数理解析研究所講究録
第 1868 巻 2013 年 74-82

74

NEGATIVELY CURVED MOBIUS STRIPSON A GIVEN KNOTKOSUKE NAOKAWA

1. INTRODUCTIONIn Euclidean 3-space , a space curve is called regular if its velocity
vector never vanishes.
closed regular space curve inis called acurve)
if
it has no self-intersections. We consider
knot (or simple closed
the following problem:$\mathbb{R}^{3}$$A$$\mathbb{R}^{3}$$\gamma$

Problem. Give a necessary and sufficient condition for the existenceof closed strips along a given knot which have prescribed negative
Gaussian curvature at each point on$\gamma$$\gamma.$

The positively curved case has been studied by Gluck-Pan [3]. It has
been proved that any positively curved strip along admits only one
isotopy type. On the other hand, the flat (zero Gaussian curvature)case has been studied in Chicone-Kalton [1], Rgen [8], and in theauthor’s previous work [5]. The purpose of this paper is to report on
the outline of the answer [6] for the above problem in the case thatis real-analytic.
$\gamma$$\gamma$

2. PRELIMINARIES:
We first define some terminologies. Let
be aparameter
arc-length,-knot withwhere
and is the$:=$total arc-length of . We assume that the curvature functionof is positive everywhere. The torsion function of is definedby$\gamma=\gamma(s)$$C^{\infty}$$s$$\mathbb{S}^{1}$$\mathbb{S}^{1}arrow \mathbb{R}^{3}$$l$$:=\mathbb{R}/l\mathbb{Z}$$\kappa(\mathcal{S})$$\gamma$$|\gamma"(s)|$$\gamma$$\gamma$$\tau:=\frac{\det(\gamma’,\gamma",\gamma"’)}{\kappa^{2}}.$

Under this definition, for example, the torsion function of the clockwise
helix $t\mapsto(\cos t, \sin t, t)$ is positive. We fix a sufficiently small
The integer$\epsilon>0.$

(2.1)
$*$

$SL(\gamma)$

$:=$

Link

$(\gamma(s), \gamma(\mathcal{S})+\epsilon n(\mathcal{S}))\in \mathbb{Z}$Partly supported by the Grant-in-Aid for JSPS Fellows.

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is called the self-linking number of , where Link
number between two knots and (cf. [7], [8]).
(resp.
Let) be the equivalence relation inby$\gamma$is the linking$\beta$$\alpha$$\sim-$$\sim_{+}$$(\alpha, \beta)$$\mathbb{R}\cross(-\epsilon, \epsilon)$

generated,and setThen,(resp.
) is homeomorphic$M$M\"obius
to a cylinder (resp. a
strip). Suppose thatis eitheror$F:Marrow\mathbb{R}^{3}$. For a sufficiently small, a-embedding$F(s,0)=\gamma(s)$satisfying
for eachis called a closed strip along$B$
Let be the boundary of the image of a closed strip $F$ along . The
number of connected components of $B$ is one or two, and the formercase occurs if and only if $M$ is non-orientable. We assign the orientationinduced by to all (one or two) connected components of $B$ . Then,$(s, u)\sim\pm(s+l, \pm u) (\mathcal{S}\in \mathbb{R}, |u|<\epsilon)$$M_{\pm}:=\mathbb{R}\cross(-\epsilon, \epsilon)/\sim\pm\cdot$$M_{-}$$M_{+}$$M_{+}$$M_{-}$$\epsilon>0$$C^{\infty}$$\mathcal{S}\in \mathbb{S}^{1}$$\gamma.$$\gamma$$\gamma$(2.2)

Mtn $(F)$$:= \frac{1}{2}$Link$(\gamma, B)\in(1/2)\mathbb{Z}$is called the topological twisting number, or the twisting number for
short, of $F$ , where Linkis the sum of linking numbers of all$B$connected components of . The isotopy type of $F$ is determined by
the isotopy type of and its twisting number. The twisting number
Mtn $(F)$ becomes a half-integer if and only lf $M$ is non-orientable. If thetwisting direction of $F$ is clockwise, then the twisting number Mtn $(F)$is positive.
$(\gamma, B)$$\gamma$

3. MAIN THEOREM
Let

be a

-knot (i.e. real-analytic knot) whosecurvature function is positive everywhere. In this case, we can give anecessary and sufficient condition for the isotopy types of negatively
curved strips on , using a prescribed negative Gaussian curvaturefunction $K=K(s)$ along . The following assertion is an answer tothe problem in the introduction:
Main Theorem([6]). Let $K=K(s)$ :be a negatively
valued -function. Define two -functions$\gamma=\gamma(s):\mathbb{S}^{1}arrow \mathbb{R}^{3}$$C^{\omega}$$\gamma$$\gamma$$\mathbb{S}^{1}arrow(-\infty, 0)$$C^{\omega}$$C^{\omega}$$b_{K}^{-}(s):=-\tau(s)-\sqrt{|K(s)|},$$b_{K}^{+}(s):=-\tau(s)+\sqrt{|K(s)|}$$(\mathcal{S}\in \mathbb{S}^{1})$.

(resp.
Denote by) the minimum of(resp. maximum of, and define the open interval.
(1) Suppose. Then,(i) for each $n\geq SL(\gamma)(n\in(1/2)\mathbb{Z})$ , there exists a.real-analytic
closed strip along such that its Gaussian curvature at eachpoint on
is $K(s)$ and its topological twisting number is. On the other hand,$B_{K}^{+}$$B_{K}^{-}$$b_{K}^{-}(s)$$b_{K}^{+}(s))$$I_{K}$$I_{K}\subset(0, \infty)$$\gamma$$\gamma(6)$$n$$:=(B_{K}^{-}, B_{K}^{+})$

76(ii) for each $n<SL(\gamma)(n\in(1/2)\mathbb{Z})$ , there does not exist such a
strip.

(2) Suppose $0\in I_{K}$ . Then, for each $n\in(1/2)\mathbb{Z}$ , there exists a realanalytic closed strip along such that its Gaussian curvature at
is $K(s)$ and its topological twisting number iseach point on$\gamma$$\gamma(s)$$\gamma\iota.$(3) Suppose $I_{K}\subset(-\infty, 0)$ . Then,(i) for each $n\leq SL(\gamma)(n\in(1/2)\mathbb{Z})$ , there exists a real-analytic
closed strip along such that its Gaussian curvature at each
is $K(s)$ and its topological twisting number ispoint on. On the other hand,(ii) for each $n>SL(\gamma)(\prime n\in(1/2)\mathbb{Z})$ , there does not exist such a
strip.
Moreover, in (1) $-(i),$ (2) and (3) $-(i)$ , the closed strip can be constructed$\gamma$$\gamma(\mathcal{S})$$r\iota$

as a ruled surface.

-function satisfies
Any pair of -knot and negatively valued$0\in I_{K}$ or $I_{K}\subset(-\infty, 0)$ . On theone of the three cases of, then it seems, instead ofother hand, if and $K$ are of classdifficult to describe such a necessary and sufficient condition. However,, then the same assertion as the Main Theorem
ifandholds, as follows:$f$$C^{\omega}$$C^{\omega}$$\gamma$$I_{K}\subset(0, \infty),$$C^{\infty}$$C^{\omega}$$\gamma$$B_{K}^{-}\neq 0$$B_{K}^{+}\neq 0$

be a-knot with positive curvature, andCorollary 3.1. Let$K(s)$ a negatively valued
and-function. Ifthen the item (1) holds as in the Main Theorem. Similarly, if$(-\infty, 0)$ and, then the item (3) holds. On the other hand, if$0\in I_{K}$ , then the item (2) holds without any
modifications.
$C^{\infty}$$\gamma(s)$$C^{\infty}$$I_{K}\subset(0, \infty)$$B_{K}^{-}\neq 0,$$I_{K}\subset$$B_{K}^{+}\neq 0$

Example 3.2. We consider twisting numbers of closed strips along theunit circle$\gamma_{0}(s) :=(\cos s, \sin s, 0) (s\in \mathbb{R})$which have Gaussian curvature $K(s)=-1$ at each point on . Since
the torsion function of vanishes identically, we can calculate$-1$ and $b_{K}^{+}(s)=1$ for. Therefore, we have $I_{K}=(-1,1)$ . By
, we can(2) of the Main Theorem, for any twisting number
construct a strip containing and having Gaussian curvature-l along. For example, the four pictures in Figure 1 are all such examples
which have twisting number $n=-1/2,0,1/2,1$ , respectively. The
first figure is the reflection of the M\"obius strip of the third example(which will be defined soon) with respect to the $xy$ -plane. The second$\gamma_{0}$$b_{K}^{-}(s)=$$\gamma_{U}$$s\in \mathbb{R}$$n\in(1/2)\mathbb{Z}$$\gamma_{0}$$\gamma_{0}$77figure is the image of$F_{)}(b, u):=\gamma_{(},(s)+u(\sin s_{\rangle}-\cos s, 1) (s’\in \mathbb{R}, |u|<1/5)$,

which is a part of a one-sheeted hyperboloid. The fourth figure is theimage of
$F_{1}(s, u)=\gamma_{0}(s)+u(-\cos^{2}s, -\cos s\sin s, \sin s)$.
This example is a ruled closed strip generated by a ruling vector which
twists just once clockwise at constant speed when is traversed once.
The third figure is a ruled $M6bius$ strip defined by$(s\epsilon \mathbb{R}, |u|<1/5)$$\gamma_{U}$$F_{1/2}(s, u)=\gamma_{0}(s)+u\xi(s) (s\in \mathbb{R}, |u|<1/5)$

,

where
$\xi(6):=p(s\rangle e\langle s)+\cos\theta(6^{\backslash })n(s)+\sin\theta(s)b(6^{\cdot}\rangle_{2}$$p(s):= \frac{-1+\cos s}{2\sin\theta(s\rangle}, \theta(s);=\frac{s+\sin s}{2},$and $\{e, n, b\}$ is the Frenet frame of . Since the pair of the two functions
andsatisfies the three conditions (4.9), (4.11), 4.12),
this strip is a $M6bius$ strip with Gaussian curvature-l at each point on. We remark that $p(s)$ can be also real-analytic at $s=0(mod 2\pi\rangle,$
so $p(s)$ is well-defined on . To find such functions $p(s)$ and, wemust control the jets of these functions at all points where $\sin\theta(s)=0.$$\gamma$$p(s\cdot)$$\theta(s)$$\langle$$\gamma_{0}$$\mathbb{R}$$\theta(s\rangle$

FIGURE 1. Closed ruled strips containing which have
Gaussian curvature-l along . Their twisting numbers$\gamma_{0}$$\gamma_{0}$$are-1/2,0,1/2,1$ , respectively.

Example 3.3. The torus knot defined by$\gamma(t)$$:=( \cos t-\frac{1}{10}\cos t\cos 5t, \sin t-\frac{1}{10}\cos 5t\sin t, -\frac{1}{10}\sin 5t)$

coils 5 times anti-clockwise around a solid torus as in Figure 2. The
curvature functionof does not vanish, and the torsion functionof is less than-l. Since the knot
is isotopic to theunit circle centered the origin in the -plane, the self-linking number
is equal to $-5$ , whereis sufficiently small and is theprincipal normal vector field of$\kappa(s)$$\tau$$\gamma$$t\mapsto\gamma(t)+\epsilon n(s)$$\gamma$$/xy$$SL(\gamma)$$\epsilon>0$$\gamma,$

$n$

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FIGURE 2. The first figure is the image of . The second and third figures are an orientable strip and a $M6bius$
strip along which have Gaussian curvature-l at eachpoint on and twisting numbers $-5$ and $-9/2$ , respectively. We can see a joint of the M\"obius strip in the top
center of the third figure.
$\gamma$$\gamma$$\gamma$Then, the function
Now, we set $K(s)=-1$ foris positive everywhere. Hence, the pair of and $K$ satisfies. By the item (1) $-(i)$ in the Main Theorem, for each $n\geq SL(\gamma)=$$-5(n\in(1/2)\mathbb{Z})$ , there exists a closed strip $F$ alongsuch that theis equal to-l and the twisting number
Gaussian curvature of $F$ on
Mtn $(F)$ is.n. For example, the second and third surfaces of Figure 2 arean orientable strip and a M\"obius strip along in the cases of $n=-5$
and $n=-9/2$ , respectively. On the other hand, by (1) $-(ii)$ , for each$n<-5(n\in(1/2)\mathbb{Z})$ , there does not exist such a strip along
whose
Gaussian curvature is-l at each point on$\mathcal{S}\in \mathbb{S}^{1}$$b_{K}^{\pm}(s)$$I_{K}\subset$$\gamma$$(0, \infty)$$\gamma$$\gamma(s)$$\gamma$$\gamma$$\gamma.$

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4. OUTLINE OF THE PROOF
Let be a given-knot with arc-length parameter whose curvature function
never vanishes. Any closed strip :along can be expressed by$C^{\omega}$$s$$\gamma$$F$$\kappa(s)$$Marrow \mathbb{R}^{3}$$\gamma$(4.1)$F(s, u)=\gamma(s)+u\xi(s)+u^{2}\eta(s, u) (s\in \mathbb{R}, |u|<\epsilon)$

,

where
is a
-vector field along andis a-vector fieldalong . The strip is ruled if
vanishes identically. The periodicity condition that the strip is closed is equivalent to(4.2) $\xi(s+l)=\pm\xi(s)$ , $\eta(s+l, u)=\eta(s, \pm u)$$\xi(s)$$C^{\omega}$

$F$

$F$

$C^{\omega}$$\eta(\mathcal{S}, u)$$\gamma$$\eta(s, u)$$F$$(s\in \mathbb{R}, |u|<\epsilon)$

for $M=M_{\pm}$ , respectively. The condition that$F$

is equivalent to(4.3)$\gamma’(s)\cross\xi(s)\neq 0 (s\in \mathbb{R})$gives an embedding,

where is the vector product in
Given
and
satisfying(4.2) and (4.3), a closed strip can be constructed by (4.1).
From now on, we may assume that any strip is expressed as in theform in (4.1). By a suitable parameter change, we can normalize
that
$x$

$\mathbb{R}^{3}$$\eta(s, u)$$\xi(s)$$F$$\xi(s)$

$SO$

(4.4)

$|\gamma’(\mathcal{S})\cross\xi(s)|=1 (s\in \mathbb{R})$Then, the Gaussian curvature $K(s)$ of $F$ along
to be
(4.5)

.

$\gamma(\mathcal{S})$can be calculated$K=2\det(\gamma’, \xi, \gamma")\det(\gamma’, \xi, \eta_{0})-\det(\gamma’, \xi, \xi’)^{2},$$:=\eta(s, 0)$ for
whereframe of . The vector fieldtion of $\{e(s), n(s), b(s)\}$ by$\eta_{0}(s)$$s\in \mathbb{R}$$\gamma$(4.6)$\xi(s)=p(s)e(s)+\cos\theta(s)n(s)+\sin\theta(s)b(s) (s\in \mathbb{R})$where $p(s)$ and(4.7)

for some$\xi(s)$. Let $\{e(s), n(s), b(s)\}$ be the Frenetcan be expressed as a linear combina-$\theta(s)$are certain$C^{\omega}$,-functions satisfying$p(s+l)=(-1)^{m}p(s) , \theta(s+l)=\theta(s)+’rr\iota\pi (s\in \mathbb{R})$$\prime/r\iota\in \mathbb{Z}$. The number
Gtn $(F)$ $:=m/2\in(1/2)\mathbb{Z}$is called the geometwic twisting number of . The topological twisting
number Mtn $(F)$ and geometric twisting number Gtn $(F)$ satisfy
$F$

(4.8)

$Mtn(F)=SL(\gamma)+Gtn(F)$ .

Namely, we can use the geometric twisting number Gtn $(F)$ , instead of
the topological twisting number Mtn $(F)$ when we consider the isotopy

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type ofto be$F$. Using (4.6), the right-hand side of (4.5) can be calculated(4.9)$K=-q\kappa\sin\theta-(\theta’-p\kappa\sin\theta+\tau)^{2},$where(4.10)$q(s) :=2\det(\gamma’(\mathcal{S}), \xi(\mathcal{S}), \eta_{0}(s))$.

By using (4.9), we can reduce the proof of the Main Theorem into
considering the existence and non-existence of -functions $p(s),$satisfying the following conditions:and$C^{\omega}$$\theta(s)$$q(\mathcal{S})$(4.11)$q(s)\kappa(s)\sin\theta(s)+K(s)\leq 0 (s\in \mathbb{R})$is satisfied, and for each
(4.12)$s\in \mathbb{R}$, either$\theta’(s)=p(s)\kappa(s)\sin\theta(s)-\tau(s)+\sqrt{|q(6)\kappa(s)\sin\theta(s)+K(s)|}$or(4.13)$\theta’(s)=p(s)\kappa(s)\sin\theta(s)-\tau(s)-\sqrt{|q(6)\kappa(s)\sin\theta(6)+K(s)|}$:and aholds. In fact, for a pair ofsat-functions $p(s),$ $q(s)$ andprescribed integer $m$ , if we takeby (4.6) andisfying (4.7), (4.11), (4.12) (or (4.13)), and define$:=q(s)e(s)\cross\xi(s)$ , then these three functions satisfy (4.9).
Hence, the strip
defined by (4.1) is a closed strip along so thatis equal to $K(s)$ and theits Gaussian curvature at each point on
geometric twisting number Gtn $(F)$ is $m/2$ . In order to construct such-functionfunctions, we use a special technique to approximate a
by a Fourier polynomial while we fix the jet of the function at finitely
many points. This technique was given in the author’s previous paper [5, Lemma A.4]. Fortunately, it can be applied also for negatively
curved closed strips along$(\gamma, K)$$\mathbb{S}^{1}arrow \mathbb{R}^{3}\cross(-\infty, 0)$$C^{\omega}$$\theta(s)$$\xi(s)$$\eta(s, u)$$F$$\gamma$$\gamma(s)$$C^{\infty}$$\gamma.$

5. REMAINING PROBLEMS
be two negatively curved strips. We writeLetand
if they can be deformed into one another through negatively curved
surfaces, and write
if they are isotopic. Although, its converse is not true. In fact, the following examples
implies
are two negatively curved strips which have the same isotopy type, butthey cannot be deformed into one another through negatively curved$F_{1}$$F_{1}\approx_{-}F_{2}$$F_{2}$$F_{1}\approx F_{2}$$F_{1}\approx F_{2}$surfaces:$F_{1}\approx_{-}F_{2}$81
Example 5.1 ([6]). Two negatively curved stripsby$F_{1},$$F_{2}$are defined$F_{1}(s, u).:=f_{0}(s, u) (|s|\leq\pi, |u|<1/2)$ ,$F_{2}(s, u):=f_{0}(u\cos s, u\sin s) (|s|\leq\pi, 1/2<u<1)$ ,respectively, where $f_{0}(x, y)$ $:=(\sqrt{1+y^{2}}\cos x, \sqrt{1+y^{2}}\sin x, y)$ is a
one-sheeted hyperboloid. Figure 3 shows the images ofandwhich are described with curvature lines of$F_{1}$$F_{2},$$f_{0}.$FIGURE 3. An example of$F_{1}\approx F_{2}$and$F_{1}\not\simeq {}_{-}F_{2}$If a negatively curved strip $F$ is orientable, then we can define therotational index of the principal direction of curvature of alongin the tangent planes of $F$ when is traversed once $(cf. R\emptyset gen[8])$ .
The rotational index does not change under deformations of $F$ keeping
negatively curved surfaces. On the other hand,andare isotopic,
but their rotational indices are and. Therefore, they cannot be
deformed into one another through negatively curved surfaces.
$F$$\gamma$$\gamma$$F_{1}$$0$$F_{2}$$\pm 1$conjectured that two negatively curved surfaces having
the same isotopy type can be deformed into one another through negatively curved surfaces if and only if their rotational indices are equal.
Recently, Ghomi-Kossowski [2] solved this conjecture affirmatively byusing the -principle. Therefore, it is interesting to consider the following problem:$R\emptyset gen[8]$$h$Problem. Give a necessary and sufficient condition for the existence ofnegatively curved closed strips along a given knot which have prescribed
twisting numbers and rotational indices of principal curvatures.

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REFERENCES[1] C. Chicone and N. J. Kalton: Flat embeddings of the M\"obius strip in
Commun. Appl. Nonhnear Anal. 9 (2002), 31-50.
[2] M. Ghomi and M. Kossowski: -principles for hypersurfaces with prescribedprincipal curvatures and directions, Trans. Amer. Math. Soc. 358 (2006), 4379$\mathbb{R}^{3},$

$h$

4393.

[3] H. Gluck and L.-H. Pan: Embedding and knotting of positive curvature surfaces
in 3-space, Topology 37 (1998), 851-873.
[4] Y. Kurono and M. Umehara: Flat Mobius strips of given isotopy type inwhose centerlines are geodesics or lines of curvature, Geom. Dedicata 134$R^{3}$

109-130.
[5] K. Naokawa: Extrinsically flat Mobizes strips on given knots in 3-dimensiond$(2(K)8),$

spaceforms, to appear in Tohoku Math. J.
[6] K. Naokawa, Negatively curved Mobius strips on given knots, J. Knot Theory
Ramifications 22 (2013), 1350023, 12 pp.
[7] W. F. Pohl : The self-linking number of a closed space curve, J. Math. Mech.
17 (1968), 975-985.
[8] P.
: Embedding and knotting of flat compact surfaces in 3-space, Comment. Math. Helv. 76 (2001), 589-606.
$R\emptyset gen$

DEPARTMENT OF MATHEMATICS, TOKYO INSTITUTE OF TECHNOLOGY, 2-12-1
OOKAYAMA, MEGURO-KU, TOKYO 152-8551, JAPAN-mail address: naokawalOis. titech. ac. jp$E$