数理解析研究所講究録
第 764 巻 1991 年 131-135

131

Developable of a Curve andDeterminacy Relative to Osculation-Type

Goo ISHIKAWA
Department of Mathematics, Hokkaido University, Sapporo 060, JAPANIntroduction
The ruled surface by tangent lines to a space curve is caLled the developable surface of thecurve. More generaUy, the developable of a curve in $(n+1)$ -dimensional projective spaceis defined as the hypersurface “ruled” by osculating (n–l)-subspaces to the curve.

Consider a$caU$

the germ$C^{\infty}$$\gamma_{p}\cdot at$curve: $Marrow BP^{n+1}$ , where $M$ is a l-dimensional manifold. Wea point $pEM$ of finite osculation-type (or simply, of finite type)$a=(a_{1}, a_{2}, \ldots, a_{n+1})$$(x_{1}, \ldots, x_{n+1})$$\gamma$

if there exist aof $BP^{n+1}$ centered at$x_{1}=t^{\alpha_{1}}+o(\ell^{a_{1}})$

coordinate of $(M,p)$ and an affine coordinate$C^{\infty}$$t$such that$\gamma(p)$...,,$\gamma$

is represented by$x_{n+1}=t^{a_{n+1}}+o(t^{a.+1})$,

where each : is a natural number and $1\leq a_{1}<\cdots<a_{n+1}$ .
$a$A point $p\in M$ is called an ordinary point if$\gamma_{p}$is of type $(1, 2, \ldots,n, n+1)$ , and,otherwise, it is called a special point.

For each $p\in M$ where$\gamma_{p}$is of finite type and for eachthe most osculating linear subspace toosculating i-subspace and denote by$BP^{n+1}$

through$p$$\gamma$

at

$p$

in

$O_{i}(\gamma,p)$$i,$$T_{\gamma(p)}BP^{n+1}$$(0\leq i\leq n+1)$, there existsof dimension . We call it the$i$. The corresponding projective subspace ofof dimension is also denoted by$i$$O_{i}(\gamma,p)$. The type of a curve therefore

describles the order of tangency to each osculating subspace, and it is the simplest local

projective invariant of the curve.

We can define the osculating i-bundle$O_{i}(\gamma)$

in the pullback $\gamma^{-1}TBP^{n+1}$ . The naturalparametrization$dev(\gamma)$: $O_{n-1}(\gamma)arrow BP^{n+1}$

132
defined by $(p, q)\mapsto q$ , where $q\in O_{n-1}(\gamma,p)(\subset BP^{n+1})$ , is cdled also a developable of
There are several results on the classification of developables of curves under the$\gamma$.

$C^{\infty}$

right-left equivalence.
For a space curve , at each ordinary point , the developable has cuspidal singularities$p$$\gamma$along$\gamma$and$dev(\gamma)_{p}$is equivalent to $(x,t)rightarrow(x,t^{2},t^{s})$ .

Cleave [C], Gafney-du Plessis [GP] and Shcherbak [S1] prove that, at a pointtype (1, 2,4),$dev(\gamma)_{p}$of

is equivalent to $(x,t)rightarrow(x,t^{2}, xt^{s})$ .

Mond [MI][M2] gives$r\leq 5$$p$$C^{\infty}$

normal forms of developable of curves of type $(1, 2, 2+r)$ ,, and of type (1, 3, 4).

In the case of arbitrary dimension, Shcherbak, in [S1], shows the the developable of acurve of type $(2, 3, \ldots,n+1,n+2)$ is equivalent to the (parametrization of) n-dimensional
swaUowtail, generarizing the observation of Arnol’d [A] for a curve of type (2, 3, 4) basedon the Legendre singularity theory.
In the connection with the study of projections of wave hont sets, Shcherbak, further

in [S2], gives the$C^{\infty}$

normal form of the union of the developable of a curve-germ

type $(1, 2, \ldots,n,n+2)$ and the osculating hyperplane$O_{n}(\gamma,p)$

We can notice that the type of a curve determines the local$\gamma_{p}$of. See also [K].

$C^{\infty}$class of the developableof the curve in the above mentioned cases.

Inspired with these previous results, we are led to the natural problem that whether

a type of a curve-germ$\gamma_{p}$determines the$C^{\infty}$

dass of map-germ$dev(\gamma)_{p}$or not.

If such determinacy for a type a is established once, then to have the normal formof developables of curves of type a is reduced to just a calculation of an example. The

purpose of this paper is to announce the complete solution of this determinacy problem.
THEOREM 1.

$A$

type a of a curve-germ in $BP^{n+1}$ determines $C^{\infty}dass$ of developable ifand only if a is one of foIlowing types:$(I)_{n},,$$a=(1,2, \ldots,n, n+r)$ ,$’=1,2,$$\ldots$,$(II)_{n,i}a=(1,2, \ldots , i, i+2, \ldots, n+1,n+2)$$(III)_{n}a=(3,4, \ldots, n+2, n+3)$ ,(IV) $a=(3,5)$ ,(V) $a=(1,3,5)$ .

2,$0\leq i\leq n-1$,

133$\mathbb{R}rtAer$$eq$, in this case, for any$\gamma_{p}$of type a, th map-germ$e$

is$dev(\gamma)_{p}$$C^{\infty}$right left

uivalent to $(x’, U(x‘,t), U,(\epsilon’,t)):B^{n},0arrow B^{n+1},0$, where $(x’,t)=(x_{1}, \ldots, x_{n-1},t)$ isa coordinate of $(B^{n}, 0)$ ,,$U(x’,t)= \frac{t^{a}}{a_{n}!}+x_{1}\frac{t^{a.-a_{1}}}{(a_{n}-a_{1})!}+\cdots+x_{n-1}\frac{t^{a.-a.-1}}{(a_{n}-a_{\iota-1})!}$$r=a_{n+1}-a_{n}$and$U,(ae’,t)= \int_{0}^{\ell}\frac{t’}{r!}\frac{\partial U}{\theta t}\$.

Notice that the developable apears as a component of the envelope of one-parameterfamily of osculating hyperplanes to a curvegerm
envelope also has a component$O_{n}(\gamma,p)$$\gamma_{p}$. In the case $a_{n+1}-a_{n}>1$ , theitself. In this case therefore it is natural to classify

developables by diffeomorphisms preserving$O_{n}(\gamma,p)$. Then we have

THEOREM 2. A type a $ofa$ cnrve-germ in $BP^{n+1}$ determines

osculating hyperplanes if an$(III)_{n},n\geq 2$

THEOREM 3.

$d$$C^{\infty}$class of envelope ofonJy if a is one of types $(I)_{n},,,$ $r=1,2,$$\ldots,$$(II)_{n.i}$and, in Theorem 1.
$A$developable andtype a $ofa$ curve-germ$O_{n}(\gamma,p)$$\gamma_{p}$in $BP^{n+1}$ determines

if and only ifa is one of types$(I)_{n},$$C^{\infty}$, and $(II)_{n},$ : in Theorem 1.

These results unifies and generdizes the results of [C], [G-P] on[A], [S1], [S2], on$(I)_{n,2}$class of the union ofand $(II)_{n,0}$ , and the results of [M1] [M2] on$(I)_{2,2}$, the results of$(I)_{2},,,$ $(r\leq(II).

The proofs of Theorems 1,2 and 3 will be given in a forthcoming paper.

35)$, and134
Mond’s theorem

Based on Theorem 1, we reprove the foilowing result due to Mond [M1], [M2, Corollary
0.2]:

COROLLARY. Let 7 : $B,$ $0arrow BP^{S}$ be a curve-germ of type $(1, 2, 2+r)$ . Then de$B^{2},0arrow BP^{S}$is a topologic$aI$

embedding $ifr$ is odd, and der$(\gamma)$$v(\gamma)$:

A as a single curve of

seIRn tersection $ifr$ is even.
PROOF: By Theorem 1, dev(7) isequivalent to the germ at$C^{\infty}$$-(1/2)(t_{1}+t_{2})$and$(x_{i},t_{*}\cdot)EB^{2},$$\int_{\ell}^{\iota_{1^{2}}}\ell(\ell+x_{1})d\iota=0$$i=1,2$ . Then we see $x_{1}=x_{2},$ $x_{1}=$. Thus, setting$\int_{-a}^{a}(\sigma-x_{1})\sigma d\sigma=0$of: $B^{2}arrow B^{S}$ .

$f(x,t)=(x, \frac{t^{2}}{2}+xt, \int_{0}\frac{\prime}{r!}(s+x)ds)$

Now, assume $f(x_{1},t_{1})=f(x_{2},t_{2}),$$0$$\sigma=s+x_{1}$

...

$(*)$

, we have,where $a=(1/2)(t_{2}-t_{1})$ .
If is odd, then the left hand side of$r$$(^{*})$

is equal to an integral $hom-a$ to with almost$a$everywhere positive integrand. Hence we have $a=0$ . This means that $(x_{1},t_{1})=(x_{2)}t_{2})$and that$f$

is injective.

By a similar argument, ifSince

$f$

$r$

is even, then we have $z_{1}=0$ or $(x_{1},t_{1})=(x_{2},t_{2})$ .

is a finite mapping and $f|\{ae=0\}=$ $(0,t^{2}/2,(r+1)\{t‘+2/(r+2)!\})$ , we see$f$is an embedding in the complement of a double point curve $\{a=0\}$ .
REFERBNCES[A] VJ. Arnol’d, Lagrangian $manifold\ell$ with singularities, asymptotic rays, and the open

swallowtail, Funct. Anal. Appl. 15-4 (1981), 235-246.
[B-M] E. Bierstone, P.D. $nm_{man}$ , Relations among analytic functions I, II, Ann. Inst.

Fourier 37-1, 37-2 (1987), 187-239, 49-77.

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135[C] J.P. Cleave, The form of the tangent developable at points of zero torsion on spacecurves, Math. Proc. Camb. Phil. Soc. 88 (1980), 403-407.
[G-P] T. Gaffney, A. du Plessis, More on the determinacy of smooth map-germs, Invent.
Math. 66 (1982), 137-163.
[K] M.\’E. Kazaryan, Singularities of the boundary offundamental systems, flat points ofprojective curves, and $Schuber_{t}$ cells, in “Itogi Nauki Tekh., Ser. Sorrem. Probl. Mat.
(Comtemporary Problems of Mathematics) 33,” VITINI, 1988, pp. 215-232.

[M1] D. Mond, On the tangent developable of a space curve, Math. Proc. Camb. Phil.

Soc. 91 (1982), 351-355.
[M2]

Singularities of the tangent developable sutface of a space curve, Quart.

J. Math. Oxford 40 (1989), 79-91.
[S1] O.P. Shcherbak, Projectively dual space curves and Legendre singularities, Trudy

Tbiliss. Univ. 232-233 (1982), 280-336.
[S2]

Wavefront and reflection groups, Russian Math. Surveys 43-3(1988), 149-194.

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