Abstract
We consider the Anti-de Sitter space \(\mathbb {H}^3_1\) equipped with Berger-like metrics, that deform the standard metric of \(\mathbb {H}^3_1\) in the direction of the hyperbolic Hopf vector field. Helix surfaces are the ones forming a constant angle with such vector field. After proving that these surfaces have (any) constant Gaussian curvature, we achieve their explicit local description in terms of a one-parameter family of isometries of the space and some suitable curves. These curves turn out to be general helices, which meet at a constant angle the fibers of the hyperbolic Hopf fibration.
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1 Introduction
A helix surface (or constant angle surface) is an oriented surface, whose normal vector field forms a constant angle with a fixed field of directions in the ambient space. In recent years, many authors investigated helix surfaces in different ambient spaces. Several examples of the study of helix surfaces in Riemannian settings may be found in [3,4,5,6, 9, 10, 13,14,15] and references therein.
The investigation of helix surfaces also extended to other settings. On the one hand, higher codimensional Riemannian helix surfaces were studied (see for example [7, 8, 19]). On the other hand, Lorentzian ambient spaces were considered. Lorentzian settings allow to more possibilities, as both spacelike and timelike surfaces can be studied. Some examples of the study of the geometry of helix surfaces in Lorentzian spaces are given in [11, 12, 16, 17]. In particular, helix surfaces of the anti-de Sitter space \(\mathbb {H}^3_1\) were studied in [12]. Equipping \(\mathbb {H}^3_1\) with its canonical metric of constant curvature, all left-invariant vector fields are Killing (see also [2]), so no special directions emerge. Moreover, all helix surfaces turn out to be flat [12].
In [2], the first author and D. Perrone introduced and studied a new family of metrics \({\tilde{g}}_{\lambda \mu \nu }\) on \(\mathbb {H}^3_1(\kappa / 4)\). These metrics were induced in a natural way by corresponding metrics defined on the tangent sphere bundle \(T_1 {\mathbb {H}}^2 (\kappa )\), after describing the covering map F from \({\mathbb {H}}_1 ^3 (\kappa /4)\) to \(T_1 {\mathbb {H}} ^2(\kappa )\) in terms of paraquaternions. A crucial role in this construction is played by the hyperbolic Hopf map and the hyperbolic Hopf vector field, that is, the hyperbolic counterparts of the Hopf map and vector field on \({\mathbb {S}}^3\), respectively.
In [13, 14, 16, 17], the second author et al. gave an explicit local classification of helix surfaces by means a one-parameter family of isometries of the ambient space and a suitable curve. A similar appproach will be used in this paper in order to find the explicit characterization of the helix surfaces in the anti-de Sitter space. We shall equip the anti-de Sitter space \(\mathbb {H}^3_1(\kappa / 4)\) with a special type of the Lorentzian metrics introduced in [2], that is, the ones that deform the standard metric of \(\mathbb {H}^3_1(\kappa / 4)\) only in the direction of the hyperbolic Hopf vector field, which is then a Killing vector field. Because of their analogies with the Berger metrics on \({\mathbb {S}}^3\), we shall refer to these metrics as Berger-like metrics. We shall consider the anti-de Sitter space \(\mathbb {H}^3_1(\kappa / 4)\) equipped with Berger-like metrics and completely describe their surfaces, whose normal vector field forms a constant angle with the hyperbolic Hopf vector field.
The paper is organized in the following way. In Sect. 2 we provide some needed information on the anti-de Sitter space \(\mathbb {H}^3_1(\kappa / 4)\). In Sect. 3 we describe the Levi-Civita connection and the curvature of Berger-like metrics \({g}_\tau \) on \(\mathbb {H}^3_1(\kappa / 4)\). The general equations for a surface of \((\mathbb {H}^3_1(\kappa / 4),{g}_\tau )\) are given in Sect. 4 and then are applied in Sect. 5 to the case of helix surfaces. Three different cases occur, according to the sign of some constant B, which depends on the parameter \(\tau \) of the Berger-like metric, the causal character of the surface and the constant angle. These three cases are completely described and characterized in Sect. 6, in terms of a 1-parameter family of isometries on \(\mathbb {H}^3_{1, \tau }\) and some suitable curves. As we prove in Sect. 7, these curves are general helices, which meet at a constant angle the fibers of the hyperbolic Hopf fibration.
2 Preliminaries
Let \({\mathbb {R}}^4_2 = ({\mathbb {R}}^4, g_0)\) denote the 4-dimensional pseudo-Euclidean space, of neutral signature (2, 2), equipped with the flat pseudo-Riemannian metric
For any real constant \(\kappa >0\), the anti-de Sitter (three)-space \(\mathbb {H}^3_1 (\kappa )\) is the hypersurface of \({\mathbb {R}}^4_2\) defined by
We denote by \(\langle \,,\,\rangle \) the Lorentzian metric induced from \(g_0\) on \(\mathbb {H}^3_1 (\kappa )\). This metric, known as the canonical metric of \({\mathbb {H}}^3_1\), has constant sectional curvature \(-\kappa <0\).
Consider now the algebra \({{\mathbb {B}}}\) of paraquaternionic numbers over \({{\mathbb {R}}}\) generated by \(\{1,i,j,k\}\), where \(k=ij\), \(-i^2=j^2=1\) and \(ij=-ji\). This is an associative, non-commutative and unitary algebra over \({{\mathbb {R}}}\). An arbitrary paraquaternonic number is given by \(q=x_1+x_2i+x_3j+x_4k\). The conjugate of q is \(\bar{q}=x_1-x_2i-x_3j-x_4k\) and the norm of q is given by
The scalar product induced by such norm on \({{\mathbb {R}}}^4\) is exactly \(g_0\), and \(\{1,i,j,k\}\) is an orthonormal basis with 1, i spacelike and j, k timelike. In terms of paraquaternionic numbers, we have
The above presentation was used in [2] to describe a covering map from the anti-de Sitter space \({\mathbb {H}}_1 ^3 (\kappa /4)\) to the unit tangent sphere bundle of the (Riemannian) hyperbolic two-space \({\mathbb {H}} ^2(\kappa )\), which is embedded in \({\mathbb {R}}_1 ^3\) as
Now consider each paraquaternonic number q as a pair of complex number in the following way \(q:=(z,w)=(x_1+ix_2,x_3+ix_4)\), we obtain the covering map
The composition of F with the canonical projection
yields the hyperbolic Hopf map \(h= \pi \circ F\), that is,
In particular, h is a submersion with geodesic fibers, which can be defined as orbits of the \({\mathbb {S}}^1\)-action \(\big (e^{it}, (z,w)\big )\mapsto \big (e^{it} z,e^{it} w)\) on the anti-de Sitter space \({\mathbb {H}}_1 ^3 (\kappa /4)\). The vector field
is tangent to fibers of h and satisfies \(\langle X_1, X_1 \rangle =-1\) and is called the hyperbolic Hopf vector field, in analogy with the case of the Hopf vector field and the Hopf fibration. We also consider
Then, \(\{X_1,X_2, X_3\}\) parallelize \(\mathbb {H}^3_1(\kappa / 4)\) and
3 Levi-Civita connection and curvature
In [2], the covering map F described by (2) was used to introduce a family of Lorentzian metrics on the anti-de Sitter space \(\mathbb {H}^3_1(\kappa / 4)\). These metrics were referred to as metrics of Kaluza-Klein type, because they correspond in a natural way to metrics defined on the unit tangent sphere bundle \(T_1 {\mathbb {H}} ^2 (\kappa ) \). Their general description is given by
where \(\rho ,\mu ,\nu \) are positive real numbers and \(\{\theta ^i \}\) denotes the basis of 1-forms dual to \(\{X_i\}\).
Here we shall focus our attention on a special type of metrics of Kaluza-Klein type, requiring to have a deformation only in the direction of the hyperbolic Hopf vector field \(X_1\), which in this case is a Killing vector field [2]. Up to homotheties, we may restrict to the case where \(\mu =\nu =1\) and \(\rho =\tau ^2\) and consider the one-parameter family of metrics
These metrics can be clearly considered an hyperbolic analogue of Berger metrics on the sphere \({\mathbb {S}}^3\) and so we refer to them as Berger-like metrics on \(\mathbb {H}^3_1(\kappa / 4)\). Observe that with respect to the standard metric \(\langle ,\,\rangle \) on \(\mathbb {H}^3_1(\kappa / 4)\), these metrics can be described as
and \(\{ E_1=\tau ^{-1} X_1, E_2=X_2, E_3=X_3 \}\) is an orthonormal basis for \({g}_\tau \), with \(E_1\) timelike and \(E_2, E_3\) spacelike. Computing the Lie brackets \([E_i,E_j]\), we find
Then, by the Koszul formula, we obtain the description of the Levi-Civita connection of \(\mathbb {H}^3_{1, \tau }=(\mathbb {H}^3_1(\kappa / 4), {g}_\tau )\) with respect to \(\{E_1,E_2,E_3\}\). Explicitly, we get
Observe that \(E_1\) is an unit timelike vector field tangent to the fibers of h and the Levi-Civita connection satisfies the following geometric relation
for any tangent vector field X, where \(E_1 \wedge E_2=E_3\), \(E_2 \wedge E_3=-E_1\) e \(E_3 \wedge E_1=E_2\) completely determine the wedge product \(U \wedge V\) (see [18]). We now consider the curvature tensor, taken with the sign convention
Using (3.2) we have that the non vanishing components of the Riemannian curvature tensor \({R}^\tau \) are
and we can prove the following result.
Proposition 3.1
The curvature tensor of \(\mathbb {H}^3_{1, \tau }\) is given by
for all tangent vector fields X, Y, Z.
Proof
Consider three arbitrary vector fields X, Y, Z on \(\mathbb {H}^3_{1, \tau }\) and their decompositions as
with \(\bar{X}, \bar{Y}, \bar{Z}\) orthogonal to \(E_1\) and \(x={g}_\tau (X,E_1)\) and so on. Observe that using (3.4) we have that all terms of \({g}_\tau ({R}^\tau (X,Y)Z,W)\) where \(E_1\) occurs either one, three or four times necessarily vanish. Therefore, we have
From the decompositions of \(\bar{X}\), \(\bar{Y}\), \(\bar{Z}\) and \(\bar{W}\) follows that
Moreover, we have
Thus, we conclude that
which ends the proof since W is arbitrary. \(\square \)
We end this section describing the isometries of \(\mathbb {H}^3_{1, \tau }\). Following the idea used in [14] and [17], we observe that the isometry group of \(\mathbb {H}^3_{1, \tau }\) is the four-dimensional pseudo-unitary group \(\textrm{U}_1(2)\), that can be identified with
where \(J_1\) is the complex structure of \({{\mathbb {R}}}^4\) corresponding to i, i.e., defined by
while
is the pseudo-orthogonal group, i.e., the group of \(4\times 4\) real matrices preserving the semi-definite inner product of \({{\mathbb {R}}}^4_2\).
We now consider a 1-parameter family \(A(v), v\in (a,b)\subset \mathbb {R}\), consisting of \(4\times 4\) pseudo-orthogonal matrices commuting (anticommuting, respectively) with \(J_1\). In order to describe explicitly the family A(v), we consider the two product structures \(J_2\) and \(J_3\) of \({{\mathbb {R}}}^4\) corresponding to j and k respectively, that is,
Since A(v) is a pseudo-orthogonal matrix, the first row must be a unit vector \({{\textbf{r}}}_1(v)\) of \({{\mathbb {R}}}^4_2\) for all \(v\in (a,b)\). Thus, without loss of generality, we can take
for some real functions \(\xi _1,\xi _2\) and \(\xi _3\) defined in (a, b). Since A(v) commutes (anticommutes, respectively) with \(J_1\) the second row of A(v) must be \({{\textbf{r}}}_2(v)=\pm J_{1}{{\textbf{r}}}_1(v)\). Now, the four vectors \(\{{{\textbf{r}}}_1, J_1{{\textbf{r}}}_1, J_2{{\textbf{r}}}_1,J_3{{\textbf{r}}}_1\}\) form an orthonormal basis of \({{\mathbb {R}}}^4_2\), thus the third row \({\textbf{r}}_3(v)\) of A(v) must be a linear combination of them. Since \({{\textbf{r}}}_3(v)\) is unit and it is orthogonal to both \({\textbf{r}}_1(v)\) and \(J_1{{\textbf{r}}}_1(v)\), there exists a function \(\xi (v)\) such that
Finally, the fourth row of A(v) is \({ {{\textbf{r}}}_4(v)=\pm J_1{{\textbf{r}}}_3(v)=\mp \sin \xi (v) J_2 {{\textbf{r}}}_1(v)\pm \cos \xi (v) J_3 {{\textbf{r}}}_1(v)}\). This means that any 1-parameter family A(v) of \(4\times 4\) pseudo-orthogonal matrices commuting (anticommuting, respectively) with \(J_1\) can be described by four functions \(\xi _1,\xi _2,\xi _3\) and \(\xi \) as
4 Structure equations for surfaces in \(\mathbb {H}^3_{1, \tau }\)
Consider a pseudo-Riemannian oriented surface M immersed in \(\mathbb {H}^3_{1, \tau }\). Let N denote the unit vector field normal to M in \(\mathbb {H}^3_{1, \tau }\). We set \(\lambda :={g}_\tau (N,N)\), so that M is spacelike if \(\lambda =-1\) and timelike if \(\lambda =1\).
In the following, we compute the Gauss and Codazzi equations for M, using the metric induced by \({g}_\tau \) on M, the shape operator A, the tangent projection of \(E_1\) on M and the angle function \(\nu :={g}_\tau (N,E_1){g}_\tau (N,N)=\lambda {g}_\tau (N,E_1)\). The vector field \(E_1\) decomposes as
where T is tangent to M, whence,
Denoting by X and Y two vector fields tangent to M we have,
where \(\nabla \) is the Levi-Civita connection of M and \(\alpha \) the second fundamental form with respect to the immersion in \(\mathbb {H}^3_{1, \tau }\). Thus, we conclude that
Observe that
On the other hand, by (3.3) we have
where \(JX:=N\wedge X\) satisfies
Comparing the above expressions for \({\nabla }^\tau _XE_1\), we find
We now prove the following.
Proposition 4.1
Denoting by X and Y two vector fields tangent to M, with K the Gaussian curvature of M and with \(\bar{K}\) the sectional curvature in \(\mathbb {H}^3_{1, \tau }\) of the plane tangent to M, we have
and
Proof
Recall that for a pseudo-Riemannian surface, one has
Consider a local orthonormal basis \(\{ X,Y\}\) on M, i.e., \({g}_\tau (X,X)=1\), \({g}_\tau (X,Y)=0\) and \({g}_\tau (Y,Y)=-\lambda \). Then, by Proposition 3.1, we have
and so,
Taking into account \({g}_\tau (X,T)^2-\lambda {g}_\tau (Y,T)^2={g}_\tau (T,T)=-(1+\lambda \nu ^2)\), we have
that gives
whence Eq. (4.6) follows.
Consider the Codazzi equation
From Proposition 3.1, we have
that leads to Eq. (4.7) by the arbitrarity of Z. \(\square \)
5 Basic properties of helix surfaces in \(\mathbb {H}^3_{1, \tau }\)
We start with the following.
Definition 5.1
Let M be an oriented pseudo-Riemannian surface immersed in \(\mathbb {H}^3_{1, \tau }\) and N the unit vector field normal to M, with \({g}_\tau (N,N)=\lambda \). The surface M is called a helix surface (or a constant angle surface) if the angle function \(\nu =\lambda \,{g}_\tau (N,E_1)\) is constant on M.
Remark 5.1
If M is spacelike (respectively, timelike), then T is spacelike (respectively, timelike) and JT is spacelike. From (4.1) and (4.4) we get
Observe that in the case \(\lambda =1\), if \(\nu =0\) then \(E_1\) is tangent to M at each point. Therefore, M is a Hopf tube. On the other hand, if \(\lambda =-1\), then \(|\nu |>1\) and so \(\nu \ne 0\).
Taking into account Remark 5.1 from now on we assume \(\nu \ne 0\).
Proposition 5.1
Let M be a helix surface in \(\mathbb {H}^3_{1, \tau }\) and N the unit vector field normal to M. Then the following hold:
-
(i)
with respect to the tangent basis \(\{T, JT\}\), the matrix of the shape operator is given by
$$\begin{aligned} A= \begin{pmatrix} 0&{} -\dfrac{\sqrt{\kappa }}{2}\lambda \tau \\[4pt] \dfrac{\sqrt{\kappa }}{2}\tau &{} \mu \end{pmatrix}, \end{aligned}$$for some smooth function \(\mu \) on M;
-
(ii)
the Levi-Civita connection \(\nabla \) of M is described by
$$\begin{aligned} \begin{array}{ll} \nabla _TT=\nu \tau \sqrt{\kappa }\,JT, &{} \nabla _{JT}T=\mu \nu \,JT,\\[4pt] \nabla _TJT=\lambda \nu \tau \sqrt{\kappa }\, T, &{} \nabla _{JT}JT=\lambda \mu \nu \,T; \end{array} \end{aligned}$$ -
(iii)
the Gaussian curvature of M is given by
$$\begin{aligned} K=\lambda \kappa \nu ^2(1-\tau ^2); \end{aligned}$$ -
(iv)
the function \(\mu \) satisfies equation
$$\begin{aligned} T(\mu )+\nu \mu ^2+\kappa \nu B=0, \end{aligned}$$(5.1)where \(B:=\nu ^2(\tau ^2-1)-\lambda \).
Proof
Taking into account Remark 5.1 for the tangent basis \(\{T, JT \}\) we have
which, by the symmetry of the shape operator, yields (i).
For the Levi-Civita connection \(\nabla \) of M, using (4.5), we have
and using the compatibility of \(\nabla \) with the metric \({g}_\tau \), follows
So,
From (4.6) we have that Gaussian curvature is given by
Remark 5.2
It is worthwhile to remark that the Gaussian curvature K is a constant, which depends on the causal character of the surface and on the sign of \((1-\tau ^2)\) and it can assume any real value. In the special case of \(\tau =1\) the Gaussian curvature vanishes and helix surfaces for the standard metric on the anti-de Sitter space are flat, coherently with the results obtained in [12] for surfaces forming a constant angle with an unit Killing vector field.
Finally, we calculate
By Proposition 4.1, we get
and so, by comparing,
which ends the proof. \(\square \)
We recall that \(E_1\) is a timelike vector field and \({g}_\tau (E_1,N)=\nu \lambda \). Thus, there exists a smooth function \(\varphi \) on M, such that
whence,
and we can calculate
Moreover, we have
Therefore,
But \(A(T)=\dfrac{\sqrt{\kappa }}{2}\tau JT\) and so, we conclude that
that is,
Moreover,
and so,
But \(A(JT)=-\dfrac{\sqrt{\kappa }}{2}\tau \lambda \, T+\mu \, JT\), so that we obtain
In this way, we determined the following system
whose compatibility condition is given by
Recalling that
by comparison we obtain
that is, Eq. (5.1), which we already established, so that the system is compatible.
We can choose a system of local coordinates (x, y) on M, such that
for some smooth functions \(a=a(x,y)\), \(b=b(x,y)\) on M. Requiring that \([\partial _x, \partial _y]=0\), we get
So, Eq. (5.1) can be rewritten as follows:
In order to integrate equation (5.3), we need to consider separately the following cases
-
(i)
If \(B>0 \), we find
$$\begin{aligned} \mu (x,y)=\sqrt{\kappa B}\tan (\eta (y)-\nu \sqrt{\kappa B} \,x). \end{aligned}$$ -
(ii)
If \(B=0 \), we have
$$\begin{aligned} \mu (x,y)=\frac{1}{\nu x+\eta (y)}. \end{aligned}$$ -
(iii)
If \(B<0 \), we get
$$\begin{aligned} \mu (x,y)=\sqrt{-\kappa B}\tanh (\eta (y)+\nu \sqrt{-\kappa B}\,x). \end{aligned}$$
In all the above cases, \(\eta (y)\) is an arbitrary smooth function.
As we are interested in only one coordinate system (x, y) on the surface M, we only need one admissible solution for a and b in each case.
5.1 Case \(B>0\)
Since \(b_x=b\mu \nu \), we deduce
which admits as solution \(b=\cos (\eta (y)-\nu \sqrt{\kappa B}\, x)\). Then, we have
and so, we can take \(a=\dfrac{\lambda \tau }{\sqrt{B}}\sin (\eta (y)-\nu \sqrt{\kappa B}\, x)\). Now, from
we get \(\varphi (x,y)=-\dfrac{\sqrt{\kappa }}{\lambda \tau }Bx+c\), for some real constant c.
5.2 Case \(B=0\)
From \(b_x=b\mu \nu \), we now have
and a solution is given by \(b=\nu x+\eta (y)\). Moreover, we have:
which holds for \(a=-\nu \frac{\sqrt{\kappa }}{2}\,\lambda \tau x (\nu x + 2\eta (y))\). Then,
whence, \(\varphi (x,y)=y+c\) for some real constant c.
5.3 Case \(B<0\)
Recalling that \(b_x=b\mu \nu \), we have
which is satisfied by \(b=\cosh (\eta (y)+\nu \sqrt{-\kappa B}\, x)\). Moreover, we find:
and so, we take \(a=-\dfrac{\lambda \tau }{\sqrt{-B}}\sinh (\eta (y)+\nu \sqrt{-\kappa B}\, x)\). Finally,
and we obtain \(\varphi (x,y)=-\dfrac{\sqrt{\kappa }}{\lambda \tau }Bx+c\), for some real constant c.
Using the above results, we have the following.
Proposition 5.2
Let M be a helix surface in \(\mathbb {H}^3_{1, \tau }\) with constant angle function \(\nu \). With respect to the local coordinates (x, y) defined above, the position vector F of M in \(\mathbb {R}^4_2\) satisfies the following equation:
-
(a)
if \(B=0\),
$$\begin{aligned} \frac{\partial ^2F}{\partial x^2}=0, \end{aligned}$$(5.4) -
(b)
if \(B\ne 0\),
$$\begin{aligned} \frac{\partial ^4 F}{\partial x^4}+(\tilde{b}^2+2\tilde{a})\frac{\partial ^2 F}{\partial x^2}+\tilde{a}^2 F=0, \end{aligned}$$(5.5)where
$$\begin{aligned} \tilde{a}=\frac{\kappa }{4}\frac{B}{\tau ^2}(\lambda +\nu ^2), \qquad \tilde{b}=-\sqrt{\kappa }\frac{B}{\lambda \tau }. \end{aligned}$$
Proof
Let M be a helix surface and let
denote the position vector of M in \(\mathbb {R}^4_2\), described with respect to the local coordinates (x, y) defined before. By definition of position vector, we get
Then, if we consider the expressions of \(E_1, E_2\) and \(E_3\) with respect to the coordinates of \(\mathbb {R}^4_2\), we can express the above equation as
Therefore, if \(B = 0\) taking the derivative of (5.7) with respect to x, we get (5.4).
If we suppose \(B\ne 0\), we obtain
where
In conclusion, taking twice the derivative of (5.8) with respect to x and using the previous relations we find (5.5). \(\square \)
Remark 5.3
From \(\langle F, F \rangle =-4/\kappa \) and relations (5.7), (5.8), we get:
where
6 Characterization theorems for helix surfaces in \(\mathbb {H}^3_{1, \tau }\)
In order to give conditions under which an immersion defines a helix surface in \(\mathbb {H}^3_{1, \tau }\) we observe that, if F is a position vector of a helix surface in \(\mathbb {H}^3_{1, \tau }\) we have that:
and, using the (5.9), we have the following identities:
We now use these relations to prove the following key result.
Proposition 6.1
Let \(F: \Omega \rightarrow \mathbb {H}^3_{1, \tau }\subset {\mathbb {R}}^4_2\) be an immersion from an open set \(\Omega \subset \mathbb {R}^2\), with local coordinates (x, y) such that the projection of \(E_1=\dfrac{\sqrt{\kappa }}{2\tau }J_1F\) is \(F_x\). Then \(F(\Omega )\subset \mathbb {H}^3_{1, \tau }\) defines a helix surface of constant angle function \(\nu \) if and only if
and
Proof
Suppose that \(F(\Omega )\) is a helix surface in \(\mathbb {H}^3_{1, \tau }\) of constant angle function \(\nu \), then we have:
In a similar way we find:
that leads to the equation (6.2). In addition, we have
that is equation (6.3).
To prove the converse, as vector fields \(F_x,F_y\) are tangent to \(F(\Omega )\) (and so, orthogonal to normal vector field N), we consider
and we get the orthogonal basis \(\{F_x, \tilde{T}, N\}\) for the tangent space to \(\mathbb {H}^3_{1, \tau }\) along \(F(\Omega )\). Moreover, from (6.2) and (6.3) we have:
This leads to \(E_1=c_1F_x+c_2N\). Next, from (6.2) we have \(c_1=1\) that means \((E_1)^T=F_x\) and also
whence, \({g}_\tau (E_1,N)=\lambda \vert \nu \vert \) is constant and so, \(F(\Omega )\) is a helix surface. \(\square \)
In order to get explicit solutions of Eqs. (5.5) and (5.4) we consider three different cases, depending on the different possibilities for B.
6.1 Helix surfaces of \(\mathbb {H}^3_{1, \tau }\) in the case \(B>0\)
Integrating (5.5) we prove the following.
Proposition 6.2
Let M be a helix surface in \(\mathbb {H}^3_{1, \tau }\) with constant angle function \(\nu \) such that \(B>0\). Then, with respect to the local coordinates (x, y) defined above, the position vector F of M in \(\mathbb {R}^4_2\) is explicitly given by
where
are real constants and \(w^i(y)\), \(i=1,2,3,4\), are mutually orthogonal vector fields in \(\mathbb {R}^4_2\), depending only on y, such that, setting \(w_{ij}=\langle w^i(y), w^j(y) \rangle \) for all indices i, j, we have
Proof
If \(B>0\), then \(\tilde{b}^2+2\tilde{a}>0\) and \(\tilde{b}^2+4\tilde{a}>0\). Integrating Eq. (5.5), we then obtain
where
are two real constants and \(w^i(y)\), \(i=1,2,3,4\), are vector fields in \(\mathbb {R}^4_2\), depending only on y.
Using the expressions of \(\tilde{a}\) and \(\tilde{b}\), we have
Setting \(w_{ij}=\langle w^i(y), w^j(y)\rangle \) and evaluating relations (5.9) at (0, y), we get
From (6.7), (6.8), (6.12) and (6.13), it follows that
Moreover, (6.5), (6.9) and (6.10) yield
Next, by (6.6), (6.11) and (6.14) we get
\(\square \)
We now prove the following result.
Theorem 6.1
(Of characterization for \(B>0\)) Let M be a helix surface in \(\mathbb {H}^3_{1, \tau }\subset {{\mathbb {R}}}^4_2\) with constant angle function \(\nu \) so that \(B>0\). Then, locally, the position vector of M in \({{\mathbb {R}}}^4_2\), with respect to the local coordinates (u, v) on M defined in (5.2), is
where
is a twisted geodesic of the Lorentzian torus \(\mathbb {S}^1(\sqrt{w_{11}})\times \mathbb {S}^1(\sqrt{-w_{33}}) \subset \mathbb {H}^3_{1, \tau }\), \(w_{11}\), \(w_{33}\), \(\alpha _1\), \(\alpha _2\) are the four constants given in Proposition 6.2, and \(A(y)=A(\xi ,\xi _1,\xi _2,\xi _3)(y)\) is a 1-parameter family of \(4\times 4\) pseudo-orthogonal matrices commuting with \(J_1\), as described in (3.6), where \(\xi \) is a constant and
Conversely, a parametrization \(F(x,y)=A(y)\,\gamma (x)\), with \(\gamma (x)\) and A(y) as above, defines a helix surface in \(\mathbb {H}^3_{1, \tau }\) with constant angle function \(\nu \).
Proof
With respect to the local coordinates (x, y) on M defined in (5.2), Proposition 6.2 implies that the position vector of the helix surface in \({{\mathbb {R}}}^4_2\) is given by
where the vector fields \(\{w^i(y)\}_{i=1}^4\) are mutually orthogonal and
Thus, if we put \(e_i(y)=w^i(y)/||w^i(y)||\), \(i\in \{1,\dots ,4\}\), we can write:
The identities (6.1), evaluated in (0, y), yield:
We point out that to obtain the previous identities we have divided by \(\alpha _1^2-\alpha _2^2=\kappa \tau ^{-1}|\nu |\sqrt{B^3}\), which, by the assumption on \(\nu \), is always different from zero. From (6.21) and (6.22), taking into account \(\alpha _1^2-\alpha _2^2\ne 0\), we find
Therefore,
Substituting (6.23) in (6.17) and (6.19), we obtain the system
a solution of which is given by
Consequently, \(\langle J_1e_1,e_2\rangle =\langle J_1e_3,e_4\rangle =-\lambda \) and \(J_1e_1=-\lambda e_2\), \(J_1e_3=\lambda e_4\). Then, if we fix the orthonormal basis of \({{\mathbb {R}}}^4_2\) given by
there must exists a 1-parameter family of \(4\times 4\) pseudo-orthogonal matrices \(A(y)\in \textrm{O}_2(4)\), with \(J_1A(y)=A(y)J_1\), such that \(e_i(y)=A(y)\tilde{E}_i\) for all indices \(i=1,\dots ,4\).. Replacing \(e_i(y)=A(y)\tilde{E}_i\) in (6.16) we obtain
where
is a twisted geodesic of the Lorentzian torus \(\mathbb {S}^1(\sqrt{w_{11}})\times \mathbb {S}^1(\sqrt{-w_{33}})\subset \mathbb {H}^3_{1, \tau }\).
Let now examine the 1-parameter family A(y) that, according to (3.7), depends on four functions \(\xi _1(y)\), \(\xi _2(y)\), \(\xi _3(y)\) and \(\xi (y)\). From (5.2), it follows that \(\langle F_y, F_y\rangle =(\lambda +\nu ^2)=\textrm{constant}\). The latter implies that
Now, if we denote by \({{\textbf{c}}_1},{{\textbf{c}}_2},{\textbf{c}_3},{{\textbf{c}}_4}\) the four colons of A(y), (6.24) implies that
where by \('\) we mean the derivative with respect to y. Replacing in (6.25) the expressions of the \({{\textbf{c}}_i}\)’s as functions of \(\xi _1(y),\xi _2(y),\xi _3(y)\) and \(\xi (y)\), we obtain
where \(h_1(y)\) and \(k_1(y)\) are two functions such that
From (6.26) we have two possibilities:
-
(i)
\(\xi =\textrm{constant}\);
or
-
(ii)
\(4 (\xi _1')^2+\sinh ^2(2\xi _1)\, (-\xi '+\xi _2'+\xi _3')^2=0\).
We will show that case (ii) cannot occurs. Indeed, if (ii) occurs, then the parametrization \(F(x,y)=A(y)\gamma (x)\) defines a Hopf tube, that is, the hyperbolic Hopf vector field \(E_1\) is tangent to the surface. To this end, we write the unit normal vector field N to the parametrization F(x, y) as
where
A long computation then gives
Now, case (ii) occurs if and only if either \(\xi _1=\textrm{constant}=0\) or \(\xi _1=\textrm{constant}\ne 0\) and \(-\xi '+\xi _2'+\xi _3'=0\). In both cases, we conclude that \(N_1=0\) and this implies that
i.e. the Hopf vector field is tangent to the surface.
Thus, we are left with the case where \(\xi =\textrm{constant}\). In this case, (6.3) is equivalent to
whence we conclude that condition (6.15) is satisfied.
The converse of the theorem follows immediately from Proposition 6.1, since a direct calculation shows that \(g_{\tau }(F_x,F_x)=g_{\tau }(E_1,F_x)=-\lambda (\lambda +\nu ^2)\) (and so, (6.2) holds), while (6.15) is equivalent to (6.3). \(\square \)
Corollary 6.1
Let M be a helix spacelike (respectively, timelike) surface in \(\mathbb {H}^3_{1, \tau }\subset {{\mathbb {R}}}^4_2\) with constant angle function \(\nu \) such that \(B>0\). Then, there exist local coordinates on M such that the position vector of M in \({{\mathbb {R}}}^4_2\) is given by
where
is a twisted geodesic in the Lorentzian torus \({{\mathbb {S}}}^1\left( \frac{2}{\sqrt{\kappa }}\frac{1}{\sqrt{d^2-1}}\right) \times {{\mathbb {S}}}^1\left( \frac{2}{\sqrt{\kappa }}\frac{d}{\sqrt{d^2-1}}\right) \subset \mathbb {H}^3_{1, \tau }\) parametrized by arc length, whose slope is given by
In addition, \(A(y)=A(\xi ,\xi _1,\xi _2,\xi _3)(y)\) is a 1-parameter family of \(4\times 4\) pseudo-orthogonal matrices commuting with \(J_1\), as described in (3.7), with \(\xi =\textrm{constant}\) and
Conversely, a parametrization \(F(s,y)=A(y)\,\gamma (s)\), with \(\gamma (s)\) and A(y) as above, defines a helix surface in \(\mathbb {H}^3_{1, \tau }\).
Proof
We consider the curve \(\gamma (x)\) given in the Theorem 6.1. Since \(\langle \gamma '(x),\gamma '(x)\rangle =\dfrac{4}{\kappa }\alpha _1\,\alpha _2\), considering
from Eq. (6.4) and taking into account the equation (6.5) with \(w_{13}=0\), we get
Observe that \(d>1\). Therefore, we can consider the arc length reparametrization of the curve \(\gamma \) given by
Finally, we observe that d represents the slope of the geodesic \(\gamma \). \(\square \)
6.2 Helix surfaces of \(\mathbb {H}^3_{1, \tau }\) in the case \(B=0\)
Integrating (5.4) and taking into account \(\partial _xF=T\), we prove at once the following.
Proposition 6.3
Let M be a helix surface in \(\mathbb {H}^3_{1, \tau }\subset \mathbb {R}^4_2\) with constant angle function \(\nu \) such that \(B=0\). Then, with respect to the local coordinates (x, y) defined in (5.2), the position vector F of M in \(\mathbb {R}^4_2\) is given by
where w(y) is a timelike unit vector field in \(\mathbb {R}^4_2\), depending only on y.
We are now ready to prove the following result.
Theorem 6.2
(Of characterization for \(B=0\)) Let M be a helix surface in \(\mathbb {H}^3_{1, \tau }\subset \mathbb {R}^4_2\) with constant angle function \(\nu \) such that \(B=0\). Then, with respect to the local coordinates (x, y) defined in (5.2), the position vector F of M in \(\mathbb {R}^4_2\) is given by
where
is a straight line of \(\mathbb {H}^3_{1, \tau }\) (contained in the plane \(x_2=x_3-\frac{2}{\sqrt{\kappa }}=0\)) and \(A(y)=A(\xi ,\xi _1,\xi _2,\xi _3)(y)\) is a 1-parameter family of \(4\times 4\) pseudo-orthogonal matrices commuting with \(J_1\) as described in (3.7), with
Conversely, a parametrization
with A(y) as above, defines a helix surface in the anti-De Sitter space \(\mathbb {H}^3_{1, \tau }\) with constant angle function \(\nu \).
Proof
From Proposition 6.3 we say that
where w(y) is a vector field in \(\mathbb {R}^4_2\), depending only on y. Using (6.28) and evaluating the first three equations of (5.9) and the second equation of (6.1) at (0, y), we get the following identities:
Moreover, evaluating (5.6) in (0, y), setting
and using (6.29), we have
In particular, setting
we have that \(\{g^1(y),J_1g^1(y),g^3(y),J_1g^3(y)\}\) is an orthonormal basis of \(\mathbb {R}^4_2\). Consequently, if we fix the orthonormal basis \(\{\hat{E}_i\}^4_{i=1}\) of \(\mathbb {R}^4_2\) given by
there exists a 1-parameter family of matrices \(A(y)\in O_2(4)\), with \(J_1A(y)=A(y)J_1\) such that
Then, (6.28) becomes
Finally, according to (3.7), the 1-parameter family A(y) depends on four functions \(\xi _1(y)\), \(\xi _2(y)\), \(\xi _3(y)\) and \(\xi (y)\) and, in this case, condition (6.3) reduces to \(\langle F_u, F_v\rangle =0\) which is equivalent to (6.27).
In order to prove the converse, let
be a parametrization, where \(A(y)=A(\xi (y),\xi _1(y),\xi _2(y),\xi _3(y))\) is a 1-parameter family of pseudo-orthogonal matrices with functions \(\xi (y),\xi _1(y),\xi _2(y),\xi _3(y)\) satisfying (6.27). Since A(y) satisfies (6.27), F satisfies (6.3). Thus, in virtue of Proposition 6.1, we only have to show that (6.2) is satisfied. We put
Now, using (3.1) and taking into account the fact that A(y) commutes with \(J_1\), we get
and similarly,
which ends the proof. \(\square \)
6.3 Helix surfaces of \(\mathbb {H}^3_{1, \tau }\) in the case \(B<0\)
In this case, we start from (5.5) and prove the following result.
Proposition 6.4
Let M be a helix surface in \(\mathbb {H}^3_{1, \tau }\) with constant angle function \(\nu \) such that \(B<0\). Then, with respect to the local coordinates (x, y) defined above, the position vector F of M in \(\mathbb {R}^4_2\) is given by
where
are real constants and \(w^i(y)\), \(i=1,2,3,4\), are linearly independent vector fields in \(\mathbb {R}^4_2\), depending only on y, such that:
Proof
As \(B <0\), we have \(\tilde{b}^2+4\tilde{a}<0\). Integrating Eq. (5.5), we obtain
where
are real constants and \(w^i(y)\), \(i=1,2,3,4\), are vector fields in \(\mathbb {R}^4_2\), depending only on y. Moreover, using the definition of \(\tilde{a}\) and \(\tilde{b}\), we get
Defining \(w_{ij}=\langle w^i(y), w^j(y)\rangle \), for all indices i, j and evaluating the relations (5.9) in (0, y), we find:
From (6.31), (6.35) and (6.36), it follows that
Also, from (6.33) and (6.39), we obtain
and, therefore, from (6.34) and (6.38),
Moreover, using (6.32), (6.37) and (6.40), we get
\(\square \)
We now prove the following.
Theorem 6.3
(Of characterization for \(B<0\)) Let M be a helix surface in \(\mathbb {H}^3_{1, \tau }\) with constant angle function \(\nu \) so that \(B<0\). Then, locally, the position vector of M in \({{\mathbb {R}}}^4_2\), with respect to the local coordinates (x, y) on M defined in (5.2), is given by
where the curve \( \gamma (x)=(\gamma _1(x),\gamma _2(x),\gamma _3(x),\gamma _4(x)) \) has components
with
and \(A(y)=A(\xi ,\xi _1,\xi _2,\xi _3)(y)\) is a 1-parameter family of \(4\times 4\) pseudo-orthogonal matrices commuting with \(J_1\), as described in (3.6), where \(\xi \) is a constant and
Conversely, a parametrization \(F(x,y)=A(y)\,\gamma (x)\), with \(\gamma (x)\) and A(y) as above, defines a helix surface in \(\mathbb {H}^3_{1, \tau }\) with constant angle function \(\nu \ne 0\).
Proof
From (6.30), we can define the following orthonormal basis in \({{\mathbb {R}}}^4_2\):
with \(\langle e_1,e_1\rangle =1=\langle e_2,e_2\rangle \) and \(\langle e_3,e_3\rangle =-1=\langle e_4,e_4\rangle \). Evaluating the identities (6.1) in (0, y), and taking into account that:
we conclude that
We point out that to obtain the previous identities, we divided by
which is always different from zero. Then,
Therefore, we have
Consequently, if we consider the orthonormal basis \({\{\hat{E}_i\}_{i=1}^{4}}\) of \({{\mathbb {R}}}^4_2\) given by
there must exists a 1-parameter family of matrices \(A(y)\in \textrm{O}_2(4)\), with \(J_1A(y)=A(y)J_1\), such that \(e_i(y)=A(y)\hat{E}_i\) for all indices \(i=1,\dots ,4\). As
computing \(\langle F,e_i\rangle \) and substituting \(e_i(y)=A(y)\hat{E}_i\), we obtain that \(F(x,y)=A(y)\,\gamma (x), \) where \(\gamma (x)\) is the curve of \(\mathbb {H}^3_{1, \tau }\) described in the statement. Proceeding as in the proof of Theorem 6.1, we now examine the 1-parameter family A(y) that, according to (3.7), depends on four functions \(\xi _1(y)\), \(\xi _2(y)\), \(\xi _3(y)\) and \(\xi (y)\). from \(\langle F_y, F_y\rangle =\lambda + \nu ^2=\textrm{constant}\) we have
If we denote by \({{\textbf{c}}_1},{{\textbf{c}}_2},{{\textbf{c}}_3},{\textbf{c}_4}\) the four columns of A(y), Eq. (6.42) implies that
where \('\) denotes the derivative with respect to y. Replacing in (6.43) the expressions of the \({{\textbf{c}}_i}\)’s as functions of \(\xi _1(y),\xi _2(y),\xi _3(y)\) and \(\xi (y)\), we obtain
where \(h_2(y)\) and \(k_2(y)\) are given by
From (6.44) we have two possibilities:
-
(i)
\(\xi =\textrm{constant}\);
or
-
(ii)
\(h_2=k_2=0\).
As
it results that if the case (ii) happens than the parametrization \(F(x,y)=A(y)\gamma (x)\) defines a Hopf tube. Thus, we can assume that \(\xi =\textrm{constant}\) and in this case (6.3) is equivalent to (6.41).
The converse easily follows from Proposition 6.1, since a direct calculation shows that \(g_{\tau }(F_x,F_x)=g_{\tau }(E_1,F_x)=-\lambda (\lambda +\nu ^2)\) (and so, (6.2) holds), while (6.41) is equivalent to (6.3). \(\square \)
7 Characterization of the helix surfaces of \(\mathbb {H}^3_{1, \tau }\) by general helices
As a consequence of Proposition 6.1 and the characterization Theorems 6.1, 6.2 and 6.3, in the next result we will prove that the curves used to describe helix surfaces in \(\mathbb {H}^3_{1, \tau }\), are general helices with axis the infinitesimal generator of the Hopf fibers. We recall that a general helix is a non-null curve \(\alpha \) in a Lorentzian manifold (N, h), admitting a Killing vector field V of constant length along \(\alpha \), such that the function angle between V and \(\alpha '\) is a non-zero constant (for the concept of general helix in a Lorentzian ambient space, we may also refer to [1]). We say that V is an axis of the general helix \(\alpha \). We now prove the following.
Proposition 7.1
The curves \(\gamma :{{\mathbb {R}}}\rightarrow \mathbb {H}^3_{1, \tau }\) used in the Theorems 6.1, 6.2 and 6.3 to characterize a constant angle spacelike (respectively, timelike) surface M, are spacelike (respectively, timelike) general helices in \(\mathbb {H}^3_{1, \tau }\) with axis \(E_1\), so that they meet at constant angle the fibers of the Hopf fibration. This angle is the same in all the three cases.
Proof
We first observe that in the three cases the position vector of M has been expressed as
where \(A(y)=A(\xi ,\xi _1,\xi _2,\xi _3)(y)\) is a 1-parameter family of \(4\times 4\) pseudo-orthogonal matrices commuting with \(J_1\) and \(\gamma (x)\) is a curve on \(\mathbb {H}^3_{1, \tau }\). Therefore, as \(F_x=A(y)\gamma '\), from (6.2) we get
thus we conclude that if M is a spacelike (respectively, timelike) surface, then \(\gamma \) is a spacelike (respectively, timelike) curve. In both cases, the above equation yields
Moreover, as \(J_1A(y)=A(y)J_1\), we have
and then, from (6.2), we obtain
Therefore, the angle function that \(\gamma \) forms with the hyperbolic Hopf vector field is given by
that is, in the three cases described in Theorems 6.1, 6.2 and 6.3, \(\gamma \) is a general helix, forming the same constant angle with its axis \(E_1\). \(\square \)
Remark 7.1
As we observed in Remark 5.2, when \(\tau =1\) we get flat helix surfaces in \(\mathbb {H}^3_1(\kappa / 4)\) equipped with its standard metric. The results we obtained are consistent with the ones deduced in [12], under the requirement of constant angle between N and \(E_1\). In this case, \(B=-\lambda \) and so:
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The second and the fourth authors were supported by a grant of Fondazione di Sardegna (Project GoAct). The second author was also supported by the Thematic Project: Topologia Álgebrica, Geométrica e Diferencial, Fapesp process number 2016/24707-4. The third author was supported by a scolarship of INdAM (Istituto Nazionale di Alta Matematica ”Francesco Severi”) reserved for master students in Mathematics.
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Calvaruso, G., Onnis, I.I., Pellegrino, L. et al. Helix surfaces for Berger-like metrics on the anti-de Sitter space. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 54 (2024). https://doi.org/10.1007/s13398-024-01550-x
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DOI: https://doi.org/10.1007/s13398-024-01550-x