In this section, we give the basic notions and familiar results in Lorentzian geometry which we need in this paper (for more details, see [7–12]).
Hyperbolic 2-space
Let R3={(x1,x2,x3)∣x1,x2,x3∈R} be a three-dimensional vector space and x=(x1,x2,x3) and y=(y1,y2,y3) be two vectors in R3. The pseudo-scalar product of x and y is defined by 〈x,y〉=−x1y1+x2y2+x3y3. (R3,〈,〉) is called a three-dimensional pseudo-Euclidean space or Minkowski 3-space. We write \({E}_{1}^{3}\) instead of (R3,〈,〉). We say that a vector x in \({E}_{1}^{3}\) is spacelike, lightlike, or timelike if 〈x,x〉>0,〈x,x〉=0 or 〈x,x〉<0, respectively. We now define spheres in \({E}_{1}^{3}\) as follows:
$$\left\{\begin{array}{l} {H}_{+}^{2}=\{x\in {E}_{1}^{3}\mid -x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=-1,x_{1}\geq 1\} \\ {H}_{-}^{2}=\{x\in {E}_{1}^{3}\mid -x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=-1,x_{1}\leq -1\} \\ {S}_{1}^{2}=\{x\in {E}_{1}^{3}\mid -x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\}. \end{array}\right. $$
We call \({H}_{\pm }^{2}\) a hyperbola and \({S}_{1}^{2}\) a pseudo-sphere. Now, we discuss some basic facts of curves in hyperbolic 2-space, which are needed in the sequel.
Let \(\alpha :I\longrightarrow {H}_{+}^{2}\subset {E}_{1}^{3};\quad \alpha (t)=(x_{1}(t),x_{2}(t),x_{3}(t))\) be a smooth regular curve in \( {H}_{+}^{2} (i.e.,\alpha ^{\prime }(t)\neq 0)\) for any t∈I, where I is an open interval. It is easy to show that 〈α′(t),α′(t)〉>0, for any t∈I. We call such a curve a spacelike curve. The norm of the vector \(x\in {E}_{1}^{3}\) is defined by \(\Arrowvert x\Arrowvert =\sqrt {\arrowvert \langle x,x\rangle \arrowvert }\). The arc-length of a spacelike curve α, measured from α(t0),t0∈I is \(s(t)=\int _{t_{\circ }}^{t}\Arrowvert \alpha ^{\prime }(t)\Arrowvert dt\). Then, the parameter s is determined such that \( \lVert \dot {\alpha }(s)\rVert =1\), where \(\dot {\alpha }(s)=\frac {d\alpha (s)}{ds}\). So, we say that a spacelike curve α is parameterized by arc-length, if it satisfies \(\Arrowvert \dot {\alpha }(s)\Arrowvert =1\). Throughout the remainder in this paper, we denote the parameter s of α as the arc-length parameter. Let us denote \(\mathbf {T} (s)=\dot {\alpha }(s)\), and we call T(s) a unit tangent vector of α at s.
For any \(x=(x_{1},x_{2},x_{3}),y=(y_{1},y_{2},y_{3})\in {E}_{1}^{3}\), the pseudo-vector product of x and y is defined as follows:
$$x\wedge y=\left| \begin{array}{ccc} -e_{1} & e_{2} & e_{3} \\ x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \end{array} \right| =(-(x_{2}y_{3}-x_{3}y_{2}),x_{3}y_{1}-x_{1}y_{3},x_{1}y_{2}-x_{2}y_{1}). $$
We remark that 〈x∧y,z〉=det(x y z). Hence, x∧y is pseudo-orthogonal to x,y. We now set a vector E(s)=α(s)∧T(s). By definition, we can calculate that 〈E(s),E(s)〉=1 and 〈α(s),α(s)〉=−1. We can also show that T(s)∧E(s)=−α(s) and α(s)∧E(s)=−T(s). Therefore, we have a pseudo-orthonormal frame {α(s),T(s),E(s)} along α(s). We have the following hyperbolic Frenet-Serret formula of plane curves:
$$ \left\{\begin{array}{l} \dot{\alpha}(s)=\mathbf{T}(s) \\ \dot{\mathbf{T}}(s)=\alpha(s)+\kappa_{g}(s) \mathbf{E}(s) \\ \dot{\mathbf{E}}(s)=-\kappa_{g}(s) \mathbf{T}(s), \end{array}\right. $$
(1)
or in the matrix form:
$$ \left[ \begin{array}{ccc} \dot{\alpha}(s)\\ \dot{\mathbf{T}}(s) \\ \dot{\mathbf{E}}(s) \end{array} \right]=\left[ \begin{array}{ccc} 0 & 1 & 0 \\ 1 & 0 & \kappa_{g} \\ 0 & -\kappa_{g} & 0 \end{array} \right] \left[ \begin{array}{ccc} \alpha (s) \\ \mathbf{T}(s) \\ \mathbf{E}(s) \end{array} \right] $$
(2)
where κg is the geodesic curvature of the curve α in \( {H}_{+}^{2} \), which is given by
$$ \kappa_{g}(s)=det(\alpha(s)\ \mathbf{T}(s)\ \dot{\mathbf{T}}(s)). \notag $$
Hyperbolic 3-space
Let R4 be a four-dimensional vector space. For any x=(x1,x2,x3,x4),y=(y1,y2,y3,y4)∈R4, the pseudo-scalar product of x and y is defined by 〈x,y〉=−x1y1+x2y2+x3y3+x4y4. (R4,〈,〉) is called a Minkowski 4-space and denoted by \({E}_{1}^{4}\). We say that a vector \(x\in {{E}_{1}^{4}}\) is spacelike, lightlike, or timelike if 〈x1,x2〉>0,〈x1,x2〉=0 or 〈x1,x2〉<0, respectively. The norm of the vector \(x\in {E}_{1}^{4}\) is defiend by \(\lVert x\rVert =\sqrt { \lvert \langle x,x\rangle \rvert }\). For a non-zero vector \(\upsilon \in E_{1}^{4}\) and a real number c, we define a space with pseudo-normal υ by
$${S}(\upsilon,c)=\{x\in {{E}_{1}^{4}}\mid \langle x,\upsilon \rangle =c\}. $$
The space S(υ,c) is called a spacelike space, a timelike space, or a lightlike space if υ is timelike, spacelike, or lightlike, respectively.
Now, we define a hyperbolic 3-space by
$${H}_{+}^{3}(-1)=\left\{x\in {{E}_{1}^{4}}\mid \langle x,x\rangle =-1,x_{1}>0\right\}. $$
For any x=(x1,x2,x3,x4), y=(y1,y2,y3,y4) and \( z=(z_{1},z_{2},z_{3},z_{4}) \in {{E}_{1}^{4}}\), the pseudo-vector product of x, y, and z is defined as follows:
$$\begin{array}{@{}rcl@{}} x\wedge y\wedge z &=&\left\vert \begin{array}{cccc} -i & j & k & l \\ x_{1} & x_{2} & x_{3} & x_{4} \\ y_{1} & y_{2} & y_{3} & y_{4} \\ z_{1} & z_{2} & z_{3} & z_{4} \end{array} \right\vert \notag \\ &=&\left(-\left\vert \begin{array}{ccc} x_{2} & x_{3} & x_{4} \\ y_{2} & y_{3} & y_{4} \\ z_{2} & z_{3} & z_{4} \end{array} \right\vert,-\left\vert \begin{array}{ccc} x_{1} & x_{3} & x_{4} \\ y_{1} & y_{3} & y_{4} \\ z_{1} & z_{3} & z_{4} \end{array} \right\vert,\left\vert \begin{array}{ccc} x_{1} & x_{2} & x_{4} \\ y_{1} & y_{2} & y_{4} \\ z_{1} & z_{2} & z_{4} \end{array} \right\vert,-\left\vert \begin{array}{ccc} x_{1} & x_{2} & x_{3} \\ y_{1} & y_{2} & y_{3} \\ z_{1} & z_{2} & z_{3} \end{array} \right\vert \right). \notag \end{array} $$
We now prepare some basic facts of curves in hyperbolic 3-space.
Let \(\beta :I\longrightarrow {H}_{+}^{3}\subset {E}_{1}^{4};\quad \beta (t)=(x_{1}(t),x_{2}(t),x_{3}(t),x_{4}(t))\) be a smooth regular curve in \( {H}_{+}^{3} (i.e.,\beta ^{\prime }(t)\neq 0)\) for any t∈I where I is an open interval. So that 〈β′(t),β′(t)〉>0 for any t∈I. The arc-length of β measured from β(t∘),t∘∈I is \(s(t)=\int _{t_{\circ }}^{t}\Arrowvert \beta ^{\prime }(t)\Arrowvert dt\). Then, the parameter s is determined such that \(\Arrowvert \dot {\beta }(s)\Arrowvert =1\), where \(\dot {\beta }(s)=\frac {d\beta (s)}{ds}\). So, we say that a spacelike curve β is parameterized by arc-length if it satisfies that \(\Arrowvert \dot {\beta }(s)\Arrowvert =1\). Let us denote \(\mathbf {T}(s)=\dot {\beta }(s)\), and we call T(s) a unit tangent vector of β at s.
Here, we construct the explicit differential geometry on curves in \( {H}_{+}^{3}(-1)\). Let \(\beta :I\longrightarrow {H}_{+}^{3}(-1)\) be a regular curve. Since \({H}_{+}^{3}(-1)\) is a Riemannian manifold, we can reparameterize β by the arc-length. Hence, we may assume that β(s) is a unit speed curve. So, we have the tangent vector \(\mathbf {T}(s)=\dot {\beta }(s)\) with ∥T∥=1. In case when \( \left \langle \dot {\mathbf {T}}(s),\dot {\mathbf {T}}(s)\right \rangle \neq -1\), then we have a unit vector
$$ \mathbf{N}(s)=\frac{\dot{\mathbf{T}}(s)-\beta (s)}{\lVert \dot{\mathbf{T}}(s)-\beta (s)\rVert }.\notag $$
Moreover, define E(s)=β(s)∧T(s)∧N(s), then we have a pseudo-orthonormal frame {β(s),T(s),N(s),E(s)} of \({E}_{1}^{4}\) along β. By standard arguments, under the assumption that \(\left \langle \dot {\mathbf {T}}(s),\dot {\mathbf {T}}(s)\right \rangle \neq -1\), we have the following Frenet formula:
$$ \left\{\begin{array}{l} \dot{\beta}(s)=\mathbf{T}(s), \\ \dot{\mathbf{T}}(s)=\beta (s)+\kappa_{g}\mathbf{N}(s), \\ \dot{\mathbf{N}}(s)=-\kappa_{g}\mathbf{T}(s)+\tau_{g}\mathbf{E}(s), \\ \dot{\mathbf{E}}(s)=-\tau_{g}\mathbf{N}(s). \end{array}\right. $$
(3)
In another form:
$$ \left[ \begin{array}{c} \dot{\beta}(s) \\ \dot{\mathbf{T}}(s) \\ \dot{\mathbf{N}}(s) \\ \dot{\mathbf{E}}(s) \end{array} \right] =\left[ \begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & \kappa_{g} & 0 \\ 0 & -\kappa_{g} & 0 & \tau_{g} \\ 0 & 0 & -\tau_{g} & 0 \end{array} \right] \left[ \begin{array}{c} \beta (s) \\ \mathbf{T}(s) \\ \mathbf{N}(s) \\ \mathbf{E}(s) \end{array} \right] \notag $$
where
$$\begin{array}{@{}rcl@{}} \kappa_{g} &=&\lVert \dot{\mathbf{T}}(s)-\beta (s)\rVert, \\\\ \tau_{g} &=&-\frac{det\left(\beta (s),\dot{\beta}(s),\ddot{\beta}(s),\dddot{\beta}(s)\right) }{(\kappa _{g}(s))^{2}} \end{array} $$
(4)
are the geodesic curvature and geodesic torsion of the curve β in \({H}_{+}^{3}(-1)\), respectively.
Since \(\left \langle \dot {\mathbf {T}}(s)-\beta (s),\dot {\mathbf {T}}(s)-\beta (s)\right \rangle =\left \langle \dot {\mathbf {T}}(s),\dot {\mathbf {T}}(s)\right \rangle +1\), the condition
$$\left\langle \dot{\mathbf{T}}(s),\dot{\mathbf{T}}(s)\right\rangle \neq -1, $$
is equivalent to the condition κg(s)≠0. Moreover, we can show that the curve β(s) satisfies the condition κg(s)≡0 if and only if there exists a lightlike vector c such that β(s)−c is a geodesic. Such a curve is called an equidistant curve (see [9, 12]).
De Sitter 3-space
Let \(\gamma :I\longrightarrow {S}_{1}^{3}\) be a smooth and regular spacelike curve in \({S}_{1}^{3}\). We can parameterize it by arc-lengths. Hence, we may assume that γ(s) is a unit speed curve and we have the tangent vector \(\mathbf {T}(s)=\dot {\gamma }(s)\) with ∥T∥=1. In this case, we call γ a unit speed spacelike curve. If \(\left \langle \dot {\mathbf {T}}(s),\dot {\mathbf {T}}(s)\right \rangle \neq 1\), then \(\lVert \dot {\mathbf {T}}(s)+\gamma (s)\rVert \neq 0\), and we define the unit vector \(\mathbf {N}(s)=\frac { \dot {\mathbf {T}}(s)+\gamma (s)}{\lVert \dot {\mathbf {T}}(s)+\gamma (s)\rVert }\). Moreover, define E(s)=γ(s)∧T(s)∧N(s), then we have a pseudoorthonormal frame {γ(s),T(s),N(s),E(s)} of \({E}_{1}^{4}\) along γ. By standard arguments, under the assumption that \(\left \langle \dot {\mathbf {T}}(s),\dot {\mathbf {T}}(s)\right \rangle \neq 1\), we have the following Frenet-Serret type formula:
$$ \left\{\begin{array}{l} \dot{\gamma}(s)=\mathbf{T}(s) \\ \dot{\mathbf{T}}(s)=-\gamma (s)+\kappa_{g}\mathbf{N}(s) \\ \dot{\mathbf{N}}(s)=-\delta (\gamma)\kappa_{g}\mathbf{T}(s)+\tau_{g} \mathbf{E}(s) \\ \dot{\mathbf{E}}(s)=\tau_{g}\mathbf{N}(s). \end{array}\right. $$
(5)
It can be written as:
$$ \left[ \begin{array}{ccc} \dot{\gamma}(s)\\ \dot{\mathbf{T}}(s) \\ \dot{\mathbf{N}}(s) \\ \dot{\mathbf{E}}(s) \end{array} \right] =\left[ \begin{array}{cccc} 0 & 1 & 0 & 0 \\ -1 & 0 & \kappa_{g} & 0 \\ 0 & -\delta (\gamma)\kappa_{g} & 0 & \tau_{g} \\ 0 & 0 & \tau_{g} & 0 \end{array} \right] \left[ \begin{array}{cccc} \gamma (s) \\ \mathbf{T}(s)\\ \mathbf{N}(s) \\ \mathbf{E}(s) \end{array} \right] \notag $$
where δ(γ)=sign (N(s)) (which we shall write as simply δ) and
$$ \left\{\begin{array}{l} \kappa_{g}=\lVert\dot{\mathbf{T}}(s)+\gamma (s)\rVert, \\ \\ \tau_{g}=\frac{\delta \det \left(\gamma (s),\dot{\gamma}(s),\ddot{\gamma}(s),\dddot{\gamma}(s)\right) }{(\kappa _{g}(s))^{2}}, \end{array}\right. $$
(6)
are the geodesic curvature and geodesic torsion of the curve γ in \({S}_{1}^{3}\), respectively.
Since \(\left \langle \dot {\mathbf {T}}(s)+\gamma (s),\dot {\mathbf {T}}(s)+\gamma (s)\right \rangle =\left \langle \dot {\mathbf {T}}(s),\dot {\mathbf {T}}(s)\right \rangle -1\), the condition \(\left \langle \dot {\mathbf {T}}(s),\dot {\mathbf {T}}(s)\right \rangle \neq 1\) is equivalent to the condition κg(s)≠0 (see [4]).