数理解析研究所講究録
第 1868 巻 2013 年 1-8

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Functions on singular surfaces
Masaru HASEGAWA
Department of Mathematics, Saitama University
1. Introduction. The purpose of this note is to announce my recent results in [8] with
T. Fukui. Refer to [8] for details.
Letbe a smooth map-germ which defines a surface (possibly withsingularities) in . We define families
andoffunctions$f:(\mathbb{R}^{2},0)arrow(\mathbb{R}^{3},0)$$H:(\mathbb{R}^{2},0)\cross S^{2}arrow \mathbb{R}$$\mathbb{R}^{3}$$D:(\mathbb{R}^{2},0)\cross \mathbb{R}^{3}arrow \mathbb{R}$

on by
$f$

$H(u, v,w)=\langle f(u, 1,),w\rangle$, and $D(u, n,p)=\Vert p-f(u, v)\Vert^{2}-t^{2},$respectively, where is a unit sphere indenotes the dot product in and
We define $h_{w}(u, lJ)=H(u, \iota,,w)$ and, which are the heightfunction on $f$in the direction wand the distance squaredfiunction on $f$ from the point , respectively. The
family $H$ is a 2-parameter unfolding of , and the family is 3-parameter unfolding of
The analysis of the contact of a submanifold with degenerate objects (lines, planes, circles,
spheres, etc.) is important to understand geometric properties of the submaifold. The contacts of a surface inwith planes and spheres are measured by singularities of the heightfunctionsand the distance squared functions
on the surface, respectively. By using
these kinds of techniques of the contacts, several researchers have studied the differential
geometry of submanifolds in Euclidean space (see, for example [2, 3, 5, 15, 22]) and otherambient spaces.
$S^{2}$$\mathbb{R}^{3},$$\langle,$$\mathbb{R}^{3}$$t\in \mathbb{R}\backslash \{O\}.$$\rangle$$d_{p}(u, \iota,)=D(\mathcal{U}, l),p)$

$p$

$D$

$h_{w}$$d_{p}.$$\mathbb{R}^{3}$$h_{w}$$d_{p}$FIGURE. 1: The Whitney umbrella, $(u, v)arrow(u, uv, v^{2})$H. Whitney [26] showed that smooth maps of into can have a singularity which are
not avoidable by small perturbation. This singularity is called Whitney umbrella or crosscap (Figure 1). Since ffie 珂珂 hitney umbrella is a stable singularity, it is natural to seek its
geometry. The extrinsic differential geometry of the 珂珂 hitney umbrella is investigated in[4, 6, 7, 9, 10, 19, 20, 21, 24, 25], and in [11] its intrinsic properties are considered. It is shownfor instance in [4, 25] that there are generically two types ofWhitney umbrellas, labeled hyperbolic Whitney umbrella and elliptic 珂珂 hitney umbrella (Figure 2), and ffiese are characterized by the singularity type oftheirparabolic set in the source. The change from an elliptic to$\mathbb{R}^{2}$$\mathbb{R}^{3}$

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a hyperbolic Whitney umbrella occurs at a parabolic Whitney umbrella. In [6, 7], the singularities of the height and distance squared functions on Whitney umbrellas were studied interms of its differential geometric properties via a blowing up. In [6], the singularities of sections ofthe hitney umbrella by planes were studied and we showed the analogous theorem$\mathcal{W}$of . A. Montaldi [17] for Whitney umbrellas. In [7], the criteria of singularities ofwave-fronts
and caustics of the Whitney umbrella were obtained, and the focal conic of the Whitney umbrella was introduced, which is the counterpart of focal points ofsmooth surfaces (Figure 2).
The (generic) type of the focal conic determines the (generic) type of the Whitney umbrella,and vice-versa.
$J$

FIGURE. 2: The three types of the Whitney umbrella and their focal conics: hyperbolic Whitney umbrella (left), parabolic Whitney umbrella (center), elliptic Whitney umbrella (right).

In [16], D. Mond showed that every -simple germ of a map from a 2-manifold to a3manifold is equivalent to one of the germs in Table 1.
$\mathscr{A}$TABLB 1.

Name
Immersion
Whitney umbrella

Normal form$(S_{0})$$\mathscr{A}$-codim.

$(u, \iota,,0)$$0$$(u, v^{2}, uv)$2

$S_{k}^{\pm} (u, v^{2}, v^{3}\pm u^{k+1}v), k\geq 1 k+2$
$B_{k}^{\pm}C_{k}^{\pm} (u, v^{2},u^{2}v\pm v^{2k+1}),k\geq 2 kk:_{2}2$$(u, v^{2}, u^{2}v\pm v^{k}v), k\geq 3$$F_{4} (u, v^{2}, u^{3}v+v^{5}) 6$$(u, uv+u^{3k-1}, v^{3}), k\geq 2$$\frac{H_{k}k+.2}{(Whenkiseven,S_{k}^{+}isequivalenttoS_{k}^{-},andC_{k}^{+}toC_{k}^{-})}$

It is also of interest to investigate the singularities of the height and distance squared functions on singular surfaces defined bywith -simple $singu4$ arities moredegenerate than the Whimey umbrella (see Table 1). Our main objective is to investigate therelations between the differential geometric properties of singular surfaces which have one, orof
singularity (shown in Table 1) and singularities of the height and distancesquared functions on these singular surfaces. Work in the direction ofthe understanding thedifferential geometry of singular surfaces with a singularity of corank 1 was carried out, forinstance, in [11, 13, 14, 21].
$f:(\mathbb{R}^{2},0)arrow(\mathbb{R}^{3},0)$$S_{k},$$B_{k},$$C_{k}$$F_{4}$$\mathscr{A}$3
2. Preliminaries. In order to analyze the differential geometry of a surface, relevant parameterization of the surface are essential. The Whitney umbrella case was obtained in [25](see also [7]).

.

Proposiuon 2 Let
be a map-germ of corank 1 at the origin. Then, afterusing rotations in the target and changes of coordtnates in the source, we can reduce $j^{k}f(0,0)$to$.1$$f:(\mathbb{R}^{2},0)arrow(\mathbb{R}^{3},0)$(2.1)$(u, \frac{1}{2}\iota^{2}+\sum_{i=2}^{k}\frac{b_{i}}{i!}u^{i}, \frac{1}{2}a_{20}u^{2}+\sum_{i+j=3}^{k}\frac{a_{ij}}{i!j!}u^{i}v^{j})$if$j^{2}f(0,0)$is$\mathscr{A}$-equivalent to $(u, \iota)2,0)$ , or
(2.2)$(u, uv+ \sum_{i=3}^{k}\frac{b_{i}}{i!}v^{i}, \frac{1}{2}a_{20}u^{2}+\sum_{i+j=3}^{k}\frac{a_{ij}}{l!j!}u^{i}v^{j})$if$j^{2}f(0,0)$is$\mathscr{A}$-equivalent to $(u, un,0)$ .

Proposition 2.2. Let $f$ :be a smooth map-germ of corank 1 and let $j^{k}f(0,0)$be given in the form (2.1). $77len$ the conditions for $f$ to$C_{k}orF_{4}$ are-equtnalent to$(\mathbb{R}^{2},0)arrow(\mathbb{R}^{3},0)$$be\mathscr{A}$$S_{k},$$B_{k},$

as follows:$S_{1}$$S_{k\geq 2}$$B_{2}$$B_{k\geq 3}$

:
:
:
:

$a_{21}\neq 0,$$a_{03}\neq 0,$$a_{21}=\cdots=a_{k,1}=0,$$a_{k+1,1}\neq 0,$$a_{03}\neq 0,$$a_{03}=0,$ $a_{21}\neq 0,3a_{05}a_{21}-5a_{13^{2}}\neq 0,$$a_{03}=0,$ $a_{21}\neq 0,3a_{05}a_{21}-5a_{13^{2}}=0,$$\xi_{3}=\cdots=\xi_{k-1}=0, \xi_{k}\neq 0,$$C_{k}$$F_{4}$

where$\xi_{m}$::$a_{03}=0,$ $a_{21}=\cdots=a_{k-1,1}=0,$$a_{03}=0,$ $a_{21}=0,$$a_{31}\neq 0,$$a_{k,1}\neq 0,$$a_{13}=0,$$a_{13}\neq 0,$$a_{05}\neq 0,$

depends on the $(2m+1)-iet$ of the third component of (2.1).

Remark2.3. Each of these criteria of Proposition 2.2 includes the condition that$a_{21}\neq 0$

or $a_{m,1}=0(2(\leq m\leq n),$$a_{n+1,1}\neq 0$

for some

$n$

.

(2.3)

Let
be a smooth map-germ of corank 1 at the origin . At the singularpoint , the tangent plane degenerates to aline, that is, the image of $(df)_{0}$ is aline, which is
the tangent line. The plane passing through the singular point perpendicular to the tangent
line is called the normal plane. There exists non-zero vector
such that $(df)_{0}(\eta)=0.$
We call the null vector (cf. [12, 23]). regular plane curve in the parameter space passing
through $(0,0)$ whose tangent vector is transverse to is called a tangential curve. Letbethe parameterization of a tangential curve with $7(0)=(0,0)$ . Clearly,
is tangent to thetangent line of $f$ at the singular point . If $j^{2}f(0,0)$ is -equivalent to $(u, v^{2},0)$ , then there
exists a plane passing through the singular point spanned by the tangent line and,where
is the twice times directional derivative of $f$ with respect to . We call the plane$f:(\mathbb{R}^{2},0)arrow(\mathbb{R}^{3},0)$

$0$

$0$

$0$

$\eta\in T_{0}\mathbb{R}^{2}$$\eta$$A$$\gamma(t)$$\eta$$f\circ\gamma$$0$$\mathscr{A}$$0$$\eta\eta f(0,0)$$\eta\eta f$$\eta$

4
the principal plane. We remark that the definitions of the tangent line, normal plane andprincipal plane are independent of the choice of coordinates on the source. Let $j^{2}f(0,0)$ be
have at least 3-equivalent to. The singular point is a inflection point if(inflectionaltangent) with the principal plane at . We denote by a family
point contactof the family is a characteristic tangential curve if themembercurvestangentialofcurvature, of the projection of
to the principal plane, at has the extremum value
Note that tangential curves tangent to the characteristic tangential curve are characteristic
tangential curves. If the singular point is a inflection point, the singular point is a degener. We remark that the definitions of the inflection and degenerate
ate inflection point if
independentof the choice of the coordinates on the Source. We also
pointsinflection
areremark that the ways of definitions of the inflection and degenerate inflection points differ
here from those in [21].
$(u, \iota^{2},0)$$\mathscr{A}$$0$$f\circ\gamma$$0$$\gamma.$$A$$\Gamma$$\Gamma_{0}$$0$$f\circ\Gamma_{0}$$\kappa_{0}.$

$0$

$0$

$\kappa_{0}=0$

FIGURE. 3: The tangent line, the normal plane and the principal plane of$S_{1}^{-}$(left) and$S_{1}^{+}$(right).

be a smooth map-germ of corank 1 at the origin,
Proposition 2.4. Let $f$ :and let $j^{k}f(0,0)$ be given in the form (2.1). Then the origin is an inflection (resp. degenerate
inflection) point if and only if $a_{20}=0$ resp. $a_{20}=b_{2}=0)$ . Moreover, tangential curves tangent
to at $(0,0)$ are the characteristic tangential curves.
$(\mathbb{R}^{2},0)arrow(\mathbb{R}^{3},0)$$($$\partial_{u}$be a smooth map-germ of corank 1 at the origin, and let $j^{k}f(0,0)$Let
with
be given in the form (2.1) with the assumption (2.3). We set the map$f:(\mathbb{R}^{2},0)arrow(\mathbb{R}^{3},0)$$\Pi_{n+1};\mathbb{R}\cross S^{1}arrow \mathbb{R}^{2}$$\Pi_{n+1}(r,\theta)=(r\cos\theta,r^{n+1}\cos^{n}\theta\sin\theta) (n=1 if a_{21}\neq 0)$

The unit normal vector
and$\tilde{n}=n\circ\Pi_{n+1}$

of $f$ in.

is extendible near the exceptional set$(r,\theta)$$\{(r,\theta)\in$$\mathbb{R}xS^{1}|r\cos\theta=0\}$$\tilde{n}(0,\theta)=\frac{(0,-a_{n+1,1}\cos\theta,(n+1)!\sin\theta)}{\mathscr{A}(\theta)},$

where
in

. The principal curvatures$\mathscr{A}(\theta)=\sqrt{(a_{n+11}\cos\theta)^{2}+((n+1)!\sin\theta)^{\angle}}$$(r,\theta)$

are expressed as follows:$\tilde{\kappa}_{1}(r,\theta)=\frac{-a_{n+1,1}b_{2}\cos\theta+(n+1)!a_{20}\sin\theta}{\mathscr{A}(\theta)}+O(r)$$\tilde{\kappa}_{2}(r,\theta)=\frac{1}{r^{2n+2}}(\frac{-((k+1)!)^{2}a_{n+1,1}}{\cos^{2n-1}\theta \mathscr{A}(\theta)^{3}}+O(r))$.

,$\tilde{\kappa}_{i}=\kappa_{i}\circ\Pi_{n+1}$

of $f$

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The vector $v_{i}=(N-\kappa_{i}G)\partial_{u}-(M-\kappa_{i}F)\partial_{v}(t=1,2)$ is a principal vector relative toprincipal vectors$\tilde{v}_{i}$in$(r,\theta)$$\kappa_{i}$. The

are given by$\tilde{v}_{1}=(-\frac{a_{n+1,1}}{\mathscr{A}(\theta)}+O(r))\partial_{r}$$+(- \frac{a_{n+2,1}\cos^{2}\theta\sin\theta+(n+2)(n+1)!a_{12}\cos\theta\sin^{2}\theta}{(n+2)\mathscr{A}(\theta)}+O(r))\partial_{\theta},$$\tilde{v}_{2}=-\frac{(n+1)!a_{n+1,1}(a_{20}a_{n+1,1}\cos\theta+(n+1)!b_{2}\sin\theta)\cos^{2-n}\theta}{\mathscr{A}(\theta)^{3}r^{2n+1}}$$((\sin\theta r+O(r^{2}))\partial_{r}+(\cos\theta+O(r))\partial_{\theta})$.

We can describe the asymptotic behavior of the ridge and sub-parabolic curves. Here, theruge (resp. sub-parabolic) curve is the locus of points where one principal curvature has anextremum along lines of the same (resp. other) principal curvature. It is known that the ridgeand sub-parabolic curves of a regular surface in correspond, respectively, to the cuspidal
edges and parabolic set of caustics of the surface (see, for instance [3, 18]).
$\mathbb{R}^{3}$

3. Singularities of height hmcuons. Even at a singular point of a singular surface $f$ , wecan define, in the parameter space, the parabolic set of $f$ as the zero set of.

$(\langle f_{uu},f_{u}\cross f_{U}\rangle\langle f_{vv/}f_{u}\cross f_{v}\rangle-\langle f_{uv},f_{u}xf_{U}\rangle^{2})(u, \iota,)$] arity The singularities of the parabolic set of singular
The parabolic set of $f$ has asurfaces with one of -simple singularities of -codim.
are investigated in [21].
$\sin\varphi$$\mathscr{A}$$\leq 3$$\mathscr{A}_{e}$Theorem 3.1. Let a smooth map-germ $f$ :beor singularity, and let the origin be not tnflection point.
$(\mathbb{R}^{2},0)arrow(\mathbb{R}^{3},0)$$C_{k}$$\mathscr{A}$-equtvalent to one of$S_{k},$$B_{k},$$F_{4}$(1)

There is a branch , of the parabolic set of$f$ , which is a chamcteristic tangential curve.

(2)

Let
be a parameterization $ofP$ on $f$ wtth $\mathscr{L}(0)=(0,0,0)$ , and let $b(t)$ and, respectively. Then the height
denote by the unit binormal vector and the torsion of$w=\pmb(0)$$fi4$ nction h$(0,0)$$f$on in
has a singularity atof type$P$$\mathscr{L}(t)$$\tau(t)$$\mathscr{L}(t)$$A_{2}\Leftrightarrow\tau(0)\neq 0$$A_{3}=\tau(0)=0, \tau’(0)\neq 0,$$A_{\geq 4}\Leftrightarrow\tau(0)=\tau’(0)=0.$(3)

The bmnch has at least -point contact withshown in the following table:
$P$

$m$

$tts$

tangent ltne at (0,0), where

$m$

is as

For Whitney umbrellas, an analogs theorem was obtained in [21]. If we replace the assumption that the origin is not inflection point with the assumption that the origin is degenerate
inflection point and has 3-point contact with the principal plane at the origin, Theorem3
holds except thaton $f$ in $w=\pm b(0)$ does not have an singularity at (0,0).
$\hat{\gamma}$$h_{w}$$A_{2}$

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4. Singularities ofdistance squared iunctions. Let :be a smooth map$(u,$j^{2}f(O,0)$germ of corank 1 at the origin , and let
be -equivalent to v^{2},0)$ . The locus
of points where the distance squared function
has a degenerate singularity at (0,0) is
called the focal locus. The focal locus can be considered as an anal gy of the focal conic ofWhitney umbrellas (cf. [7, Lemma 3.3]).
$f$

$0$

$(\mathbb{R}^{2},0)arrow(\mathbb{R}^{3},0)$$\mathscr{A}$$d_{p}$$p$$0$

be a smooth map-germ of corank 1 at the origin , andTheorem 4.1. Letf:
letj2 $f(0,0)$ be -equivalent to $(u, v^{2},0)$ .
$(\mathbb{R}^{2},0)arrow(\mathbb{R}^{3},0)$$0$$\mathscr{A}$(1)

The origin is not an inflection point if and only if the focal locus is a pair of two intersecting lines (a unique line and a line).

(2)

The origin is a non-degenerate inflection point if and only if the focal locus is a twoparallel lines (a unique line and a line).

(3)

The origin is a degenerate inflection point if and only if then the focal locus is a uniqueline.

FIGURE. 4: Focal loci of$S_{1}^{+}.$

Wave-fronts are described by Huygens’s principle as follows: The wave-frontofa surface
at a instant is the envelope of spherical wavelets from points on the surface at the instant.
Even if a surface is smooth its wave-fronts in general have singularities. The caustic of asurface is the envelope of the normal rays to the surface. The caustic can be thought of theset of the singularities of all wave-fronts. The wave-front and caustic of a surface are thediscriminant and bifurcation sets of the family of the distance squared function on thesurface, respectively.
$D$$d_{p}$

Theorem 4.2. Assume that $f$ :is a smooth map-germ of corank 1 at the origin$k-jetj^{k}f(0,0)$
whose
is given in the form (2.1). Moreover, let $f$ be -equtvalent to one ofor singularity, and let the origin be not degenerate inflection point. The singularities offronts and caustics off atare shown in Table 2 and 3,
respectively.
TABLE 2. Singularities offronts.
$(\mathbb{R}^{2},0)arrow(\mathbb{R}^{3},0)$$\mathscr{A}$$B_{k},$$C_{k}$$F_{4}$$p= \tilde{n}(0,\theta)\int\tilde{\kappa}_{1}(0,\theta)(cos\theta\neq 0)$$\frac{conditionfor(0,\theta)singularity}{notrugerelati\nu eto\tilde{v}_{1}cuspidaledge}$

lst ridge relative$to\tilde{v}_{1},$notsub-parabolic relative

swallowtail$to\tilde{v}_{2}$$S_{k},$

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TABLB 3. Singularities ofcaustics.
$\frac{conditionfor(0,\theta)singularity}{notridgerelati\iota_{!}/eto\tilde{v}_{1}non-singular}$

lstridge relativeHere,$(0,\theta)$

is lst ridge relative$a$$to\tilde{v}_{1}$$to\tilde{v}_{1}$cuspidal edge

if the ridge curve relative$to\tilde{v}_{1}$is transverse$to\tilde{v}_{1}$at$(0,\theta)$.

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$J$

Department of Mathematics,
Saitama University,
Saitama 338-8570, Japan.
-mails: mhasegawa@mail.saitama-u.ac.jp, masaruhasegawa@hotmail.com$E$$n$