}M)=:T_{x}$

$f$

$f$

$\Sigma(f)$,$(x’\in M-\Sigma(f))$ .

in the Grassmann bundle $Gr(m, TN)$ , such that the correspondence$C^{\infty}$$x\mapsto T_{x}$is of class.

Examples of frontal mappings are (0) submanifolds, (1) singular curves with noinfinitely flat point, (2) their arbitrarily intermediate developables, (3) wave front setsin the ordinary sense and (4) varieties of irregular orbits of finite reflection groups [17].
If we take atransverse intersection of wave ffont sets, then we get a“kontal variety” ,which does not necessarily admit aparametrization by asingle non-singular manifold.
: $Marrow N$ be afrontal mapping. Then $f$ lifts naturally to amappingLet$\tilde{f}:Marrow Gr(m, TN)$ , which is called the Nash lifting of .
Let $D\subset TGr(m, TN)$ be the tautological subbundle (or the canonical systemin the sense of [43] of codimension $n+1-m=:r$ . Notice that, if $r=1$ , then$Gr(m, TN)=Gr(r, T^{*}N)=PT^{*}\underline{N}$ , and $D$ is the canonical contact distribution
over $PT^{*}N$ . Then the Nash lifting $f$ : $Marrow(Gr(m, TN),$ $D)$ is a(not necessarily
maximal dimensional) integral mapping of the distribution $D$ on $Gr(m, TN)$ . The
Nash lifting is characterized as the unique integral lifting of the frontal mapping .
$f$

$f$

$)$

$f$

$\tilde{f}$6Relation to poly-symplectic singularity theory.
Let

sum

$B$

be amanifold of dimension $m$ . For apositive integer , consider the Whitney$r$$T^{*(r)}B=T^{*}B\oplus\cdots\oplus T^{*}B-^{\pi}B$$1\leq i\leq r$ , whereis theendowed with the system of closed 2-forms
Liouville 1-form on the -th factor [6].
is called
mapping : $M^{m}arrow T^{*(f)}B$ from an $m$-dimensionalA$\varphi’\omega_{i}=0,1\leqi\leqr$$Marrow
M$ of. If we take the universal covering :isotropic if$M$ , then there exist functions$1\leq
i\leq
r$
. We
:
such that$\omega_{i}=d\theta_{i},$$\theta_{i}$$i$$C^{\infty}$$\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{l}\underline{\mathrm{d}}M$$\varphi$$\rho$$\overline{M}arrow \mathrm{R}$$e_{\mathrm{t}}$$de_{i}=(\varphi\circ\rho)^{*}\theta_{i},$137define the graph of$\varphi$by$f=(\pi 0\varphi 0\rho, e)$

If$\Sigma(f)$is nowhere dense in, then$\overline{M}$$f$:$\overline{M}arrow B\cross \mathrm{R}^{r}=:N$.

is frontal: The Nash lifting is: $\overline{M}arrow T^{*(r)}B\cross \mathrm{R}^{f}arrow Gr(m,TN)$ .

$\tilde{f}=(\varphi\circ\rho, e)$

We compare equivalence relations for isotropic mappings, integral mappings andfrontal mappings.
Two isotropic mappings and: $Marrow T^{*(r)}B$ are called Lagrange equivalent
if there exist diffeomorphisms : $Marrow M$ and : $\Gamma^{(r)}Barrow T^{*(r)}B$ such that
$1\leq i\leq r,$
covers adiffeomorphism : $Barrow B$ with respect to$T^{*(r)}Barrow
B$ , and that:.
Two integral mappings
and $F’$ :are called s-Legendre
equivalent if there exist diffeomorphisms: $Marrow M$ and :such that preserve the distribution and the fibration :and that.
Two bontal mappings and:are called -equivalent if there
exist diffeomorphisms: $Marrow M$ and pc :of the form$\varphi’$$\varphi$$\sigma$$\tau^{*}\{v_{i}=\omega.\cdot,$$\tau$$\overline{\tau}$$\tau$$\tau\circ\varphi=\varphi’\circ\sigma$$\pi$$Marrow T^{*(r)}B\cross \mathrm{R}^{r}$$F$$\Gamma^{(\mathrm{r})}B\cross \mathrm{R}^{f}arrow$$\tilde{\tau}$$\sigma$$T^{*(r)}B\cross \mathrm{R}^{r}$$\Pi$$\overline{\tau}$$?^{r(\mathrm{r})}B\cross \mathrm{R}^{r}arrow$$\overline{\tau}\mathrm{o}F=F’\circ\sigma$$B\cross \mathrm{R}^{f}$$Marrow B\cross \mathrm{R}^{r}$

$f’$

$f$

$\kappa(y, z)=(\overline{\tau}(y), z+\rho(y))$$s$$B\cross \mathrm{R}^{r}arrow B\cross \mathrm{R}^{r}$$\sigma$and that to$\circ f=f’\circ\sigma$.

Then we have
Proposition 6.1 Let : $Marrow T^{*(r)}B$ be an isotropic mapping with nowhere dense
singular set. Then the follouting conditions are equivalent to each other:(1) Isotropic mappings and: $Marrow T^{*(r)}B$ are Lagrange equivalent.
$\varphi$$\Sigma(\pi\circ\varphi)$$\varphi’$$\varphi$(2) Nash liftings$\tilde{f}$and(3) FVontal mappings$f$$\overline{f’}$:$\overline{M}arrow T^{*(r)}B\cross \mathrm{R}^{r}$

and $f’$ :$\overline{M}arrow B\cross \mathrm{R}^{f}$are -Legendre equivalent.
are s-equivalent.
$s$

It holds also the local version of this result. The concrete classification of isotropic
mappings to apoly-symplectic manifold under the Lagrange equivalence will be given
in aforthcoming paper.

7Projective duality and Grassmannian duality.
The projective duality plays an essential role, for instance, to formulate the famous
Pliicker-Klein’s formula, to analyze generic projective hypersurface (Bruce, Platonova,
Landis [4] , tangent developables (Scherbak [40], I[20] [22]) and Monge-Amp\‘e equations ([26]).
$)$$\mathrm{r}\mathrm{e}$

138

:

(e.g. aparametrization of asubbe afrontal$Marrow
Gr(n, T\mathrm{R}P^{n+1})=PT^{*}\mathrm{R}P^{n+1}$ .
manifold). Then we have the Nash lifting :, the manifold of incident pairs. ThenSet
is endowed with acontact structure and contact diffeomorphisms. Then we get the projective dual : $Marrow \mathrm{R}P^{n+1*}$ of by the. If is sufficiently
composition of with the projection.
generic, thenis also frontal, and we get the presumable equality
With the notion of frontal mappings, we are naturally led to the following generalization of the projective duality.
be afrontal mapping of codimension $r=n+1-m$ . Then,Let :consider the Nash lifting of :

Let

$f$

$M^{n}arrow \mathrm{R}P^{n+1}$$\mathrm{m}\underline{\mathrm{a}}\mathrm{p}\mathrm{p}\mathrm{i}\mathrm{n}\mathrm{g}$$f$$Q=\{(p, q)\in \mathrm{R}P^{n+1}\cross \mathrm{R}P^{n+1*}|p\subseteq q^{\vee}\}$$P\Gamma \mathrm{R}P^{n+1}\cong$$Q$$Q\cong PT^{*}\mathrm{R}P^{n+1*}$$f$$f^{\vee}$$PT^{*}\mathrm{R}P^{n+1*}arrow \mathrm{R}P^{n+1*}$$\tilde{f}$$f$$f^{\mathrm{v}\mathrm{v}}=f$$f^{\vee}$$f$$M^{m}arrow \mathrm{R}P^{n+1}$$f$$\overline{f}:Marrow Gr(m, T\mathrm{R}P^{n+1})$$Gr(1, \mathrm{R}^{n+2})\cross Gr(m+1, \mathrm{R}^{n+2})$$arrow$$Gr(1, \mathrm{R}^{n+2})\cross Gr(r, \mathrm{R}^{n+2*})$$\cong$.

The image is again $Q=\{(p, q)|p\subseteq q^{\vee}\}$ . Therefore we naturally define the Grassman”,: $Marrow Gr(r, \mathrm{R}^{n+2*})$ of : $Marrow \mathrm{R}P^{n+1}$ . The equality “nian dual
however does not have any meaning, even if is affont mapping in the meaning ofprevious definition. Therefore, for amapping into aGrassmannian, it seems natural
to specialize the definition of frontal mappings as follows:
mappings with $m+r\leq n+1$ . Set: $NI^{m}arrow Gr(r, \mathrm{R}^{n+2})$ be aLet$s=n+2-m–r$ . Then is called Grassrnann-frontal if there exists aunique integral
lifting : $Marrow(Q, D)$ of with respect to afibration : $Qarrow Gr(r, \mathrm{R}^{n+2})$ and a
distribution on defined as follows: Set first$f$$f^{\vee}$$f^{\mathrm{v}\mathrm{v}}=f$$f^{\vee}$$C^{\infty}$

$f$

$f$

$f$

$f$

$D$

$\pi$

$Q$

$Q=\{(p, q)\in Gr(r, \mathrm{R}^{n+2})\cross Gr(s, \mathrm{R}^{n+2*})|p\subseteq q^{\vee}\}$,and$P=\{(p, q,p’)\in Gr(r, \mathrm{R}^{n+2})\cross Gr(s, \mathrm{R}^{n+2*})\cross Gr(r, \mathrm{R}^{n+2})|p\subseteq q^{\vee},p’\subseteq q^{\vee}\}$Then we get the special divergent diagram$(\rho, \pi\circ\rho)$.

:$P\rhoarrow Qarrow Gr(\pi r, \mathrm{R}^{n+2})$,(resp. ) is the projection to the first and second factors (resp. to the first
factor). To define the tautological subbundle $D\subset TQ$ of codimension $rs$ , for each$c=(p, q)\in Q$ , we set, where $Gr(r, \mathrm{R}^{r+m})=$byis embedded in
as $\{p’\in Gr(r, \mathrm{R}^{n+2})|p’\subseteq q^{\vee}\}$ . Notice that,
if $r\neq 1$ , or, $r\neq n+1$ , then the “system of tangential linear subspaces” on thedefined by $D$ does not represent general tangential linear
Grassmannian
subspaces of the Grassmannian.

where$\rho$$\pi$$D_{c}\subset T_{c}Q$$\pi(\rho^{-1}(c))$$D_{c}=\pi_{*}^{-1}(T_{\mathrm{p}}(Gr(r, \mathrm{R}^{r+m})))$$Gr(r, \mathrm{R}^{n+2})$$Gr(r, \mathrm{R}^{n+2})$

139
If we take local coordinates
andof, then Q is defined by the system of equations$(a_{ij})_{1\leq i\leq r,1\leq j\leq m+s}$of$Gr(r, \mathrm{R}^{n+2})$$(b_{k\ell})_{1\leq k\leq m+r,1\leq\ell\leq s}$$Gr(s, \mathrm{R}^{n+2*})$$b_{ij}+a_{i1}b_{r+1g}+\cdots+a_{im}b_{r+mj}+a_{im+j}=0,1\leq i\leq r,$ $1\leq j\leq s$,

and $D$ is defined by the system of l-forms$b_{r+1j}da_{i1}+\cdots+b_{r+mj}da_{im}+d\mathfrak{R}.m+j=0,1\leq i\leq r,$ $1\leq j\leq s$.

The integral lifting is called the Grassmann-Nash lifting of . The relation tothe original definition of frontal mappings is as follows:$f$$\overline{f}$be an integral rnap-germ. Then $f=\pi\circ F$ :isis Grassmann-frontal if and only
nowhere dense, for some projection

Lemma 7.1 Let$\mathrm{R}^{m},$$F:\mathrm{R}^{m},$ $\mathrm{O}arrow Q,$ $(p_{0}, q_{0})$$\iota f\Sigma(\rho\circ f)\subset \mathrm{R}^{m},$ $0$$\mathrm{O}arrow Gr(r, \mathrm{R}^{n+2}),p_{0}$$\rho:Gr(r, \mathrm{R}^{n+2}),p_{0}rightarrow Hom(\mathrm{R}^{r}, \mathrm{R}^{m+s}),$ $0arrow.Hom(|^{\mathrm{r}}\mathrm{R}, \mathrm{R}^{m+s}),$ $0arrow \mathrm{R}P^{m+s-1}$

induced from a linear inclusiori$\pi’$$i:\mathrm{R}rightarrow \mathrm{R}^{r}$,.

Now, from the duality, we have another distribution $D’\subset TQ$ from the projection: $Qarrow Gr(s, \mathrm{R}^{n+2*})$ to the second factor, setting$P’=\{(q’,p, q)\in Gr(s, \mathrm{R}^{n+2*})\cross Gr(r, \mathrm{R}^{n+2})\cross Gr(s, \mathrm{R}^{n+2*})|q\subseteq p^{\vee}, q’\subseteq p^{\vee}\}$.

Then the fundamental result is the following:
Proposition 7.2 Tuto distributions $D$ and $D’$ on the incidental manifold$Q$

coincide.

Based on this fact and aversion of the transversality theorem, we have the following
Grassmannian duality theorem:in the space of integral mappings
with $m+r+s=n+2$ of kernel rank at
most one, with the following property: For any $F:Marrow Q$ belonging to $O,$ $F$ is theunique integral lifting of $\pi\circ F=:f$ and of $\pi’\circ F=:f’$ respectively, and the singularlociandare both nowhere dense in M. In particular, in this case, we have

Theorem 7.3 There eists an open dense subset$O$$M^{m}arrow Q\subset Gr(r, \mathrm{R}^{n+2})\cross Gr(s, \mathrm{R}^{n+2*})$$\Sigma(f)$

that

$f$

$\Sigma(f’)$

and $f’$ are both Grassmann-frontal, $f’=f^{\vee},$$f=f^{\prime\vee}$

and that$f^{\mathrm{v}\mathrm{v}}=f$.

The proofs of these results will be given in forthcoming papers. We conclude thissurvey by giving just several illustrative examples.

140
Example 7.4 If f :. Thenof acurve indual curve).

$M^{2}arrow \mathrm{R}P^{4}$$\mathrm{R}P^{4}$$f^{\vee}:$is the natural parametrization of the l-developable
collapses to acurve (Grassmannian$M^{2}arrow Gr(2, \mathrm{R}^{5*})$be the Veronese embedding. Then the dual
Example 7.5 Let $f$ :composed with the Pliicker
is also an embedding. In fact,:
is decomposed into the Veronese embedding
embedding.
and alinear embedding$\mathrm{R}P^{2}arrow \mathrm{R}P^{5}$$f^{\vee}$$\mathrm{R}P^{2}arrow Gr(3, \mathrm{R}^{6*})$$f^{\vee}$$\mathrm{R}P^{2}arrow$$Gr(3, \mathrm{R}^{6})arrow \mathrm{R}P^{19}$$\mathrm{R}P^{9}arrow \mathrm{R}P^{19}$$\mathrm{R}P^{9}$

Example 7.6 Let : $M^{2}arrow Gr(2, \mathrm{R}^{5})$ be an embedding. If $f(M)\subset Gr(2, \mathrm{R}^{3})\subset$, thenhas infinitely many integral liftings : $Marrow Q$ . The “dual”$Marrow \mathrm{R}P^{4*}$.
collapses to apoint on the projective line dual to$f$$Gr(2, \mathrm{R}^{5})$$\tilde{f}$$f$$\mathrm{R}^{3}\subset \mathrm{R}^{5}$$f^{\vee}:$

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