数理解析研究所講究録
第 1664 巻 2009 年 71-74

71Singularities of nullsphere Gauss map for spacelike
surface in nullcone 3-space
D.H. Pei and L.L. Kong

Introduction

1The nullcone in Minkowski 4-spaoe is one kind of Minkowski pseudo-sphere, which is similar with thesphere in Euclidean 4-space. In [6], Izumiya has studied the details of spacelike hypersurfaoe in thenullcone by Legendrian dualities. Our aim in this article is to study spacelike surfaces in nullcone 3-spaceby the method similar to that in [5].
We shall assume throughout the whole article that all maps and manifolds are
unless the contrary
is explicitly stated.
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$y\}=-x_{1}y_{1}+\sum_{i=2}^{4}x_{i}y_{i}$ .
is called a Minkowski 4-spaoe and written by. A vector inis called spacelike, lightlike or timelike if $\{x,x\}$ is positive, zero or negative respectively. The
norm of a vector
is defined by. For any $x,$, we say pseudo-perpendicular$y\rangle=0$ . For a vector
to if
and a real number , a hyperplane with pseudo normal is definedby $HP(v, c)=\{x\in \mathbb{R}_{1}^{4}|\langle x, v\rangle=c\}$. $HP(v, c)$ is called a timelike hyperplane, a spacelike hyperplane or alightlike hyperplane if is timelike, spacelike or lightlike respectively. Now, hyperbolic 3-space is definedby $H_{1}^{3}=\{x\in \mathbb{R}_{1}^{4}|\{x, x\}=-1\},$ de Sitter S-space is defined by $S_{1}^{3}=\{x\in \mathbb{R}_{1}^{4}|\langle x, x\}=1\}$ and thenullcone 3-space is defined by $NC^{3}=\{x=(x_{1}, x_{2}, x_{3}, x_{4})\in \mathbb{R}_{1}^{4}|x_{1}\neq 0,$ $\{x, x\rangle=0\}$ . The 3-dimension
nullcone with vertex inis defined by. If $x=(x_{1}, x_{2}, x_{3}, x_{4})$ is alightlike vector, then. Therefore we have $\tilde{x}=(1,\frac{x}{l},x1’ 1l_{1}\in S_{+}^{2}=\{x\in \mathbb{R}_{1}^{4}|\langle x, x\rangle=0, x_{1}=1\}$ .
is called the nullcone unit 2-sphere.
, X3
For any, we define a vector
by$C^{\infty}$$\mathbb{R}^{4}=\{(x_{1}, x_{2}, x_{3}, x_{4})|x_{1}, x_{2}, x_{3}, x_{4}\in \mathbb{R}\}$$\mathbb{R}^{4}$$\langle x,$$(\mathbb{R}^{4},$$\{,$$x$$\rangle)$

$y$

$x$

$\mathbb{R}_{1}^{4}$$\mathbb{R}_{1}^{4}\backslash \{0\}$$||x||=\sqrt{|\langle x,x\rangle|}$$x\in \mathbb{R}_{1}^{4}$$y$$\langle x,$$v\in \mathbb{R}_{1}^{4}$$y\in \mathbb{R}_{1}^{4}$

$x$

$c$

$v$

$v$

$\lambda$$NC_{\lambda}^{3}=\{x\in \mathbb{R}_{1}^{t}|\langle x-\lambda, x-\lambda\rangle=0\}$$\mathbb{R}_{1}^{4}$$x_{1}\neq 0$$S_{+}^{2}$$x_{1},$ $x_{2}$$\in \mathbb{R}_{1}^{4}$$x_{1}\wedge x_{2}\wedge x_{3}$$x_{1}\wedge x_{2}\wedge x_{3}=|\begin{array}{llll}-e_{1} e_{2} e_{8} e_{4}x_{1}^{1}x_{2)}^{1}x_{3|}^{1} x^{2}x_{2}^{2}x_{3}^{2}1 x^{3}x_{2}^{s^{l}}x_{3}^{3}1 x^{4}x_{2}^{4}x_{3}^{4}1\end{array}|$,is the canonical basis ofand. It can easily check that, so thatis pseudo orthogonal to any $x_{i}(i=1,2,3)$ .
We fix an orientation and timelike orientation of(i.e., a 4-volume form $dV$ , and future time-likevector field, have been chosen). Let $X$ ; $Uarrow NC^{3}$ be an embedding, where is an open subset of .
Denote $M=X(U)$ and identify $M$ with by the embedding $X$ . We say $X$ a spacelike surface if
andare spacelike vectors. Therefore, the tangent space $T_{p}M$ of $M$ is a spacelike subspace(i.e., consists ofspacelike vectors) for any point $p\in M$ . In this case, the pseudo-normal space $N_{p}M$ is a timelike plane(i.e.,
Mathematics Subject classiflcation.Primary 53A35;Secondary $58K05$ .

where$e_{1},$ $e_{2},$ $e_{3},$ $e_{4}$$x_{i}=(x_{*}^{i}, x_{i}^{2}, x_{i}^{3}, x_{i}^{4})$$\mathbb{R}_{1}^{4}$$(x,$ $x_{1}\wedge x_{2}\wedge x_{3}\}=\det(x, x_{1}, x_{2}, X3)$

$x_{1}\wedge x_{2}\wedge x_{3}$$\mathbb{R}_{1}^{4}$

$U$

$U$

$X_{u_{2}}$$\overline{2000}$Work
NSF of China No.10871035 and NCET of China No.05-0319.

partially supported byKey Words and Phrases: spacelike surface, nullsphere Gauss map, nullsphere height function.
E-mail:konglllll@nenu.edu.cn, peidh340@nenu.edu.cn$\mathbb{R}^{2}$$X_{u1}$

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Lorentz plane). Denote by $N(M)$ the pseudo-normal bundle over $M$ . Since this is a trivial bundle, we canarbitrarily choose a future directed unit timelike normal section $n^{T}(u)\in N_{p}M\cap H_{1}^{3}$ , where $p=X(u)$ .
Therefore we can define a spacelike unit normal section $n^{s}(u)$ by,$n^{s}(u)= \frac{n^{T}(u)\wedge X_{u_{1}}(u)\wedge X_{u_{2}}(u)}{||n^{T}(u)\wedge X_{u_{1}}(u)\wedge X_{u_{2}}(u)||}\in S_{1}^{3}$$n^{s}\rangle=0$ . Although
and we have
we could also choose $-n^{S}(u)$ as a spacelike unit normal section
with the above properties, we fix the direction $n^{S}(u)$ throughout this article. $(n^{T}, n^{S})$ is called a future
directed normal frame along $M=X(U)$ . Clearly, the vector $n^{T}\pm n^{S}(u)$ is lightlike. Since
isa basis of $T_{p}M$ , the system $\{X_{u}1’ X_{u_{2}}, n^{T}, n^{S}\}$ provides a basis for.
$X\in N_{p}M,$ $N_{p}M$ is a Lorentzian plane and $X(U)$ is a regular surface, so $\tilde{X}(u)=n^{\overline{T}}+n^{S}(u)$ for any$\langle n^{T},$$\{X_{u_{1}}, X_{u_{2}}\}$$T_{p}\mathbb{R}_{1}^{4}$

or $\tilde{X}(u)=n^{T}-n^{S}(u)$ for any $u\in U$ .
Here, we only consider the case of$u\in U$$\tilde{X}(u)=n^{\overline{T}}-n^{S}(u)$for $u\in U$. The case of$\tilde{X}(u)=n^{\overline{T}}+n^{S}(u)$can be discussed similarly, Define two maps of $M=X(U)$ as$NG_{M}^{\pm}:Uarrow S_{+}^{2}$,$NG_{M}^{\pm}(u)=n^{\overline{T}}\pm n^{S}(u)$,each one of these maps is called
Gauss map. Under the identification of $M$ with through $X$ ,we have the linear mapping $d_{p}(n^{T}\pm n^{S})$ : $T_{p}Marrow\tau_{p}R=T_{p}M\oplus N_{p}M$ . Consider the orthogonal projec:
tions: $T_{p}M\oplus N_{p}Marrow N_{p}M$ . Define $d_{p}(n^{T}\pm n^{S})^{t}=\pi^{t}0\underline{d_{p}(n}^{T}\pm n^{S})$and$U$$\underline{nullsp}here$$\pi^{t}$$\pi^{n}$$\underline{T_{p}M}\oplus N_{p}Marrow T_{p}M$and$d_{p}(n^{T}\pm n^{S})^{n}=\pi^{n}\circ d_{p}(n^{\overline{T}}\pm n^{S})$. The linear transformations$S_{p}^{\pm}(n^{T}, n^{S})=-d_{p}(n^{T}\pm n^{S})^{t}$andare respectively called the-shape operator and the normal connection with respect
toof $M=X(U)$ at $p=X(u)$ .
The eigenvalues ofdenoted by $\{\kappa_{i}^{\pm}(n^{T}, n^{S})(p)\}(i=1,2)$ are called the-nullsphereprincipal curvature with respect toat . Then the nullsphere Gauss-Kronecker curvature withrespect toat $p=X(u)$ is defined as$(n^{T}, n^{S})$$d_{p}(n^{T}\pm n^{S})^{n}$$(n^{T}, n^{S})$$(n^{T}, n^{S})$$S_{p}^{\pm}(n^{T}, n^{S})$$(n^{T}, n^{S})$$p$$(n^{T}, n^{S})$$K_{n}^{\pm}(n^{T}, n^{S})(p)=\det S_{p}^{\pm}(n^{T}, n^{S})$.

We say that a point $p=X(u)$ is a $(n^{T}, n^{S})$ -umbilic point if all the principal curvatures coincide atand thusfor some function. We say that $M=X(U)$ is totally$(n^{T}, n^{S})$$(n^{T},n^{S})$$M$-umbilic if all points onare-umbilic.
We deduce now the nullcone Weingarten formula. Since
andare spacelike vectors, wehave a Riemannian metric (the first fundamental form) on $M$ defined by $ds^{2}= \sum_{i=1}^{2}g_{tj}du_{i}du_{j}$ , where$g_{1j}(u)=\langle X_{u}.,$for any $u\in U$. We also have a nullcone second fundamental invanant with respect tothe normal vector field $(n^{T}, n^{s})$ defined by $h_{j}^{\dot{\pm}}(n^{T}, n^{S})(u)=\{-(n^{T}\pm n^{S})_{u_{*}}(u), X_{u}!(u)\}$ for any $u\in U$.

$p$$S_{p}^{\pm}(n^{T}, n^{S})=\kappa^{\pm}(n^{T}, n^{S})I|_{T_{p}M}$$\kappa^{\pm}$$X_{u}2$$X_{u_{1}}$$X_{u_{j}}\rangle$

Proposition 1.1. Under the above notations, we have the following nullcone Weingarten formula withrespect to $(n^{T}, n^{S})$ :$(a)(n^{\overline{T}}\pm n^{S})_{u}$.

$= \frac{\mp(n_{1}^{T}\pm n_{1}^{s})(n_{u}^{s}:’.

$(b)\pi^{t}\circ(n^{\overline{T}}\pm n^{S})_{u}$n^{T}\rangle-(n_{1}^{T}\pm n_{1}^{s})_{u}:}{(n_{1}^{T}\pm n_{1}^{s})^{2}}(n^{T}\pm n^{S})-\sum_{j=1}^{2}h_{i}^{j\pm}(n^{T}, n^{S})X_{u}j$$=- \sum_{j=1}^{2}h_{i}^{j\pm}(n^{T},n^{S})X_{u_{i}}$;.

Here, $h_{:}^{j\pm}(n^{T}, n^{S})=h_{ik}^{\pm}(n^{T}, n^{S})g^{kj},$ $g^{kj}=(g_{kj})^{-1}$ and $n^{i}=(n:, n_{2}^{i}, n_{3}^{i}, n_{\dot{4}})(i=T, S)$ .

As a corollary of the above proposition, we have an explicit expression of the nullsphere GaussKronecker curvature by Riemannian metric and the nullcone second fundamental invariant.
Corollary 1.2. Under the same notations as in the above proposition, the nullsphere Gauss-Kronecker
curvature is given by$K_{n}^{\pm}(n^{T}, n^{S})(u)= \frac{\det(h_{1j}^{\pm}(n^{T},n^{S})(u))}{\det(g_{\alpha\beta})}$.

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If $K_{n}^{\pm}(n^{T}, n^{S})(u_{0})=0$ , the point $p_{0}=X(u_{0})$ is called a $(n^{T}, n^{S})$ -nullcone parabolic point of$X$ : $Uarrow NC^{3}$ . And we say that a pointis a $(n^{T}, n^{s})$ -nullcone flat point if it is a $(n^{T}, n^{s})$ -nullcone$K_{n}^{\pm}(n^{T},n^{s})(u_{0})=0$.
umbilical point and$p_{0}$Theorem 1.3. $K_{n}^{-}(n^{T}, n^{s})(u)\not\equiv 0$.

2

Nullsphere height function
The nullsphere height function family on $M=X(U)$ is defined by$H$:$H(u,v)=\langle X(u),$$U\cross S_{+}^{2}arrow \mathbb{R},$$v\rangle$.

The Hessian matrix of the nullsphere height function $h_{v0}=H(u, v_{0})$ at$u_{0}$is denoted by $Hess(h_{v_{0}})(u_{0})$ .

Proposition 2.1. Let $H$ be a nullsphere height function on M. Then.
(1) $\partial h_{v_{0}}/\partial u_{i}(u_{0})=0(i=1,2)$ if and only if$=0(i = 1,2)$ if and only if(2)
detHess$v_{0}=n^{\overline{T}}\pm n^{S}(u_{0})$$K_{n}^{\pm}(n^{T}, n^{S})(u_{0})=0$(3)$p_{0}$$(h_{v_{0}}(u_{0}))$$=$$\partial h_{v_{0}}/\partial u_{i}(u_{0})$$=n^{\overline{T}}\pm n^{S}(u_{0})$$v_{0}$.

andis a nullcone flat point if and only if rankHess $(h_{v0})(u_{0})=0$ .

Corollary 2.2. For a point $p_{0}=X(u_{0})\in M$ , the following conditions are equivalent;(1) The point $p_{0}\in M$ is a $(n^{T}, n^{S})$ -nullcone parabolic point.
.
(2) The point $p_{0}\in M$ is a singular point of the nullsphere Gauss map(3) $K_{n}^{\pm}(n^{T}, n^{s})(u_{0})=0$.
$NG_{M}^{\pm}$(4) detHess $(h_{v0})(u_{0})=0$ for

Corollary 2.3.

$NG_{M}^{-}$$v_{0}=n^{\overline{T}}\pm n^{S}(u_{0})$.

is a regular nullsphere Gauss map.

(resp.
)
Consider now the particular case of a surface $M\subset NC^{3}$ . Given a vector
and a number , denoted by $S(v, c)$ the null hyperhorosphere(resp. null equidistant hyperplane, nullhypersphere) determined by the intersection of the hyperplane $HP(v,c)$ with $NC^{3}$ .
$v\in S_{+}^{2}$$S_{1}^{3},$$H_{1}^{3}$$c$

Proposition 2.4. Let $M$ be a spacelike surfaoe in $NC^{3}$ . Ifstraight line.

$NG_{M}^{-}$

is constant, then $M$ degenerate to a

We now define a family of functions$\tilde{H}$; $UxNC^{3}arrow \mathbb{R}\tilde{H}(u, v)=\{X(u),\tilde{v}\}-v_{1}$ ,where $v=(v_{1},v_{2},v_{3},v_{4}).\tilde{H}$ is called the extended nullsphere height function of $M=X(U)$ . The Hessian.
atis denoted bymatrix of the extended nullsphere height function$\tilde{h}_{v0}=\tilde{H}(u, v_{0})$Proposition 2.5. Let $M$ be a spacelike surface inM. For $v_{0}\in NC^{3}$ , we have the following:(1)
if and only if
(2)

$=$

$\underline{\partial}\overline{h_{v}\theta}uarrow(p_{0})$$v_{1}=\langle X(u_{0}),$ $n^{T}\pm n^{S}(u_{0})\}$

is the extended nullsphere height function of$\tilde{v}_{0}=n^{\overline{T}}\pm n^{S}(u_{0})$$\tilde{h}_{vo}(p_{0})=\frac{\partial}{\theta}arrow(p_{0})\tilde{h}_{v}=0$$\tilde{h}_{v_{0}}(p_{0})$$NC^{3}.\tilde{H}$$=$$detHess\tilde{h}_{v_{0}}(p_{0})$$=$$Hess(\tilde{h}_{v_{0}})(u_{0})$$u_{0}$$0$

and $v_{1}=\langle X(u_{0}),$if and only if$n^{\overline{T}}\pm n^{S}(u_{0})\}$$\tilde{v}_{0}$$=$.

$n^{\overline{T}}\pm n^{S}(u_{0})$,and $K_{n}^{\pm}(n^{T}, n^{S})(p_{0})=0$ .

The assertions of proposition 2.5 means that the discriminant set of the extended nullsphere height. Therefore we now define a pairfunction is given by, each one ofis calledof singular surfaces in $NC^{3}$ by $NP_{M}^{\pm}(u)=\langle X(u),$the nullcone pedal surface of $X(U)=M$ . A singularity of the nullcone pedal surface exactly corresponds
to a singularity of the nullsphere Gauss map.
$\tilde{H}$$D_{\overline{H}}=\{v|v=\{X(u), n^{\overline{T}}\pm n^{S}(u)\rangle(n^{\overline{T}}\pm n^{S}\underline{)(u})\}$$n^{\overline{T}}\pm n^{S}(u)\rangle(n^{T}\pm n^{S})(u)$Corollary 2.6.

$NP_{\overline{M}}$$NP_{M}^{\pm}$is a zero map.

This work is only a preparation for further studying, in the following, we will discuss some geometricalproperties of spacelike curve from singularity theory viewpoint.

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References[1] V. I. Amol’d, S. M. Gusein-Zade and A. N. Varchenko, Singularities of differentiable maps vol. I, Momogr.
Math. 82, Birkh\"auser Boston, Inc., Boston, MA, 1985.
[2] T. Banchoff, T. Gaifney and C. McCrory, Cusps of Gauss mappings, Res. Notes in Math. 55, Pitman,London, 1982.

[3] J. W. Bruce and P. J. Giblin, Curves and singularities (second edition), Cambridge University press,(1992).

[4] S. Izumiya, D. Pei and T. Sano, Singularities of hyperbolic Gauss maps, Pro. London Math. Soc. 86(2003), 485-512.

[5] S. Izumiya, D. Pei and M. C. Romero-Fuster, The lightcone Gauss map of a spacelike surface in Minkowski
4-space, Asian Math. 8 (2004), 511-530.
[6] S. Izumiya, Legendrian dualities and spacelike hypersurfaces in the lightcone, to appear in Mosc. Math.
J.
[7] E. J. N. Looijenga, Structural stability of smooth families of1974.

$C^{\infty}$-functions, Thesis, Univ. Amsterdam,[8] J. Martinet, Singulareties of smooth functions and maps, London Math. Soc. Lecture Note Ser. 58.
Cambridge Univ. Press, Cambridge-New York, 1982.

[9] J. N. Mather, Stability of$C^{\infty}$-mappings IV, Classification of stable $ge s$ bytudes Sci. Publ. Math. No. 37 (1969) 223-248.

$\mathbb{R}$algebras, Inst. Hautes[10] J. A. Montaldi, On contact between submanifolds, Michigan Math. J. 33 (1986), 195-199.

[11] J. A. Montaldi, On genere c composites of maps, Bull. London Math. Soc., 23 (1991), 81-85.
[12] G. Wassermann, Stability of Caustics, Math. Ann. 216 (1975), 43-50.
[13] H. Whitney, On singulareties of mappings of Euclidean spaces I, Ann. of Math. 62 (1955), 374-410.

[14] V. M. Zakalyukin, Lagrangian and Legendre singular;ties, Funct. Anal. Appl., 10 (1976) 26-36.
[15] V. M. Zakalyukin, Reconstructions of fronts and caustics depending on a parameter, and versality ofmappings, (Russian) Current problems in mathematics, Vol. 22, (1983) 56-93,Donghe Pei, School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R.Chinae-mail: peidh340Qnenu. edu. cn
Lingling Kong, School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, P.R.Chinae-mail: konglllllQnenu. edu. cn