Pseudo-hyperbolic Gauss maps of Lorentzian surfaces in anti-de Sitter space

TL;DR

This paper classifies the type numbers of pseudo-hyperbolic Gauss maps for Lorentzian surfaces in anti-de Sitter space, exploring their behavior across families and higher-dimensional hypersurfaces.

Contribution

It provides a comprehensive analysis of the type numbers of pseudo-hyperbolic Gauss maps for various Lorentzian surfaces in anti-de Sitter space, including constant curvature and B-scroll type hypersurfaces.

Findings

Type numbers determined for all oriented Lorentzian surfaces with specific curvature conditions.

Behavior of type numbers along parallel families analyzed.

Type number of Lorentzian B-scroll hypersurfaces in higher-dimensional anti-de Sitter space studied.

Abstract

In this paper, we determine the type numbers of the pseudo-hyperbolic Gauss maps of all oriented Lorentzian surfaces of constant mean and Gaussian curvatures and non-diagonalizable shape operator in the $3$-dimensional anti-de Sitter space. Also, we investigate the behavior of type numbers of the pseudo-hyperbolic Gauss map along the parallel family of such oriented Lorentzian surfaces in the $3$-dimensional anti-de Sitter space. Furthermore, we investigate the type number of the pseudo-hyperbolic Gauss map of one of Lorentzian hypersurfaces of B-scroll type in a general dimensional anti-de Sitter space.

Equations

$$\mathbf{x}(s,t,z)=\left(1+\frac{|z{|}^2}{2}\right)\gamma (s)+tB(s)+\sum _{j=1}^{n-2}{z}_j{Z}_j(s)-\frac{k_2|z{|}^2}{2}C(s),$$

$$\left(\begin{tabular}[]{l}$D:=\pi(S^{1}_{\mathbb{C}}(\kappa))$ ($\pi\,:\,$ the natural projection of $\mathbb{R}^{4}_{2}=\mathbb{R}^{2}_{0}\times\mathbb{R}^{2}_{2}$ onto $\mathbb{R}^{2}_{0}$)\\ $\displaystyle{L_{\kappa}:=\begin{cases}\pi^{-1}(\partial D)\cap S^{1}_{\mathbb{C}}(\kappa)&(\mathrm{Im}\kappa\neq 0)\\ \pi^{-1}((0,0))\cap S^{1}_{\mathbb{C}}(\kappa)&(\mathrm{Im}\kappa=0)\end{cases}}$\\ $l_{i}:=\pi^{-1}(\gamma_{i})\cap S^{1}_{\mathbb{C}}(\kappa)\quad(i=1,2)$\\ \end{tabular}\right)$$

$$⟨v,w⟩=-\sum _{i=1}^s{v}_i{w}_i+\sum _{j=s+1}^m{v}_j{w}_j,$$

$$\tilde{g}=-\sum _{i=1}^{s}d{x}_i^2+\sum _{j=s+1}^{m}d{x}_j^2,$$

$${\mathbb{S}}_s^{m-1}(x_0,c)=\{x=(x_1,\cdots ,x_m)\in {\mathbb{E}}_s^m|⟨\overrightarrow{x_{0}x},\overrightarrow{x_{0}x}⟩=\frac{1}{c}\}$$

$${\mathbb{H}}_{s-1}^{m-1}(x_0,-c)=\{x=(x_1,\cdots ,x_m)\in {\mathbb{E}}_s^m|⟨\overrightarrow{x_{0}x},\overrightarrow{x_{0}x}⟩=-\frac{1}{c}\}$$

$${\tilde{\nabla }}_X{e}_A=\sum _{B=1}^m{ϵ}_B{\tilde{\omega }}_{AB}(X)e_B$$

$${\nabla }_X{e}_i=\sum _{j=1}^n{ϵ}_j{\omega }_{ij}(X)e_j$$

$${\nabla }_X^{\perp }{e}_r=\sum _{s=n+1}^m{ϵ}_s{\omega }_{rs}^{\perp }(X)e_s$$

$${\tilde{\nabla }}_{e_k}{e}_i=\sum _{j=1}^n{ϵ}_j{\omega }_{ij}(e_k)e_j+\sum _{r=n+1}^m{ϵ}_r{h}_{ki}^r{e}_r$$

$${\tilde{\nabla }}_{e_k}{e}_r=-A_r(e_k)+\sum _{s=n+1}^m{ϵ}_s{\omega }_{rs}^{\perp }(e_k)e_s$$

$$H=\frac{1}{n}\sum _{r=n+1}^m{ϵ}_r\mathrm{tr}{A}_r{e}_r$$

$$S=n^2⟨H,H⟩-\Vert h{\Vert }^2,$$

$$({\hat{\nabla }}_{e_k}h)(e_i,e_j)=\sum _{r=n+1}^m{ϵ}_r{h}_{ij;k}^r{e}_r.$$

$$h_{ij;k}^r=h_{jk;i}^r,$$

$$h_{jk;i}^r=e_i(h_{jk}^r)-\sum _{l=1}^n{ϵ}_l(h_{lk}^r{\omega }_{jl}(e_i)+h_{lj}^r{\omega }_{kl}(e_i))+\sum _{s=n+1}^m{ϵ}_s{h}_{jk}^s{\omega }_{sr}^{\perp }(e_i),$$

$$R^{\perp }(e_j,e_k;e_r,e_s)=⟨[A_r,A_s](e_j),e_k⟩=\sum _{i=1}^n{ϵ}_i(h_{ik}^r{h}_{ij}^s-h_{ij}^r{h}_{ik}^s),$$

$$H$$

$$h$$

$$S=ϵcn(n-1)+n^2⟨\hat{H},\hat{H}⟩-\Vert \hat{h}{\Vert }^2,$$

$$\varphi ={\varphi }_1+{\varphi }_2+\cdots +{\varphi }_k$$

$$0={\Delta }^2\varphi ={\lambda }_1^2{\varphi }_1+\cdots +{\lambda }_k^2{\varphi }_k.$$

$$⟨z,w⟩=\mathrm{Re}\left(\sum _{i=1}^{n+1}{z}_i{w}_i\right)(z=(z_1,\cdots ,z_{n+1}),w=(w_1,\cdots ,w_{n+1})\in \mathbb{C})$$

$$S_{\mathbb{C}}^n(\kappa )=\{(z_1,z_2,\cdots ,z_{n+1})\in {\mathbb{C}}^{n+1}={\mathbb{E}}_{n+1}^{2n+2}|\sum _{i=1}^{n+1}{z}_i^2=\kappa \},$$

$$S^{n}_{\mathbb{C}}(\kappa)\subset\left\{\begin{array}[]{ll}\mathbb{S}^{2n+1}_{n+1}\left(\mathrm{Re}(\kappa)\right)&(\mbox{if}\quad\mathrm{Re}(\kappa)>0)\\ &\\ \mathbb{H}^{2n+1}_{n}\left(\mathrm{Re}(\kappa)\right)&(\mbox{if}\quad\mathrm{Re}(\kappa)<0)\end{array}\right.$$

$$\begin{array}[]{lll}{\it spacelike}&{\rm if}&\langle v,v\rangle>0\quad{\rm or}\quad v=0,\\ {\it timelike}&{\rm if}&\langle v,v\rangle<0,\\ {\it null}&{\rm if}&\langle v,v\rangle=0\quad{\rm and}\quad v\neq 0.\end{array}$$

$$\displaystyle\begin{array}[]{l}\dot{\gamma}(s)=A(s),\\ \dot{A}(s)=\tilde{\nabla}_{\dot{\gamma}}A(s)+\langle A(s),A(s)\rangle\hat{H}_{\gamma}=k_{1}(s)C(s),\\ \dot{C}(s)=\tilde{\nabla}_{\dot{\gamma}}C(s)+\langle A(s),C(s)\rangle\hat{H}_{\gamma}=k_{2}A(s)+k_{1}(s)B(s),\\ \dot{B}(s)=\tilde{\nabla}_{\dot{\gamma}}B(s)+\langle A(s),B(s)\rangle\hat{H}_{\gamma}=k_{2}C(s)+\epsilon\gamma,\end{array}$$

$$⟨⟨{\tilde{e}}_{i_1}\wedge \cdots \wedge {\tilde{e}}_{i_{n+1}},{\hat{e}}_{j_1}\wedge \cdots \wedge {\hat{e}}_{j_{n+1}}⟩⟩=\det \left(⟨{\tilde{e}}_{i_l},{\hat{e}}_{j_k}⟩\right),(l,k=1,\cdots ,n+1).$$

$$\tilde{\nu }(p)=\mathbf{x}(p)\wedge {\mathbf{x}}_{\ast }(e_1^p)\wedge {\mathbf{x}}_{\ast }(e_2^p)\wedge \cdots \wedge {\mathbf{x}}_{\ast }(e_n^p)(p\in M_t),$$

$$e_i\tilde{\nu }=\sum _{k=1}^n\sum _{r=n+1}^{m-1}{ϵ}_r{h}_{ik}^r\mathbf{x}\wedge e_1\wedge \cdots \wedge \underset{k-th}{\underbrace{e_r}}\wedge \cdots \wedge e_n.$$

$$\begin{array}{r}\Delta \tilde{\nu }=\Vert \hat{h}{\Vert }^2\tilde{\nu }+n\hat{H}\wedge e_1\wedge \cdots \wedge e_n-n\sum _{k=1}^n\mathbf{x}\wedge e_1\wedge \cdots \wedge \underset{k-th}{\underbrace{D_{e_k}\hat{H}}}\wedge \cdots \wedge e_n\\ +\sum _{\begin{array}{c}i,k=1\\ j\ne k\end{array}}^n\sum _{\begin{array}{c}r,s=n+1\\ r<s\end{array}}^{m-1}{ϵ}_r{ϵ}_s{R}_{sjk}^r\mathbf{x}\wedge e_1\wedge \cdots \wedge \underset{j-th}{\underbrace{e_r}}\wedge \cdots \wedge \underset{k-th}{\underbrace{e_s}}\wedge \cdots \wedge e_n\end{array}$$

Full text

1 Introduction

The notion of finite typeness of isometric immersions into a Euclidean space was introduced by B.Y. Chen in the late 1970’s (see [C1], [C2], [C3], [CV] etc.). Later, the finite typeness of isometric immersions into a Euclidean space or, in more general, a pseudo-Euclidean space have been studied by many geometers (see [C1], [C2], [C3] etc.). B.Y. Chen and P. Piccinni ([CP]) extended the notion of the finite typeness to $C^{\infty}$-maps and studied the finite typeness of the Gauss map (which is not an immersion) of isometric immersions. Recently B. Bektaş, E.Ö. Canfes, U. Dursun and R. Yeǧin has been studied the finite typeness of the pseudo-spherical (resp. the pseudo-hyperbolic) Gauss maps of isometric immersions into the pseudo-sphere (resp. the pseudo-hyperbolic space) (see [BD], [BCD1], [BCD2], [YD]).

Let $M$ be an $n$-dimensional pseudo-Riemannian manifold of index $t$ and $\mathbb{E}^{m}_{s}$ be an $m$-dimensional pseudo-Euclidean space of index $s$. The smooth map $\phi:M\rightarrow\mathbb{E}^{m}_{s}$ is said to be of finite type if $\phi$ has the spectral decomposition: $\phi=\phi_{1}+\cdots+\phi_{k}$ with $\phi_{i}:M\rightarrow\mathbb{E}^{m}_{s}$’s are non-constant map such that $\Delta\phi_{i}=\lambda_{i}\phi_{i}$, where $\Delta$ is the Laplacian operator of $M$ and $\lambda_{i}$ are constants, and furthermore, if $\lambda_{i}$’s are mutually distinct, then $\phi$ is said to be of $k$-type. Denote by $\mathbb{S}^{m-1}_{s}$ the $(m-1)$-dimensional pseudo-sphere of constant curvature $1$ and index $s$ and $\mathbb{H}^{m-1}_{s}$ the $(m-1)$-dimensional pseudo-hyperbolic space of constant curvature $-1$ and index $s$. Let $M$ be an $n$-dimensional oriented pseudo-Riemannian manifold of index $t$ and ${\bf x}:M\hookrightarrow\mathbb{S}^{m-1}_{s}\subset\mathbb{E}^{m}_{s}$ be an isometric immersion. Let $G(n+1,m)_{t}$ be the Grassmannian manifold consisting of $(n+1)$-dimensional oriented non-degenerate subspaces of index $t$ of $\mathbb{E}^{m}_{s}$. Define a map $\tilde{\nu}:M\rightarrow G(n+1,m)_{t}$ by $\tilde{\nu}(p)={\bf x}(p)\wedge{\bf x}_{*}(e_{1}^{p})\wedge{\bf x}_{*}(e_{2}^{p})\wedge\cdots\wedge{\bf x}_{*}(e_{n}^{p})$ for $p\in M$, where $(e_{1}^{p},\cdots,e_{n}^{p})$ is an orthonormal frame of $T_{p}M$ compatible with the orientation of $M$. This map $\tilde{\nu}$ is called the pseudo-spherical Gauss map of ${\bf x}$. Similarly, for an isometric immersion ${\bf x}:M\hookrightarrow\mathbb{H}^{m-1}_{s}\subset\mathbb{E}^{m}_{s+1}$, the pseudo-hyperbolic Gauss map $\tilde{\nu}:M\rightarrow G(n+1,m)_{t+1}$ is defined.

D.S. Kim and Y.H. Kim([KK]) classified the Lorentzian surfaces of constant mean and Gaussian curvatures and non-diagonalizable shape operator in the $3$-dimensional de Sitter $\mathbb{S}^{3}_{1}$ and anti-de Sitter $\mathbb{H}^{3}_{1}$ space as follows.

Fact 1**.**

Let $M^{2}_{1}$ be a Lorentzian surface in $\mathbb{S}^{3}_{1}$ or $\mathbb{H}^{3}_{1}$. If the mean and Gaussian curvatures are constant and the shape operator is not diagonalizable at a point, then $M^{2}_{1}$ is an open part of a complex circle or a B-scroll.**

B. Bektaş, E.Ö. Canfes and U. Dursun ([BCD1]) determined the type number of the pseudo-spherical Gauss map of an oriented Lorentzian surface in $\mathbb{S}^{3}_{1}$ of non-zero constant mean curvature and non-diagonalizable shape operator at a point.

Fact 2**.**

An oriented Lorentzian surface in $\mathbb{S}^{3}_{1}$ of constant mean curvature and non-diagonalizable shape operator is of null 2-type pseudo-spherical Gauss map if and only if it is an open part of a non-flat B-scroll over a null curve.**

In this paper, we determined the type numbers of the pseudo-hyperbolic Gauss maps of such Lorentzian surfaces in $\mathbb{H}^{3}_{1}$.

Theorem A**.**

Let $M$ be an oriented Lorentzian surface in $\mathbb{H}^{3}_{1}$ of constant mean and Gaussian curvatures and non-diagonalizable shape operator. The following facts hold.

(i) If $M$ is the complex circle of radius $-1$, then the pseudo-hyperbolic Gauss map is of $1$-type.

(ii) If $M$ is the complex circle of radius $\kappa$ ($\mathrm{Re}(\kappa)=-1$, $\kappa\neq-1$), then the pseudo-hyperbolic

Gauss map is of infinite type.

(iii) If $M$ is a non-flat B-scroll, then the pseudo-hyperbolic Gauss is of null $2$-type.

(iv) If $M$ is a flat B-scroll, then the pseudo-hyperbolic Gauss map is of infinite type.

**

Remark 1.1*.*

The shapes of the complex circles of radius $-1$ and $\kappa$ ($\mathrm{Re}(\kappa)=-1$, $\kappa\neq-1$) are as in Figure 1.

Theorem B**.**

Let $M$ be a complex circle in $\mathbb{H}^{3}_{1}$ and $u$ be any real number. The parallel surface $M^{u}$ of $M$ at distance $u$ is a complex circle and the radius $\kappa^{u}$ of the complex circle $M^{u}$ moves over the whole of $\{z\in\mathbb{C}\,|\,\mathrm{Re}(z)=-1\}$ when $u$ moves over $\mathbb{R}$. Hence the only parallel surface of $M$ has the pseudo-hyperbolic Gauss map of $1$-type and other parallel surfaces of $M$ have the pseudo-hyperbolic Gauss map of infinite type. **

Theorem C**.**

Let $M$ be a B-scroll in $\mathbb{H}^{3}_{1}$ and $u\in\mathbb{R}$ sufficiently close to [math]. If $M$ is flat (resp. non-flat), then the parallel surface $M_{u}$ also is flat (resp. non-flat) B-scroll. Hence the type numbers of the pseudo-hyperbolic Gauss maps of the parallel surfaces of a B-scroll are equal to that of the original B-scroll. **

Furthermore, we prove the following fact for the pseudo-hyperbolic Gauss map of one of the $n$-dimensional Lorentzian hypersurfaces of B-scroll-type given by of L.J. Alias, A. Ferrandez, and P. Lucas([AFL]).

Theorem D**.**

Let $(A,B,C,Z_{1},\cdots,Z_{n-2})$ be the Cartan frame field of a null curve $\gamma(s)\subset\mathbb{H}^{n+1}_{1}$. Assume that $k_{1}(s)\neq 0$, $k_{2}^{2}=1$ and $k_{i}$ is non-zero constant for $i=3,4,\cdots,n-2$. The immersion ${\bf x}:I\times\mathbb{R}\times\mathbb{R}^{n-2}\rightarrow\mathbb{H}^{n+1}_{1}\subset\mathbb{E}^{n+2}_{2}$ given by

[TABLE]

parametrizes an oriented Lorentzian hypersurface of $\mathbb{H}^{n+1}_{1}$. The pseudo-hyperbolic Gauss map of this oriented Lorentzian hypersurface is of infinite type.**

\mathbb{R}^{2}_{2}$$S^{1}_{1}(1)$$(0,0)$$S^{1}_{0}(\hat{\kappa})$$(0,0)$$\mathbb{R}^{2}_{0}$$\mathbb{R}^{2}_{2}$$S^{1}_{1}(1)$$(0,0)($\hat{\kappa}$ : small,  radius : big)\mathbb{R}^{4}_{2}$$L_{\kappa}$$\pi(S^{1}_{\mathbb{C}}(\kappa))$$l_{2}$$l_{1}$$\gamma_{1}$$\gamma_{1}

\mathbb{R}^{2}_{2}$$S^{1}_{1}(1)$$(0,0)$$S^{1}_{0}(\hat{\kappa})$$(0,0)$$\mathbb{R}^{2}_{0}$$\mathbb{R}^{2}_{2}$$S^{1}_{1}(1)$$(0,0)($\hat{\kappa}$ : big,  radius : small)\mathbb{R}^{4}_{2}$$L_{\kappa}$$\pi(S^{1}_{\mathbb{C}}(\kappa))$$l_{2}$$l_{1}$$\gamma_{2}$$\gamma_{1}

\mathbb{R}^{2}_{2}$$S^{1}_{1}(1)$$(0,0)$$(0,0)$$\mathbb{R}^{2}_{0}$$\mathbb{R}^{2}_{2}$$S^{1}_{1}(1)$$(0,0)$$\mathbb{R}^{4}_{2}$$L_{\kappa}$$\pi(S^{1}_{\mathbb{C}}(\kappa))$$l_{2}$$l_{1}$$\gamma_{2}$$\gamma_{1}

            $0\neq|\mathrm{Im}(\kappa)|$ : big             $0\neq|\mathrm{Im}(\kappa)|$ : small                   $|\mathrm{Im}(\kappa)|\neq 0$

[TABLE]

Figure 1 : The shape of $S^{1}_{\mathbb{C}}(\kappa)$ in $\mathbb{R}^{4}_{2}$

2 Basic notions and facts

Let $\langle\,\,,\,\,\rangle$ be the non-degenerate symmetric linear form of the $m$-dimensional real vector space $\mathbb{R}^{m}$ defined by

[TABLE]

where $v=(v_{1},v_{2},\cdots,v_{m})$ and $w=(w_{1},w_{2},\cdots,w_{m})$. Let $\tilde{g}$ be the pseudo-Euclidean metric of index $s$ on the $m$-dimensional affine space $\mathbb{R}^{m}$ induced from $\langle,\rangle$ , i.e.

[TABLE]

where $(x_{1},\cdots,x_{m})$ denotes an affine coordinate of $\mathbb{R}^{m}$. Denote $(\mathbb{R}^{m},\tilde{g})$ by $\mathbb{E}^{m}_{s}$. For $x_{0}\in\mathbb{E}^{m}_{s}$ and $c>0$ , we put

[TABLE]

and

[TABLE]

Then $\mathbb{S}^{m-1}_{s}(x_{0},c)$ (resp. $\mathbb{H}^{m-1}_{s-1}(x_{0},-c)$) is the $(m-1)$-dimensional pseudo-Riemannian submanifold in $\mathbb{E}^{m}_{s}$ of constant curvature $c$ and index $s$ (resp. constant curvature $-c$ and index $s-1$), called a pseudo-sphere (resp. a pseudo-hyperbolic space). In particular, $\mathbb{S}^{m-1}_{1}(c)$ (resp. $\mathbb{H}^{m-1}_{1}(-c)$) is called a de Sitter space (resp. an anti-de Sitter space). For the simplicity, denote $\mathbb{S}^{m-1}_{s}(0,c)$ and $\mathbb{H}^{m-1}_{s-1}(0,-c)$ by $\mathbb{S}^{m-1}_{s}(c)$ and $\mathbb{H}^{m-1}_{s-1}(-c)$, where [math] is the origin of $\mathbb{E}^{m}_{s}$. Furthermore, we abbreviate $\mathbb{S}^{m-1}_{s}(1)$ (resp. $\mathbb{H}^{m-1}_{s-1}(-1)$) by $\mathbb{S}^{m-1}_{s}$ (resp. $\mathbb{H}^{m-1}_{s-1}$).

Let $M_{t}$ be an $n$-dimensional pseudo-Riemannian submanifold of index $t$ in $\mathbb{E}^{m}_{s}$ immersed by ${\bf x}$, for the simplicity, we denote ${\bf x}(M_{t})$ by $M_{t}$. Let $\nabla$ and $\tilde{\nabla}$ be the Levi-Civita connections of $M_{t}$ and $\mathbb{E}^{m}_{s}$, respectively. Also, let $\tilde{\nabla}^{\bf x}$ the induced connection of $\tilde{\nabla}$ by ${\bf x}$ and $\nabla^{\bot}$ the normal connection of $M_{t}$.

For the simplicity, we denote all metrics by the common symbol $\langle\,\,,\,\,\rangle$. We take a local orthonormal frame field $(e_{1},\cdots,e_{n})$ of the tangent bundle $TM_{t}$ of $M_{t}$ defined on an open set $U$ of $M_{t}$ and a local orthonormal frame field $(e_{n+1},\cdots,e_{m})$ of the normal bundle $T^{\bot}M_{t}$ of $M_{t}$ defined on $U$. Put $\epsilon_{A}:=\langle e_{A},e_{A}\rangle=\pm 1$ ($A=1,\ldots,m$). Let $\{\tilde{\omega}_{AB}\}_{A,B=1,\cdots,m}$ be the connection form of $\tilde{\nabla}$ with respect to $(e_{1},\cdots,e_{n},e_{n+1},\cdots,e_{m})$, $\{\omega_{ij}\}_{i,j=1,\cdots,n}$ the connection form of $\nabla$ with respect to $(e_{1},\cdots,e_{n})$ and $\{\omega_{rs}^{\bot}\}_{r,s=n+1,\cdots,m}$ the connection form of $\nabla^{\bot}$ with respect to $(e_{n+1},\cdots,e_{m})$, that is,

[TABLE]

[TABLE]

and

[TABLE]

for $X\in TM_{t}$. Also, let $h^{r}_{ij}$ be the components of the second fundamental form $h$ of $M_{t}$ with respect to $(e_{1},\cdots,e_{n},e_{n+1},\cdots,e_{m})$, that is, $h(e_{i},e_{j})=\sum_{r=n+1}^{m}h_{ij}^{r}e_{r}$ and $A_{r}$ the shape operator of $M_{t}$ for $e_{r}$, that is, $A_{r}=A_{e_{r}}$, where $A$ denotes the shape tensor of $M_{t}$. Then we have $\omega_{AB}+\omega_{BA}=0$,

[TABLE]

and

[TABLE]

The mean curvature vector $H$ and the scalar curvature $S$ are defined by;

[TABLE]

and

[TABLE]

where $\|h\|^{2}=\sum_{i,j=1}^{n}\sum_{r=n+1}^{m}\epsilon_{i}\epsilon_{j}\epsilon_{r}h_{ij}^{r}h_{ji}^{r}$. Denote by $\hat{\nabla}h$ the covariant derivative of $h$ with respect to $\nabla$ and $\nabla^{\bot}$. Let

[TABLE]

Then we have

[TABLE]

[TABLE]

[TABLE]

where $R^{\bot}$ is the normal curvature tensor of $M_{t}$.

Let ${\bf x}:M_{t}\hookrightarrow\mathbb{S}^{m-1}_{s}(c)$ or $\mathbb{H}^{m-1}_{s-1}(-c)\subset\mathbb{E}^{m}_{s}$ be an isometric immersion. Denote by $h$ and $H$ the second fundamental form and the mean curvature vector of $M_{t}$ in $\mathbb{E}^{m}_{s}$, $\hat{h}$ and $\hat{H}$ be the second fundamental form and of the mean curvature vector of $M_{t}$ in $\mathbb{S}^{m-1}_{s}(c)$ or $\mathbb{H}^{m-1}_{s-1}(-c)$.

We have

[TABLE]

and (2.4) is rewritten as

[TABLE]

where $\epsilon=+1$ if in $\mathbb{S}^{m-1}_{s}(c)$ and $\epsilon=-1$ if in $\mathbb{H}^{m-1}_{s-1}(-c)$.

The gradient vector field $\nabla f$ of $f\in C^{\infty}(M_{t})$ is defined by $\nabla f=\sum_{i=1}^{n}\epsilon_{i}e_{i}(f)e_{i}$, and Laplacian operator $\Delta$ of $M_{t}$ with respect to the induced metric is given by $\Delta=\sum_{i=1}^{n}\epsilon_{i}(\nabla_{e_{i}}e_{i}-e_{i}e_{i})$.

Definition 1**.**

Let $\phi:M_{t}\rightarrow\mathbb{H}^{m-1}_{s-1}(-c)\subset\mathbb{E}^{m}_{s}$ (resp. $\phi:M_{t}\rightarrow\mathbb{S}^{m-1}_{s}(c)\subset\mathbb{E}^{m}_{s}$) be a smooth map. Then $\phi$ is said to be of finite type in $\mathbb{H}^{m-1}_{s-1}$ (resp. in $\mathbb{S}^{m-1}_{s}$) if $\phi$ has the following spectral decomposition as follows;

[TABLE]

where $\phi_{i}:M_{t}\rightarrow\mathbb{E}^{m}_{s}$’s are non-constant map such that $\Delta\phi_{i}=\lambda_{i}\phi_{i}$ with $\lambda_{i}\in\mathbb{R}$, $i=1,2,\ldots,k$. If $\phi$ has this spectral decomposition and $\lambda_{i}$’s are mutually distinct constant, then the map $\phi$ is said to be of $k$-type, and when one of $\lambda_{i}$’s is equal to zero, the map $\phi$ is said to be of null $k$-type.

For a map of finite type, the following fact holds.

Lemma 2.1**.**

Let $\phi:M_{t}\rightarrow\mathbb{H}^{m-1}_{s-1}$ or $\mathbb{S}^{m-1}_{s}$ be a smooth map. If $\Delta^{2}\phi=0$, then $\Delta\phi=0$ or $\phi$ is of infinite type.**

Proof.

Assume that $\phi$ is of finite type and $\phi$ has the following spectral decomposition; $\phi=\phi_{1}+\phi_{2}+\cdots+\phi_{k}$ with $\Delta\phi_{i}=\lambda_{i}\phi_{i}$ for $\lambda_{i}\in\mathbb{R}$ and $i=1,2,\cdots,k$. Then we have

[TABLE]

Therefore we have $k=1$ and $\lambda_{1}=0$, that is, $\Delta\phi=0$. ∎

Let $\mathbb{C}^{n+1}$ be the $(n+1)$-dimensional complex vector space which is identified with $\mathbb{R}^{2n+2}$. Define a non-degenerate symmetric bilinear form $\langle\,\,,\,\,\rangle$ of the $\mathbb{C}^{n+1}(=\mathbb{R}^{2n+2})$ by

[TABLE]

Note that $\langle z,z\rangle=\sum_{i=1}^{n+1}x_{i}^{2}-\sum_{i=1}^{n+1}y_{i}^{2}$ when $z=(x_{1}+\sqrt{-1}y_{1},\cdots,x_{n+1}+\sqrt{-1}y_{n+1})$ ($x_{i},y_{i}\in\mathbb{R}$ $i=1,\cdots,n+1$)). Let $\tilde{g}$ be a pseudo-Euclidean metric of index $n+1$ on the $(2n+2)$-dimensional affine space $\mathbb{R}^{2n+2}(=\mathbb{C}^{n+1})$ induced from $\langle\,\,,\,\,\rangle$.

Definition 2**.**

Fix a non-zero complex number number $\kappa$. We put

[TABLE]

that is, $\sum_{i=1}^{n+1}(x_{i}^{2}-y_{i}^{2})=\mathrm{Re}(\kappa)$ and $2\sum_{i=1}^{n+1}x_{i}y_{i}=\mathrm{Im}(\kappa)$ for $z_{i}=x_{i}+\sqrt{-1}y_{i}$ ($x_{i},y_{i}\in\mathbb{R}$). This submanifold $S^{n}_{\mathbb{C}}(\kappa)$ is called a complex sphere of radius $\kappa$. In particular, when $n=1$, it is a complex circle of radius $\kappa$. **

Remark 2.1*.*

[TABLE]

The complex circle $S^{1}_{\mathbb{C}}(\kappa)\subset\mathbb{H}^{3}_{1}$ is parameterized as ${\bf x}(z)=\sqrt{\kappa}(\cos z,\sin,z)$  ($z\in\mathbb{C}$).

Definition 3**.**

A tangent vector $v$ of pseudo-Riemannian manifold $M$ is said to be**

[TABLE]

The category into which a given tangent vector falls is called its causal character. The causal character of a curve $\gamma$ in $M$ is that of the velocity $\gamma^{\prime}$.**

Definition 4**.**

Let $\gamma(s)$ be a null curve in $\mathbb{S}^{3}_{1}(\subset\mathbb{E}^{4}_{1})$ or $\mathbb{H}^{3}_{1}(\subset\mathbb{E}^{4}_{2})$. Let $(A,B,C)$ be a tangent frame field of $\mathbb{S}^{3}_{1}$ or $\mathbb{H}^{3}_{1}$ along $\gamma$ such that;

[TABLE]

where $\epsilon=+1$ if in $\mathbb{S}^{3}_{1}$, $\epsilon=-1$ if in $\mathbb{H}^{3}_{1}$, $k_{1}$ and $k_{2}$ are positive-valued functions. This frame field $(A,B,C)$ is called the Cartan frame field along $\gamma$. The the function $k_{1}$ and $k_{2}$ are called the first curvature and the second curvature of $\gamma$, respectively. Then the immersion ${\bf x}(s,t)=\gamma(s)+tB(s)$ parametrizes a Lorentzian ruled surface $M$ in $\mathbb{S}^{3}_{1}$ or $\mathbb{H}^{3}_{1}$. In this paper, if $k_{2}$ is constant, then $M$ is called the B-scroll over $\gamma$.**

3 Pseudo-spherical and Pseudo-hyperbolic Gauss map

Let $G(n+1,m)$ be Grassmannian manifold consisting of $(n+1)$-dimensional oriented non-degenerate subspaces of $\mathbb{E}^{m}_{s}$, Let $G(n+1,m)_{t+1}$ be the submanifold of $G(n+1,m)$ consisting of $(n+1)$-dimensional oriented non-degenerate subspaces of index $t+1$ of $\mathbb{E}^{m}_{s}$. Let $(\widetilde{e}_{1},\cdots,\widetilde{e}_{m})$ and $(\widehat{e}_{1},\cdots,\widehat{e}_{m})$ be two orthonormal frames of $\mathbb{E}^{m}_{s}$. Let $\widetilde{e}_{i_{1}}\wedge\cdots\wedge\widetilde{e}_{i_{n+1}}$ and $\widehat{e}_{j_{1}}\wedge\cdots\wedge\widehat{e}_{j_{n+1}}$ be two vectors in $\bigwedge^{n+1}\mathbb{E}^{m}_{s}$. Define an indefinite inner product $\langle\langle,\rangle\rangle$ on $\bigwedge^{n+1}\mathbb{E}^{m}_{s}$ by

[TABLE]

B(s_{2})$$\dot{\gamma}(s_{1})=A(s_{1})$$B(s_{3})light cone\gamma$$B(s_{1})$$\dot{\gamma}(s_{3})=A(s_{3})$$\gamma$$\rightarrownull geodesicnull geodesic$\dot{\gamma}(s_{2})=A(s_{2})$

Figure 2 : B-scroll over $\gamma$

Therefore, we may identify $\bigwedge^{n+1}\mathbb{E}^{m}_{s}$ with the pseudo-Euclidean space $\mathbb{E}^{N}_{q}$ for some positive integer $q$, where $N=\binom{m}{n+1}$. The Grassmannian manifold $G(n+1,m)_{t+1}$ can be imbedded into a pseudo-Euclidean space $\bigwedge^{n+1}\mathbb{E}^{m}_{s}\simeq\mathbb{E}^{N}_{q}$ by assigning $\Pi\in G(n+1,m)_{t+1}$ to $\widetilde{e}_{1}\wedge\cdots\wedge\widetilde{e}_{n+1}$ where $(\widetilde{e}_{1},\cdots,\widetilde{e}_{n+1})$ is an orthonormal basis of $\Pi$ compatible with the orientation of $\Pi$.

Let ${\bf x}:M_{t}\hookrightarrow\mathbb{H}^{m-1}_{s-1}\subset\mathbb{E}^{m}_{s}$ be an isometric immersion. For the immersion x, we define a map $\tilde{\nu}:M_{t}\rightarrow G(n+1,m)_{t+1}$ by

[TABLE]

where $(e_{1}^{p},\cdots,e_{n}^{p})$ is an orthonormal frame of $T_{p}M_{t}$ compatible with the orientation of $M_{t}$. This map $\tilde{\nu}$ is called the pseudo-hyperbolic Gauss map of x. In the sequel, we all rewrite ${\bf x}_{*}(e_{i})$ by $e_{i}$.

Let $(e_{1},\cdots,e_{n})$ be an local orthonormal frame field of $TM_{t}$ compatible with the orientation of $M_{t}$ and $(e_{n+1},\cdots,e_{m})$ be an local orthonormal frame field of $T^{\perp}M_{t}$ defined an open set $U$ of $M_{t}$, respectively.

The first derivative of the pseudo-hyperbolic Gauss map $\tilde{\nu}$ is given by

[TABLE]

Yeǧin and Dursun proved the following fact.

Lemma 3.1**.**

[YD]** Let $M_{t}$ be an $n$-dimensional oriented pseudo-Riemannian submanifold of index $t$ of a pseudo-hyperbolic $\mathbb{H}^{m-1}_{s}\subset\mathbb{E}^{m}_{s+1}$. Then the Laplacian of the pseudo-hyperbolic Gauss map $\tilde{\nu}:M_{t}\rightarrow G(n+1,m)\subset\mathbb{E}^{N}_{q},N=\binom{m}{n+1}$. for some $q$ is given by

[TABLE]

where $R^{r}_{sjk}=R^{D}(e_{j},e_{k};e_{r},e_{s})$.

In case of $n=m-2$, Yeǧin and Dursun have the following fact.

Lemma 3.2**.**

[YD]** For an oriented pseudo-Riemannian hypersurface $M_{t}$ with index $t$ of $\mathbb{H}^{n+1}_{s-1}\subset\mathbb{E}^{n+2}_{s}$ we have

[TABLE]

where $\hat{\mathcal{H}}$ is the mean curvature $M_{t}$ in $\mathbb{H}^{n+1}_{s-1}$, that is, $\hat{H}=\epsilon_{n+1}\hat{\mathcal{H}}e_{n+1}$.

Lemma 3.3**.**

If there exists a polynomial $P(t)=(t-\lambda_{1})(t-\lambda_{2})$ with mutually distinct roots $\lambda_{1},\lambda_{2}\in\mathbb{C}$ such that $P(\Delta)\tilde{\nu}=0$, then $\tilde{\nu}$ is of at most $2$-type or infinite type. **

Proof.

Assume that $\tilde{\nu}$ is of finite type and it has the following spectral decomposition; $\tilde{\nu}=\tilde{\nu}_{\hat{\lambda}_{1}}+\cdots+\tilde{\nu}_{\hat{\lambda}_{k}}$ with $\Delta\tilde{\nu}_{\hat{\lambda}_{i}}=\hat{\lambda}_{i}\tilde{\nu}_{\hat{\lambda}_{i}}$ $(1\leq i\leq k)$. Then we have

[TABLE]

Thus, we have $(\hat{\lambda}_{i}-\lambda_{1})(\hat{\lambda}_{i}-\lambda_{2})=0$ $(1\leq i\leq k)$. Hence, we have $\hat{\lambda}_{i}=\lambda_{1}$ or $\lambda_{2}$ for all $i$, that is, $\tilde{\nu}$ is of at most $2$-type. Hence the statement of this lemma follows. ∎

4 Proof of (i) and (ii) of Theorem A.

In this section, we prove (i) and (ii) of Theorem A.

Proof of (i) and (ii) of Theorem A. Let $M$ be the complex circle of radius $\kappa$ in $\mathbb{H}^{3}_{1}$. This surface $M$ is parameterized as

[TABLE]

where $\sqrt{\kappa}$ is one (with smaller argument) of squared roots of $\kappa$. Note that $M$ is included by $\mathbb{H}^{3}_{1}$ because of $\mathrm{Re}(\kappa)=-1$. For the convenience, we put $\sqrt{\kappa}=c+\sqrt{-1}d$. Let $z=x+\sqrt{-1}y$. By simple calculations, we have

[TABLE]

Also, we can show that the unit normal vector field $N$ of $M$ in $\mathbb{H}^{3}_{1}$ is given by $N=(d+\sqrt{-1}c)(\cos z,\sin z)$. With respect to the frame field $({\bf x}_{x},{\bf x}_{y})$, the shape operator $A_{N}$ in the direction $N$ is expressed as

[TABLE]

where $\alpha=-\frac{2cd}{c^{2}+d^{2}}$ and $\beta=\frac{1}{c^{2}+d^{2}}$. Put $e_{1}:={\bf x}_{x}$, $\tilde{e}_{2}={\bf x}_{y}-\langle{\bf x}_{y},e_{1}\rangle e_{1}$ and $e_{2}=\frac{1}{|\tilde{e}_{2}|}\tilde{e}_{2}$. Then $(e_{1},e_{2})$ forms an orthonormal tangent frame field on $M$. Note that $\langle e_{1},e_{1}\rangle=-1$, $\langle e_{2},e_{2}\rangle=1$ and $|\tilde{e}_{2}|^{2}=1+4c^{2}d^{2}$. With respect to $(e_{1},e_{2})$, the shape operator $A_{N}$ is expressed as

[TABLE]

Thus, by remarking $\alpha=-2cd\beta$ and $c^{2}-d^{2}=-1$, we obtain $\hat{\mathcal{H}}=\alpha$ and $\|\hat{h}\|^{2}=2(\alpha^{2}-\beta^{2})$, where $\hat{\mathcal{H}}$ and $\hat{h}$ denote the mean curvature and the second fundamental form of $M$ in $\mathbb{H}^{3}_{1}$. Hence, by (2.11), $M$ is flat. By (3.2) and (3.3), we have

[TABLE]

Hence,

[TABLE]

Therefore by Lemma 3.3, $\tilde{\nu}$ is either of finite type with type number $k\leq 2$ or of infinite type.

If $\kappa=-1$, then we have $c=0$ and hence $\alpha=0$. Therefore, it follows from (4.1) that $\Delta\tilde{\nu}=-2\beta^{2}\tilde{\nu}$, that is, $\tilde{\nu}$ is of $1$-type. If $\kappa\neq-1$, then we have $\alpha\neq 0$. Hence it follows from (4.1) that $\tilde{\nu}$ is of not $1$-type. Therefore, $\tilde{\nu}$ is of $2$-type or of infinite type. Suppose that it is of $2$-type, and that $\tilde{\nu}$ has decomposition $\tilde{\nu}=\tilde{\nu}_{1}+\tilde{\nu}_{2}$ ($\Delta{\tilde{\nu}_{1}}=\lambda_{1}\tilde{\nu}_{1}$, $\Delta{\tilde{\nu}_{2}}=\lambda_{2}\tilde{\nu}_{2}$). where $\lambda_{1},\lambda_{2}\in\mathbb{R}$ are mutually distinct. From (4.1) and (4.2), $\tilde{\nu}_{1}$ and $\tilde{\nu}_{2}$ can be expressed as

[TABLE]

for some constants $a$ and $b$. By substituting (4.4) and (4.5) into (4.1) and (4.2), and comparing coefficient of $\tilde{\nu}$ and $N\wedge e_{1}\wedge e_{2}$, we have

[TABLE]

Therefore, we obtain

[TABLE]

The discriminant of (4.11) is

[TABLE]

Therefore, there is no $\lambda_{2}\in\mathbb{R}$ satisfying (4.11). Thus a contradiction arises. Therefore, $\tilde{\nu}$ is of infinite type. ∎

5 Proof of (iii) and (iv) of Theorem A.

In this section, we prove (iii) and (iv) of Theorem A.

Proof of (iii) and (iv) of Theorem A. Let $(A,B,C)$ be the Cartan frame field along a null curve $\gamma(s)$ in $\mathbb{H}^{3}_{1}$ given by Definition 4. Then, the immersion ${\bf x}(s,t)=\gamma(s)+tB(s)$ parametrizes the B-scroll over a null curve $\gamma$. We have

[TABLE]

and hence

[TABLE]

The unit normal vector field of ${\bf x}$ is given by $N(s,t)=k_{2}tB(s)+C(s)$. With respect to the frame $({\bf x_{s}},{\bf x_{t}})$, the shape operator $A_{N}$ in the direction $N$ is expressed as

[TABLE]

When $|k_{2}|>1$ and $t\neq 0$, an orthonormal frame field $(e_{1},e_{2})$ on $M$ is given by

[TABLE]

By simple calculations, we have

[TABLE]

Thus, with respect to an orthonormal frame $(e_{1},e_{2})$, the shape operator $A_{N}$ is expressed as

[TABLE]

and hence we have $\hat{\mathcal{H}}=-k_{2}$, $\|\hat{h}\|^{2}=2k_{2}^{2}$. When $|k_{2}|<1$ and $t\neq 0$, we put $e_{1}=\frac{{\bf x_{s}}}{|{\bf x_{s}}|}$ and $e_{2}=e_{1}-|{\bf x_{s}}|{\bf x_{t}}$. Similarly we have $\hat{\mathcal{H}}=-k_{2}$, $\|\hat{h}\|^{2}=2k_{2}^{2}$. Hence $M$ is non-flat B-scroll by (2.11) when $k_{2}^{2}\neq 1$ and $t\neq 0$. We put $e_{3}:=N$ and

[TABLE]

It is clear that $\tilde{\nu}=\tilde{\nu}_{1}+\tilde{\nu}_{2}$. Using (3.2) and (3.3) we obtain that $\Delta\tilde{\nu}_{1}=0$ and $\Delta\tilde{\nu}_{2}=2(k_{2}^{2}-1)\tilde{\nu}_{2}.$ On the other hand, by using (3.1) and (3.3), we have

[TABLE]

[TABLE]

and hence

[TABLE]

Therefore $\tilde{\nu}$ is of null $2$-type.

When $k_{2}^{2}=1$ or $t=0$, an orthonormal frame field $(e_{1},e_{2})$ on $M$ is given by

[TABLE]

By simple calculations, we have

[TABLE]

Thus, with respect to an orthonormal frame $(e_{1},e_{2})$ ,the shape operator $A_{N}$ is expressed as

[TABLE]

and hence we have $\hat{\mathcal{H}}=-k_{2}$ and $\|\hat{h}\|^{2}=2k_{2}^{2}$. Hence $M$ is flat B-scroll by (2.11). Using (3.2) and (3.3), we obtain $\Delta\tilde{\nu}=2\tilde{\nu}-2k_{2}e_{3}\wedge e_{1}\wedge e_{2}\neq 0$ and $\Delta^{2}\tilde{\nu}=0$. Therefore $\tilde{\nu}$ is infinite type by Lemma 2.1.

Conversely, assume that the pseudo-hyperbolic Gauss map $\tilde{\nu}$ is of null $2$-type. Then, from (3.1), (3.2) and (3.3), we obtain

[TABLE]

[TABLE]

Since $\tilde{\nu}$ is of null $2$-type, we can put $\tilde{\nu}=\tilde{\nu}_{1}+\tilde{\nu}_{2}$ with $\Delta\tilde{\nu}_{1}=0$ and $\Delta\tilde{\nu}_{2}=\lambda_{2}\tilde{\nu}_{2}$ $(\lambda_{2}\neq 0)$, where $\tilde{\nu}_{1}$ is non-constant. Then we have $\Delta^{2}\tilde{\nu}=\lambda_{2}\Delta\tilde{\nu}$. This together with (5.4) implies, $e_{j}(\|\hat{h}\|^{2})=0$ (i.e. $\|\hat{h}\|^{2}$ is constant), $\lambda_{2}=\|\hat{h}\|^{2}-2$. Hence the Gaussian curvature is constant, that is, $\Delta(\|\hat{h}\|^{2})=0$. This together with (5.4) implies that $\|\hat{h}\|^{2}=2\hat{\mathcal{H}}$. Hence, from (2.11), we have

[TABLE]

From Fact 1, (i) and (ii) of Theorem A, $M$ is an open part of a non-flat B-scroll. ∎

6 Proof of Theorems B and C.

In this section, we consider the pseudo-hyperbolic Gauss map of the parallel surface of a complex circle and a B-scroll.

Definition 5**.**

Let $\bar{M}$ be a pseudo-Riemannian manifold and $M$ be a pseudo-Riemannian hypersurface of $\bar{M}$ with unit normal vector field $N$. At least locally and for $u\in\mathbb{R}$ sufficiently close to [math], for the map ${\bf x}^{u}:M\rightarrow\bar{M}$ defined by ${\bf x}^{u}(p):=\exp_{{\bf x}(p)}uN_{p}=\gamma_{N_{p}}(u)$ $(p\in M)$ is an immersion and $M_{u}:={\bf x}^{u}(M)$ is called the parallel surface of $M$ at distance $u$, where $\exp_{{\bf x}(p)}$ denotes the exponential map of $\bar{M}$ at ${\bf x}(p)$ and $\gamma_{N_{p}}$ denotes the geodesic in $\bar{M}$ with $\gamma^{\prime}_{N_{p}}(0)=N_{p}$.**

Example 1**.**

Let ${\bf x}:M\hookrightarrow\mathbb{S}^{n+1}_{1}(\subset\mathbb{E}^{n+2}_{1})$ be a Lorentzian hypersurface and $N$ be its unit normal vector field. Then ${\bf x}^{u}$ is given by ${\bf x}^{u}(p)=\cos u{\bf x}(p)+\sin uN_{p}$ $(p\in M)$. **

Example 2**.**

Let ${\bf x}:M\hookrightarrow\mathbb{H}^{n+1}_{1}(\subset\mathbb{E}^{n+2}_{2})$ be a Lorentzian hypersurface and $N$ be its unit normal vector field. Then ${\bf x}^{u}$ is given by ${\bf x}^{u}(p)=\cosh u{\bf x}(p)+\sinh uN_{p}$ $(p\in M)$. **

O{\LARGE\mathbb{H}^{3}_{1}}$${\bf x}(p)$$N(p)$${\bf x}(p^{\prime})$$\gamma_{N_{p}}(u)$$\gamma_{N_{p^{\prime}}}(u)$${\bf x}(M)$${\bf x}^{u}(M)$$\gamma_{N_{p^{\prime}}}$$\gamma_{N_{p}}

Figure 3 : parallel surface

Proof of Theorem B.  Let $M$ be a complex circle in $\mathbb{H}^{3}_{1}$, $\kappa$ be the radius of $M$. Let $\sqrt{\kappa}=c+\sqrt{-1}d$ $(c,d\in\mathbb{R})$. Since $M\subset\mathbb{H}^{3}_{1}$, we have $\mathrm{Re}(\kappa)=c^{2}-d^{2}=-1$. We remember that a unit normal vector field $N$ of $M$ is given by $N(z)=(d+\sqrt{-1}c)(\cos z,\sin z)$. Hence the parallel surface $M^{u}$ of $M$ at distance $u$ is parameterized by

[TABLE]

where $\kappa^{u}$ is the complex number satisfying $\sqrt{\kappa^{u}}:=(c\cosh u+d\sinh u)+\sqrt{-1}(d\cosh u+c\sinh u)$. Thus $M^{u}$ is the complex circle of radius $\kappa^{u}$. It is easy to show that, when $u$ moves over $(-\infty,\infty)$, $\kappa^{u}$ moves over the whole of $\{z\in\mathbb{C}\,|\,\mathrm{Re}(z)=-1\}$. There the statement of Theorem B follows from (i) and (ii) of Theorem A. ∎

Next we prove Theorem C.

Proof of Theorem C.  We consider a B-scroll $\displaystyle M(:\mathop{\Leftrightarrow_{\rm def}\,\,{\bf x}(s,t)=\gamma(s)+tB(s))}$ in $\mathbb{H}^{3}_{1}$. Since $N(s,t)=k_{2}tB(s)+C(s)$ where $k_{2}$ is the second curvature of $\gamma$, the parallel surface $M^{u}$ of $M$ is parameterized as

[TABLE]

We put $r(u):=\cosh u+k_{2}\sinh u$. By simple calculations, we have

[TABLE]

and hence

[TABLE]

If $r(u)=0$, then ${\bf x}^{u}$ is not immersion. Hence, we need to assume that

[TABLE]

The unit normal vector field $N^{u}$ of $M^{u}$ is given by $N^{u}(s,t)=\frac{k_{2}+r(u)\sinh u}{\cosh u}tB(s)+\cosh uC(s)+\sinh u\gamma(s)$. Hence the shape operator $A_{N^{u}}$ in the direction $N^{u}$ is expressed with respect to the usual frame $(\frac{\partial{\bf x}^{u}}{\partial s},\frac{\partial{\bf x}^{u}}{\partial t})$ as

[TABLE]

where $\alpha=\frac{k_{2}+r(u)\sinh u}{r(u)\cosh u}$. When $\frac{\partial{\bf x}^{u}}{\partial s}$ is non-null, we put

[TABLE]

We have $|\tilde{e}_{2}|=r(u)^{2}$. We put $e_{2}:=\frac{1}{r(u)^{2}}\tilde{e}_{2}$. Then, with respect to an orthonormal frame $(e_{1},e_{2})$, the shape operator $A_{N^{u}}$ is expressed as

[TABLE]

where $\beta=\frac{k_{1}(s)}{r(u)(-2k_{2}\sinh u+r(u)t^{2}(k_{2}^{2}-1))}$. Thus, the mean curvature $\hat{\mathcal{H}}_{u}=-\alpha$ and $\|\hat{h_{u}}\|^{2}=2\alpha^{2}$.

When $\frac{\partial{\bf x}^{u}}{\partial s}$ is null, an orthonormal frame field $(e_{1},e_{2})$ on $M_{u}$ is given by

[TABLE]

With respect to $(e_{1},e_{2})$, the shape operator $A_{N_{u}}$ is expressed as

[TABLE]

where $\beta^{\prime}=\frac{k_{1}(s)}{2r(u)^{2}}$. Thus, the mean curvature $\hat{\mathcal{H}}_{u}=-\alpha$ and $\|\hat{h_{u}}\|^{2}=2\alpha^{2}$. In both cases, it follows that $M_{u}$ has constant Gaussian curvature. Hence, by Fact 1, $M_{u}$ is a B-scroll or a complex circle. By Theorem B, $M_{u}$ is a B-scroll. By (2.11), $M_{u}$ is flat if $\alpha^{2}=1$ and $M_{u}$ is non-flat if $\alpha^{2}\neq 1$.

Assume that $M_{u}$ is flat for some $u\neq 0$. Then we have $\alpha^{2}=1$ and hence $r(u)(-\sinh u\pm\cosh u)=k_{2}$. From this relation, we have $k_{2}=\pm 1$. Hence we obtain the second-half of statement of this theorem. ∎

Remark 6.1*.*

We put

[TABLE]

If $u_{+}<\infty$ (resp. $u_{-}>-\infty$), then $M_{u_{+}}$ (resp. $M_{u_{-}}$) is a focal submanifold of $M$ by (6.1).

Remark 6.2*.*

In $\mathbb{S}^{3}_{1}$, we can derive the following fact similar to Theorem C and Remark 6.1. We put

[TABLE]

$M_{u_{+}}$ and $M_{u_{-}}$ are a focal submanifold of $M$.

when r(u)=0$${\bf x}^{0}(M)={\bf x}(M)geodesic{\bf x}^{0}(p)={\bf x}(p)$${\bf x}^{u_{2}}(p)$${\bf x}^{u_{2}}(M)$${\bf x}^{u_{1}}(M)$${\bf x}^{u_{1}}(p)$$N_{p}$${\bf x}^{r^{-1}(u)}(M)$${\bf x}^{0}(M)={\bf x}(M)$${\bf x}^{0}(p)={\bf x}(p)$${\bf x}^{u_{2}}(p)$${\bf x}^{u_{2}}(M)$${\bf x}^{u_{1}}(M)$${\bf x}^{u_{1}}(p)$$N_{p}$${\bf x}^{r^{-1}(u)}(M)in the case of k_{2}<-1$${\bf x}^{u}(p)

Figure 4 : focal submanifold of B-scroll

7 Cartan frame field of null curve of general order

E. Cartan showed that the existence of the Frenet type frame along a null curve of $\mathbb{R}^{3}_{1}$, called Cartan frame. K. L. Duggal and A. Bejancu ([DB]) constructed a Frenet type frame and equations along a null curve of a Lorentzian manifold. D. H. Jin ([J]) constructed a new type Frenet frame and equations simpler than Duggal and Bejancu’s Frenet frame and equations, by using it. Also, he proved that if we takes a special parameter (called distinguished parameter) then it becomes simpler moreover. Daggal and Bejancu-type Frenet frame is called the general Frenet frame and Jin-type one is called the natural Frenet frame. Frenet equations also are named similary. In particular, Ferrandez-Gimenez-Lucas ([FGL]) call Natural Frenet of null curves $\gamma$ parameterized by pseudo-arc parameter (that is $\langle\nabla_{\dot{\gamma}}\dot{\gamma},\nabla_{\dot{\gamma}}\dot{\gamma}\rangle=1$) the Cartan frame field. Note that the Cartan frame in this paper differ from the meaning of [FGL].

Lemma 7.1**.**

Let $\gamma(s)$ be a null curve of order $d(\geq 3)$ in $n$-dimensional Lorentzian manifold $(M,\langle,\rangle,\nabla)$, there is a frame field $(A,B,C,Z_{1},Z_{2},\cdots,Z_{d-3})$ along $\gamma(s)$ satisfying

[TABLE]

and

[TABLE]

This frame field $(A,B,C,Z_{1},\cdots,Z_{d-3})$ is called the Cartan frame field along $\gamma(s)$, where $k_{i}$ $(i=1,\cdots,d-1)$ are positive-valued functions. **

Natural Frenet equations of null curves parameterized by distinguished parameter and (7.15) equations are the same form by remarking $\langle A,B\rangle=-1$. The frame field in Lemma 7.1 is different from the Natural Frenet frame field in constructive method and parameters condition. We construct in the order of $A,C,B,Z_{1},\cdots,Z_{n-3}$, while they construct in the order of $A,B,C,Z_{1},\cdots,Z_{n-3}$. Also, we constructed the frame field directly while they constructed the natural Frenet frame field by way of a general Frenet frame fields. In [FGL], the Cartan frame in the sense of their is constructed directly. Their constructive method and that of Lemma 7.1 are the same except for a null curve parameterized by pseudo-arc or not. In this paper, a parameter of $\gamma$ is arbitrary, that is, $\langle\nabla_{\dot{\gamma}}\dot{\gamma},\nabla_{\dot{\gamma}}\dot{\gamma}\rangle$ is non-constant.

Proof.

We constitue it like the Frenet formula. First we show that $\nabla_{\dot{\gamma}}A$ is non-null. Suppose that $\nabla_{\dot{\gamma}}A$ is null. Since $\gamma$ is of order $d(\geq 3)$, $A$ and $\nabla_{\dot{\gamma}}A$ are linearly independent. Denote by $W_{1}$ the $2$-dimensional subspace spanned by $A$ and $\nabla_{\dot{\gamma}}A$. Also we have

[TABLE]

Let $(e_{1}=A,e_{2}=\nabla_{\dot{\gamma}}A,e_{3},\cdots,e_{n})$ be a frame field along $\gamma(s)$. Then we have

[TABLE]

This contradicts the fact that $\langle,\rangle$ is Lorentzian. Therefore $\nabla_{\dot{\gamma}}A$ is non-null.

We put $k_{1}(s)=\sqrt{|\langle\nabla_{\dot{\gamma}}A,\nabla_{\dot{\gamma}}A\rangle|}$ and $C(s)=\frac{1}{k_{1}(s)}\nabla_{\dot{\gamma}}A$, and $\epsilon_{C}:=\langle C,C\rangle$. We have

[TABLE]

Also we have $\langle\nabla_{\dot{\gamma}}C,C\rangle=0$ and $\langle A,C\rangle=0$, Hence

[TABLE]

Let $(\hat{e}_{1}=C,\hat{e}_{2}=A,\hat{e}_{3},\cdots,\hat{e}_{n})$ be a frame field along $\gamma$ satisfying $(\hat{e}_{2},\cdots,\hat{e}_{n})$ is a frame field of $\mathrm{Span}\{C\}^{\bot}$. Then we have

[TABLE]

Therefore we have $\epsilon_{C}=1$ and $\langle\nabla_{\dot{\gamma}}C,A\rangle=-k_{1}$ because $\langle,\rangle$ is Lorentzian. Hence we have Since $\gamma$ is of order $d(\geq 3)$, we have $\mathrm{dim}(\mathrm{Span}\{A,\nabla_{\dot{\gamma}}C\})=2$. From these facts, it follows that $\mathrm{Span}\{A,\nabla_{\dot{\gamma}}C\}$ is a $2$-dimensional Lorentzian space. There is a unique null vector $B\in\mathrm{Span}\{A,\nabla_{\dot{\gamma}}C\}$ such that $\langle A,B\rangle=-1$. It can be expressed as $\nabla_{\dot{\gamma}}C=aA+bB$ for some functions $a$ and $b$ because $\nabla_{\dot{\gamma}}C\in\mathrm{Span}\{A,B\}$. We have $b=k_{1}$ by (7.17). We put $k_{2}:=a$. Put $W_{2}=\mathrm{Span}\{A,B,C\}^{\bot}$, which is $(n-3)$-dimensional Euclidean space. We put

[TABLE]

We have

[TABLE]

If $d=3$, then we have $Z=0$. In the sequel, we consider the case of $d\geq 4$.

We put $k_{3}=|Z|$ and $Z_{1}=\frac{Z}{|Z|}$. Then $\nabla_{\dot{\gamma}}B$ can be expressed as $\nabla_{\dot{\gamma}}B=k_{2}C+k_{3}Z_{1}$. We put

[TABLE]

where $\hat{Z}\in\mathrm{Span}\{A,B,C,Z_{1}\}^{\bot}$. Then we have

[TABLE]

If $d=4$, then we have $\hat{Z}=0$. In the sequel, we consider the case of $d\geq 5$. Then we have $\hat{Z}\neq 0$. We put $k_{4}=|\hat{Z}|$ and $Z_{2}=\frac{\hat{Z}}{|\hat{Z}|}$. Then $\nabla_{\dot{\gamma}}Z_{1}$ can expressed as $\nabla_{\dot{\gamma}}Z_{1}=k_{3}A+k_{4}Z_{2}$. We put

[TABLE]

where $\tilde{Z}\in\mathrm{Span}\{A,B,C,Z_{1},Z_{2}\}^{\bot}$. Then we have $\tilde{a}=\tilde{b}=\tilde{c}=\tilde{z}_{2}=0$ and $\tilde{z}_{1}=-k_{4}$. If $d=5$, then we have $\tilde{Z}=0$. In the sequel, we consider the case of $d\geq 6$. Then we have $\tilde{Z}\neq 0$. We put $k_{5}=|\tilde{Z}|$ and $Z_{3}=\frac{\tilde{Z}}{|\tilde{Z}|}$. Then $\nabla_{\dot{\gamma}}Z_{2}$ is expressed as $\nabla_{\dot{\gamma}}Z_{2}=-k_{4}Z_{1}+k_{5}Z_{3}.$ By repeating the same discussion, we can derive the relations in Lemma 7.1. ∎

It has already been proven their uniquely exist of a null curve of $\mathbb{R}^{n}_{1}$ equipped with a general Frenet frame field for differentiable for any given functions $k_{i}:[-\epsilon,\epsilon]\rightarrow\mathbb{R}$. For a null curve of $\mathbb{S}^{n}_{1}$ (or $\mathbb{H}^{n}_{1}$) and a Cartan frame field in the sense of [FGL], respectively. See [DB], [DJ], [FGL] etc about these facts. We prove the above about the Cartan frame field in Lemma 7.1 in another method.

Proposition 7.1**.**

Let $k_{1},\cdots,k_{n}:[-\epsilon,\epsilon]\rightarrow\mathbb{R}$ be differentiable non-zero functions. There is a unique null curve $\gamma$ of order $n$ of $\mathbb{R}^{n}_{1}$ equipped with a frame field $(A,B,C,Z_{1},\cdots,Z_{n-3})$ along $\gamma$ satisfying (7.5) and (7.15). **

Proof.

We put $V:=\frac{1}{\sqrt{2}}(A+B)$ and $W:=\frac{1}{\sqrt{2}}(A-B)$. Let $\tilde{F}$ be a matrix consisting column vector fields $V,W,C,Z_{1},\cdots,Z_{n-3}$, that is $\tilde{F}=(V\quad W\quad C\quad Z_{1}\quad\cdots\quad Z_{n-3})$. We put

[TABLE]

then (7.15) holds

[TABLE]

If you give the initial value, the solution is unique. We prove that the solution satisfying (7.5). We put $E:=\mathrm{diag}(-1,1,\cdots,1)$. Let $\Phi$ be (7.22) solution with $E$ as the initial value. It is easy to check that for any initial value $\tilde{F}_{0}$ the solution expressed as $\tilde{F}=\tilde{F}_{0}E\Phi$