数理解析研究所講究録
第 1664 巻 2009 年 68-70

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On spacelike curve in nullcone 3-space
L.L. Kong and D.H. Pei
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R.China

Basic notions

1The nullcone is one kind of pseudo-sphere of Minkowski space. Our aim in this article is to develop thestudy for spacelike curve in nullcone 3-space by Bruce and Giblin’s singularity theory. In order to study
the spacelike curve of nullcone 3-space, we need to develop differential geometry of spacelike curve innullcone 3-space similarly as it was done for curves in Euclidean space [2].
Letbe a 4-dimensional vector space. For any two vectors$x=(x_{1}, x_{2}, x_{3}, x_{4})$ and $y=(y_{1}, y_{2}, y_{3}, y_{4})$ in, the pseudo-scalar product ofandis defined by$y\rangle=-x_{1}y_{1}+\sum_{i=2}^{4}x_{i}y_{i}$ .
is called a Minkowski 4-space and denoted by. A vector in$\mathbb{R}^{4}=\{(x_{1}, x_{2}, x_{3}, x_{4})|x_{1}, x_{2}, x_{3}, x_{4}\in \mathbb{R}\}$$\mathbb{R}^{4}$$(\mathbb{R}^{4},$$\langle x,$$\{,$$x$$\rangle)$$y$$\mathbb{R}_{1}^{4}$$x$$\mathbb{R}_{1}^{4}\backslash \{o\}isca11edspacelike,lightlike\circ rtimenormofavectorx\in \mathbb{R}_{1}^{4}isdefinedby\Vert x||=l|\langle x,$$x\}|Foranyx,y\in \mathbb{R}_{1}^{4},$ $wesayxpseudo- perpendicular$$y\}=0$ . For a vector
to if
and a real number , we define a hyperplane with pseudo normal
by $HP(v, c)=\{x\in \mathbb{R}_{1}^{4}|(x,$ $v\rangle=c\}$ . $HP(v, c)$ is called a timelike hyperplane, a spacelike hyperplane ora lightlike hyperplane if is timelike, spacelike or lightlike respectively. Now, define the nullcone 3-spaceby $NC^{3}=\{x=(x_{1}, x_{2}, x_{3}, x_{4})\in \mathbb{R}_{1}^{4}|x_{1}\neq 0,$ $\{x,x\rangle=0\}$ , the de Sitter 3-space by$\{x, x\}=1\}$ and the hyperbolic 3-space by $H_{1}^{3}=\{x\in \mathbb{R}_{1}^{4}|\langle x, x\}=-1\}$ . If $x=(x_{1}, x_{2}, x_{3}, x_{4})$ is alightlike vector, then $x_{1}\neq 0$ . Therefore
1,,$x_{1}=1\}$ .
iscalled the nullcone unit 2-sphere.
, we define a vector
For anyby$v\in \mathbb{R}_{1}^{4}$$\langle x,$

$y$

$c$

$v$

$v$

$S_{1}^{3}=\{x\in \mathbb{R}_{1}^{4}|$$\tilde{x}=$$x_{1},$ $x_{2},$$($$x_{3}\in \mathbb{R}_{1}^{4}$$\lrcorner xx_{1})\in S_{+}^{2}=\{x\in \mathbb{R}_{1}^{4}|\langle x, x\}=0,$$\frac{x}{x}Z,$$\frac{x}{x}A11$$S_{+}^{2}$$x_{1}\wedge x_{2}\wedge x_{3}$$x_{1}\wedge x_{2}\wedge x_{3}=|\begin{array}{llll}-e_{1} e_{2} e_{3} e_{4}x_{1}^{1}x_{2)}^{1}x_{3}^{1}’ x^{2}x_{2}^{2}x_{3)}21| x^{3}x_{3}^{3}x_{3)}^{2}1^{|} x_{4}^{4}x_{2}^{4}x_{3}1\end{array}|$

whereare the canonical basis of, so that

. It is easy to check thatis pseudo orthogonal to $x_{i}(i=1,2,3)$ .
Let : $Iarrow NC^{3};\gamma(t)=(\gamma_{1}(t), \gamma_{2}(t), \gamma_{3}(t), \gamma_{4}(t))$ be a smooth regular curve in $NC^{3}(i.e.,\dot{\gamma}(t)\neq 0$ for
any $t\in I)$ , where is an open interval. The curve is called a spa celike curve if
is positive$t_{0}\in I$ isfor any $t\in I$ . The arc-length of a spacelike curve , measured from.
, where $\gamma’(s)=d\gamma/ds(s)$ . We say that a spacelike
Then a parameter is determined such thatcurve is parametenzed by arc-length if it satisfies that. Throughout the reminder in thisarticle, will denote the arc-length parameter. Let $t(s)=\gamma’(s)$ . we call $t(s)$ an unit tangent vector ofat . The signature of is defined to be$e_{1},$$e_{2},$$e_{3},$$e_{4}$$\{x, x_{1} A x_{2} A x_{3}\}=\det(x, x_{1}, x_{2}, x_{3})$$\mathbb{R}_{1}^{4}$

and$x_{\mathfrak{i}}=(x_{f}^{i}, x_{i}^{2}, x_{i}^{3}, x_{i}^{4})$$x_{1}\wedge x_{2}\wedge x_{3}$$\gamma$$I$$\langle\dot{\gamma}(t),\dot{\gamma}(t)\rangle$$\gamma$$\gamma(t_{0}),$$\gamma$$\Vert\gamma’(s)\Vert=1$$s$$\Vert\gamma’(s)\Vert=1$$\gamma$$s$$s$$\gamma$$x$$\delta(x)=$sign$(x)=\{\begin{array}{ll}1 x:

spacelike;0 x:lightlike;-1 x:timelike.\end{array}$2000 Mathematics Subject classification. $53B30,58K05,57R70$ .
Key Words and Phrases. spacelike curve, nullsphere Gauss map, nullsphere height function
Work partially supported by NSF of China No.10871035 and NCET of China No.05-0319.
’E-mail:peidh340@nenu.edu.cn$s(t)= \int_{t_{0}}^{t}\Vert\dot{\gamma}(t)\Vert dt$

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For any nonlightlike curve : $Iarrow NC^{3}$ , which is parameterized by arc-length and satisfies $k_{1}(s)\neq 0$ .
We can construct a pseudo-orthogonal frame $\{t(s), n_{1}(s), n_{2}(s), n_{3}(s)\}$ ofalong which satisfies thefollowing Frenet-Serret type formulae:$\gamma$$\mathbb{R}_{1}^{4}$$\gamma$$\{\begin{array}{ll}t(s) = \gamma’(s);t’(s) = k_{1}(s)n_{1}(s);n_{1}’(s) = -\delta_{1}k_{1}(s)t(s)+k_{2}(s)n_{2}(s);n_{2}’(s) = \delta_{3}k_{2}(s)n_{1}(s)+k_{3}(s)n_{3}(s);n_{3}’(s) = \delta_{1}k_{3}(s)n_{2}(s),\end{array}$and $\delta_{i}=\delta(n_{i})(i=1,2,3)$ .
Let $n_{2}(s)$ be a timelike vector. Then $n_{j}(j\neq 2)$ is a spacelike vector.
Define maps

where$n_{1}= \frac{\gamma’’}{||\gamma^{t}||}=L’’k_{1}’ n_{i}=\frac{n_{i-1}^{J}+\delta_{0}\delta_{1}..\delta_{-1}k_{i-1}n_{i-2}}{\delta_{0}k_{i}},$$\delta_{0}=\delta(t)$$NG_{2,j}^{\pm}:Iarrow S_{+}^{2}$

by $NG_{2,j}^{\pm}(s)=n_{j}\overline{\pm n}_{2}(s)(j=1,3)$ . Also define a map$\eta:S_{+}^{2}arrow S_{+\prime}^{2}$by$S_{+}^{2}$$\eta(n_{1}\overline{\pm n}_{2}(s))=n_{2}\overline{\pm n}_{3}(s),$. Each one of2

and is identity on the other elements ofis called the nullsphere Gauss map of .

$\eta(n_{2}\overline{\pm n}_{3}(s))=n_{1}\overline{\pm n}_{2}(s)$$\eta$$NG_{2,j}^{\pm}(j=1,3)$$\gamma$Nullsphere height functions on spacelike curve in $NC^{3}$

Now the function$H_{1}:I\cross S_{+}^{2}arrow \mathbb{R}$is defined by $H_{1}(s, v)=\langle\gamma(s),$$v\rangle$

and the function$H_{2}:I\cross S_{+}^{2}arrow \mathbb{R}$is defined by $H_{2}(s, v)=\langle\gamma(s),$andare called the nullsphere height function on the spacelike, we denote that $h_{1,v_{0}}(s)=H_{1}(s, v_{0})$ and $h_{2,v_{0}}(s)=H_{2}(s, v_{0})$ , then wecurve . For any fixedhave the following theorem.
$\eta(v)\},$$H_{1}$$H_{2}$$v_{0}\in S_{+}^{2}$$\gamma$

Theorem 2.1. Let : $Iarrow NC^{3}$ be an unit speed spacelike curve with $k_{1}(s)\neq 0$ . Then we have thefollowing assertions:(1) $h_{1,v_{0}}’(s_{0})=0(resp. h_{2,v_{0}}’(s_{0})=0)$ if and only if there vist
(resp.
such that $v=$ fi
and$\eta(v)=\tilde{n}(s_{0})),$.
(2) $h_{1,v0}’(s_{0})=h_{1,v_{0}}’’(s_{0})=0(resp. h_{2,v_{0}}’(s_{0})=h_{2,v_{0}}’’(s_{0})=0)$ if and only if$\gamma$$e$$\lambda_{1}$$(s_{0})$$\lambda_{2}$$n(s_{0})=(\lambda_{1}n_{1}\pm\sqrt{\lambda_{1}^{2}+\lambda_{2}^{2}}n_{2}+\lambda_{2}n_{3})(s_{0})\in NC^{3}$$v=n_{3}\overline{\pm n}_{2}(s_{0})$

$($

resp.

$v=n_{1}\overline{\pm n}_{2}(s_{0}))$.

(3) $h_{1,v_{0}^{l}}(s_{0})=h_{1,v_{0}}’’(s_{0})=h_{1,\underline{v_{0}}}’’’(s_{0})=0(resp. h_{2,v_{0}}’(s_{0})=h_{2,v0}’’(s_{0})=h_{2,v_{0}}’’’(s_{0})=0)$ if andonly if $v=n_{3}\overline{\pm n}_{2}(s_{0})(resp. v=n_{1}\pm n_{2}(s_{0}))$ and $k_{2}(s_{0})=0$ .
(4) $h_{1,v_{0}}’(s_{0})=$ . . . $=\underline{h_{1,v_{0}}^{(4)}}(s_{0})=0(resp. h_{2,v_{0}}’(s_{0})= . . . =h_{2,v_{0}}^{(4)}(s_{0})=0)$ if and only if$v=n_{3}\overline{\pm n}_{2}(s_{0})(resp.

v=n_{1}\pm n_{2}(s_{0}))$and $k_{2}(s_{0})=k_{2}’(s_{0})=0$ .

Theorem 2.2. Let
be a spacelike curve in nullcone 3-space. Then:(1), thennever equal to zero.
, then $h_{2,v0}’’(s_{0})$ never equal to zero.
(2)$\gamma(s)$$Ifv_{0}=\tilde{\gamma}(s_{0})$$h_{1,v_{0}}’’(s_{0})$$If\eta(v_{0})=\tilde{\gamma}(s_{0})$Proposition 2.3. $Ij\gamma(s)$ is an unit speed spacelike curve,
are nullsphere height functions,
and$B_{H_{1}}=\{v\in S_{+}^{2}|h_{1,v}’(s)=h_{1,v}’’(s)=0\}$ and $B_{H_{2}}=\{v\in S_{+}^{2}|h_{2,v}’(s)=h_{2,v}’’(s)=0\}$ , then thefollowing conditions are equivalent:;(1) $h_{1,v_{0}}’’’(s_{0})=0$ for $v_{0}=(n_{3}\overline{\pm n}_{2})(s_{0})(resp$. $h_{2,v_{0}}’’’(s_{0})=0$ for(2)
is a singularity of nullsphere Gauss map $NG_{2,3}^{\pm}(resp. NG_{2,1}^{\pm})$ on ;(3) $k_{2}(s_{0})=0$ .
$H_{1}$$H_{2}$$v_{0}=(n_{1}\overline{\pm n}_{2})(s_{0}))$$s_{0}$$\gamma$70
Consider now the particular case of a curve. Given a vector(resp.
) and anumber , denote by $S(v, c)$ the null hyperhorosphere(resp. null hypersphere, null equidistant hyperplane)$\gamma\subset NC^{3}$$v\in S_{+}^{2}$$S_{1}^{3},$$H_{1}^{3}$$c$determined by the intersection of the hyperplane $HP(v, c)$ with $NC^{3}$ .

Proposition 2.4. Suppose that

Proof Since$\tilde{\gamma}(s)=NG_{2,j}^{\pm}(s)$. If$\tilde{\gamma}(s)=NG_{2,j}^{\pm}(s),$ $\gamma(s)=\gamma_{1}(s)NG_{2,j}^{\pm}(s)$.

$NG_{2,j}^{\pm}\iota s$

constant, then$NG_{2,j}^{\pm}(s)$$\gamma(s)$is constant, so

is a straight line.

$\gamma(s)$is a straight line.

$\square$For an unit speed spacelike curve: $Iarrow NC^{3}$ , we now define extended nullsphere height func:
tions
by $H_{1}(s, v)=H_{1}(s,\overline{v})-v_{1}=\langle\gamma(s),\overline{v}\}-v_{1}$ and:by$\tilde{H}_{2}(s, v)=H_{2}(s,\tilde{v})-v_{1}=\{\gamma(s),$, whereand
are the nullsphere height function on .
For any fixed $v_{0}\in NC^{3}$ , we denoteand.
Let $F:NC^{3}arrow \mathbb{R}$ be a submersion and : $Iarrow NC^{3}$ be a spacelike curve. We say that and $F^{-1}(0)$have k-point contact at
if $g(t)=Fo\gamma(t)$ satisfies $g(t_{0})=g’(t_{0})=--$ $=g^{(k-1)}(t_{0})=0,9^{(k)}(t_{0})\neq 0$ .
Then we have the following corollary.
$\gamma$$H_{1}$$I\cross NC^{3}arrow \mathbb{R}$$\tilde{H}_{2}$$\eta(\tilde{v})\rangle-v_{1}$$H_{1}$$I\cross NC^{3}arrow \mathbb{R}$$H_{2}$$\tilde{h}_{1,v_{0}}(s)=\tilde{H}_{1}(s, v_{0})$$\gamma$$\tilde{h}_{2,v_{0}}(s)=\tilde{H}_{2}(s, v_{0})$$\gamma$$\gamma$$t_{0}$

Corollary 2.5. Letnull hyperhorosphere$\gamma$: $Iarrow NC^{3}$ be an unit speed spacelike curve with $k_{1}(s)\neq 0$ . Thenand thehave 4-point contact at
if and only if $k_{2}(s)=0$ and $k_{2}’(s)\neq 0$ , where.

$S(v_{0}^{\pm}, c_{0}^{\pm})$$\gamma$$s_{0}$$v_{0}^{\pm}=n_{3}\overline{\pm n}_{2}(s_{0}),$ $c_{0}^{\pm}=\{\gamma(s_{0}),v_{0}^{\pm}\rangle$

This work is only a preparation for further studying, in the following, we will give the classificationof singularities of nullsphere Gausss map and discuss some geometrical properties of spacelike curve fromsingularity theory viewpoint.

References[1] J. W. Bruce and P. J. Giblin, Generic geometry, Amer. Math. Monthly 90 (1983), 529-545.
[2] J. W. Bruce and P. J. Giblin, Curves and singulareties(second edition) (Cambridge University Press, 1992).

[3] S. Izumiya, D. Pei and T. Sano, The lightcone Gauss map and the lightcone developable of a spacelike curvein Minkowski 3-space, Glasgow Math. J. 42 (2000), 75-89.
[4] L.Kong and D.Pei, On spacelike curve in nullcone 3-space, Preprint(2008).

[5] B. O’Neill, Semi-Riemannian G eometry, (Academic Press, New York, 1983).
[6] I. Porteous, The normal singularities of submanifold, J. Diff. Geom. 5, (1971), 543-564.
Lingling Kong, School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R.Chinae-mail;kong111110nenu. edu. cn
Donghe Pei, School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R.Chinae-mail:pe $idh3400nenu$ . edu. cn