Helix surfaces for Berger-like metrics on the anti-de Sitter space

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.


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 [2]- [5], [8], [9], [12], [13], [14] 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 [6], [7], [18]).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 [10], [11], [15], [16].In particular, helix surfaces of the anti-de Sitter space H 3  1 were studied in [11].Equipping H 3 1 with its canonical metric of constant curvature, all left-invariant vector fields are Killing (see also [1]), so no special directions emerge.Moreover, all helix surfaces turn out to be flat [11].
In [1], the first author and D. Perrone introduced and studied a new family of metrics gλµν on H 3  1 (κ/4).These metrics were induced in a natural way by corresponding metrics defined on the tangent sphere bundle T 1 H 2 (κ), after describing the covering map F from H 3  1 (κ/4) to T 1 H 2 (κ) 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 S 3 , respectively.In [12], [13], [15] and [16], 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 H 3  1 (κ/4) with a special type of the Lorentzian metrics introduced in [1], that is, the ones that deform the standard metric of H 3  1 (κ/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 S 3 , we shall refer to these metrics as Berger-like metrics.We shall consider the anti-de Sitter space H 3  1 (κ/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 Section 2 we provide some needed information on the anti-de Sitter space H 3 1 (κ/4).In Section 3 we describe the Levi-Civita connection and the curvature of Berger-like metrics g τ on H 3  1 (κ/4).The general equations for a surface of (H 3 1 (κ/4), g τ ) are given in Section 4 and then are applied in Section 5 to the case of helix surfaces.Three different cases occur, according to the sign of some constant B, which depends on the parameter τ of the Berger-like metric, the causal character of the surface and the constant angle.These three cases are completely described and characterized in Section 6, in terms of a 1-parameter family of isometries on H 3  1,τ and some suitable curves.As we prove in Section 7, these curves are general helices, which meet at a constant angle the fibers of the hyperbolic Hopf fibration.

Preliminaries
Let R 4  2 = (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 κ > 0, the anti-de Sitter (three)-space H 3  1 (κ) is the hypersurface of R 4 2 defined by = −1/κ}.We denote by , the Lorentzian metric induced from g 0 on H 3 1 (κ).This metric, known as the canonical metric of H 3  1 , has constant sectional curvature −κ < 0. Consider now the algebra B of paraquaternionic numbers over 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 R.An arbitrary paraquaternonic number is given by q = x 1 + x 2 i + x 3 j + x 4 k.The conjugate of q is q = x 1 − x 2 i − x 3 j − x 4 k and the norm of q is given by The scalar product induced by such norm on R 4 is exactly g 0 , and {1, i, j, k} is a pseudo-orthonormal basis with 1, i spacelike and j, k timelike.In terms of paraquaternionic numbers, we have The above presentation was used in [1] to describe a covering map from the anti-de Sitter space H 3 1 (κ/4) to the unit tangent sphere bundle of the (Riemannian) hyperbolic two-space H 2 (κ), which is embedded in R 3  1 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 In particular, h is a submersion with geodesic fibers, which can be defined as orbits of the S 1 -action e it , (z, w) → e it z, e it w) on the anti-de Sitter space H 3 1 (κ/4).The vector field is tangent to fibers of h and satisfies X 1 , X 1 = −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 X l , X m = 0, for l = m.

Levi-Civita connection and curvature
In [1], the covering map F described by ( 2) was used to introduce a family of Lorentzian metrics on the anti-de Sitter space H 3 1 (κ/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 H 2 (κ).Their general description is given by where ρ, µ, ν are positive real numbers and {θ 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 [1].Up to homotheties, we may restrict to the case where µ = ν = 1 and ρ = τ 2 and consider the one-parameter family of metrics These metrics can be clearly considered an hyperbolic analogue of Berger metrics on the sphere S 3 and so we refer to them as Berger-like metrics on H 3 1 (κ/4).Observe that with respect to the standard metric , on H 3 1 (κ/4), these metrics can be described as and Then, by the Koszul formula, we obtain the description of the Levi-Civita connection of 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 determine the wedge product U ∧ V (see [17]).We now consider the curvature tensor, taken with the sign convention Using (3.3) we have that the non vanishing components of the Riemannian curvature tensor R τ are and we can prove the following result.
Proposition 3.1.The curvature tensor of H 3  1,τ is given by for all tangent vector fields X, Y, Z.
Proof.Consider three arbitrary vector fields X, Y, Z on H 3 1,τ and their decompositions as with X, Ȳ , Z orthogonal to E 1 and x = g τ (X, E 1 ) and so on.Observe that using (3.5) we have that all terms of g τ (R τ (X, Y )Z, W ) where E 1 occurs either one, three or four times necessarily vanish.Therefore, we have From the decompositions of X, Ȳ , Z and W follows that Moreover, we have Thus, we conclude that which ends the proof since W is arbitrary.
We end this section describing the isometries of H 3 1,τ .Following the idea used in [13] and [16], we observe that the isometry group of H 3  1,τ is the four-dimensional indefinite unitary group U 1 (2), that can be identified with where J 1 is the complex structure of R 4 corresponding to i, i.e., defined by (3.7) is the pseudo-orthogonal group, i.e., the group of 4 × 4 real matrices preserving the semi-definite inner product of R 4 2 .We now consider a 1-parameter family A(v), v ∈ (a, b) ⊂ R, consisting of 4 × 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 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 r 1 (v) of R 4 2 for all v ∈ (a, b).Thus, without loss of generality, we can take for some real functions ξ 1 , ξ 2 and ξ 3 defined in (a, b).Since A(v) commutes (anticommutes, respectively) with J 1 the second row of A(v) must be r 2 (v) = ±J 1 r 1 (v).Now, the four vectors {r 1 , J 1 r 1 , J 2 r 1 , J 3 r 1 } form a pseudo-orthonormal basis of R 4  2 , thus the third row r 3 (v) of A(v) must be a linear combination of them.Since r 3 (v) is unit and it is orthogonal to both r 1 (v) and J 1 r 1 (v), there exists a function ξ(v) such that . This means that any 1-parameter family A(v) of 4 × 4 pseudo-orthogonal matrices commuting (anticommuting, respectively) with J 1 can be described by four functions ξ 1 , ξ 2 , ξ 3 and ξ as In the following, we compute the Gauss and Codazzi equations for M , using the metric induced by g τ on M , the shape operator A, the tangent projection of E 1 on M and the angle function ν := g τ (N, E 1 )g τ (N, N ) = λg τ (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 ∇ is the Levi-Civita connection of M and α the second fundamental form with respect to the immersion in H 3  1,τ .Thus, we conclude that On the other hand, by (3.4) we have where JX := N ∧ X satisfies (4.12) Comparing the above expressions for ∇ τ X E 1 , we find (4.13) 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 K the sectional curvature in H 3 1,τ of the plane tangent to M , we have and Proof.Recall that for a pseudo-Riemannian surface, one has and so, that gives whence equation (4.14) follows.
Consider the Codazzi equation From Proposition 3.1, we have that leads equation (4.15) by the arbitrarity of Z.

Basic properties of helix surfaces in H 3 1,τ
We start with the following.
Definition 5.1.Let M be an oriented pseudo-Riemannian surface immersed in H 3 1,τ and N the unit vector field normal to M , with g τ (N, N ) = λ.The surface M is called a helix surface (or a constant angle surface) if the angle function ν = λ g τ (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 (4.9) and (4.12) we get Observe that in the case λ = 1, if ν = 0 then E 1 is tangent to M at each point.Therefore, M is a Hopf tube.On the other hand, if λ = −1, then |ν| > 1 and so ν = 0.
Taking into account Remark 5.1 from now on we assume ν = 0.
Proposition 5.1.Let M be a helix surface in H 3 1,τ 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 for some smooth function µ on M ; (ii) the Levi-Civita connection ∇ of M is described by (iii) the Gaussian curvature of M is given by (iv) the function µ satisfies equation where 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 ∇ of M , using (4.13), we have and using the compatibility of ∇ with the metric g τ , follows So, From (4.14) 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 − τ 2 ) and it can assume any real value.
In the special case of τ = 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 [11] 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.
We recall that E 1 is a timelike vector field and g τ (E 1 , N ) = νλ.Thus, there exists a smooth function ϕ on M , such that and we can calculate Moreover, we have Therefore, But A(T ) = √ κ 2 τ JT and so, we conclude that √ κ that is, Moreover, In this way, we determined the following system whose compatibility condition is given by Recalling that by comparison we obtain that is, equation (5.16), which we already established, so that the system is compatible.
We can choose a system of local coordinates (x, y) on M , such that (5.17) for some smooth functions a = a(x, y), b = b(x, y) on M .Requiring that [∂ x , ∂ y ] = 0, we get So, equation (5.16) can be rewritten as follows: (5.18) In order to integrate equation (5.18), we need to consider separately the following cases In all the above cases, η(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.
which admits as solution b = cos(η(y) − ν √ κB x).Then, we have and so, we can take and a solution is given by b = νx + η(y).Moreover, we have: which holds for a = −ν whence, ϕ(x, y) = y + c for some real constant c.
which is satisfied by b = cosh(η(y) + ν √ −κB x).Moreover, we find: and so, we take a and we obtain ϕ(x, y) = − √ κ λτ 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 H 3  1,τ with constant angle function ν.With respect to the local coordinates (x, y) defined above, the position vector F of M in R 4  2 satisfies the following equation: (a) if B = 0, Proof.Let M be a helix surface and let denote the position vector of M in 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 R 4  2 , we can express the above equation as (5.22) Therefore, if B = 0 taking the derivative of (5.22) with respect to x, we get (5.19).
If we suppose B = 0, we obtain where In conclusion, taking twice the derivative of (5.23) with respect to x and using the previous relations we find (5.20).

1,τ
In order to give conditions under which an immersion defines a helix surface in H 3 1,τ we observe that, if F is a position vector of a helix surface in H 3  1,τ we have that: and, using the (5.24), we have the following identities: We now use these relations to prove the following key result.

defines a helix surface of constant angle function ν if and only if
Proof.Suppose that F (Ω) is a helix surface in H 3  1,τ of constant angle function ν, then we have: In a similar way we find: that leads to the equation (6.26).In addition, we have that is equation (6.27).
To prove the converse, consider Then, we get the orthonormal basis {F x , T , N } for the tangent space to H 3 1,τ along F (Ω).Moreover, from (6.26) and (6.27) we have: This leads to E 1 = c 1 F x + c 2 N .Moreover from (6.26) we have c 1 = 1 that means (E 1 ) T = F x and also In order to get explicit solutions of equations (5.20) and (5.19) we consider three different cases, depending on the different possibilities for B.
We now prove the following result.
Proof.We consider the curve γ(x) given in the Theorem 6.1.Since γ ′ (x), γ ′ (x) = 4 κ α 1 α 2 , considering from equation (6.28) and taking into account the equation (6.29) with w 13 = 0, we get Observe that d > 1.Therefore, we can consider the arc length reparameterization of the curve γ given by Finally, we observe that d represents the slope of the geodesic γ.
6.2.Helix surfaces of H 3 1,τ in the case B = 0. Integrating (5.19) and taking into account ∂ x F = T , we prove at once the following.Proposition 6.3.Let M be a helix surface in H 3  1,τ ⊂ R 4 2 with constant angle function ν such that B = 0.Then, with respect to the local coordinates (x, y) defined in (5.17), the position vector F of M in R 4  2 is given by F (x, y) = T (y)x + w(y), where w(y) is a timelike unit vector field in 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 H 3  1,τ ⊂ R 4 2 with constant angle function ν such that B = 0.Then, with respect to the local coordinates (x, y) defined in (5.17), the position vector F of M in R 4  2 is given by F (x, y) = A(y)γ(x), where is a 1-parameter family of 4 × 4 indefinite orthogonal matrices commuting with J 1 as described in (3.8), with Conversely, a parametrization with A(y) as above, defines a helix surface in the anti-De Sitter space H 3 1,τ with constant angle function ν.
If we denote by c 1 , c 2 , c 3 , c 4 the four columns of A(y), equation (6.66) implies that (6.67) where ′ denotes the derivative with respect to y. Replacing in (6.67) the expressions of the c i 's as functions of ξ 1 (y), ξ 2 (y), ξ 3 (y) and ξ(y), we obtain it results that if the case (ii) happens than the parametrization F (x, y) = A(y)γ(x) defines a Hopf tube.Thus, we can assume that ξ = constant and in this case (6.27) is equivalent to (6.65).The converse easily follows from Proposition 6.1, since a direct calculation shows that g τ (F x , F x ) = g τ (E 1 , F x ) = −λ(λ + ν 2 ) (and so, (6.26) holds), while (6.65) is equivalent to (6.27).
7. Characterization of the helix surfaces of H 3 1,τ 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 H 3  1,τ , are general helices with axis the infinitesimal generator of the Hopf fibers.We recall that a general helix is a non-null curve α in a Lorentzian manifold (N, h), admitting a Killing vector field V of constant length along α, such that the function angle between V and α ′ is a non-zero constant.We say that V is an axis of the general helix α.We now prove the following.
Proposition 7.1.The curves γ : R → H 3  1,τ 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 H 3  1,τ 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 F (x, y) = A(y)γ(x), where A(y) = A(ξ, ξ 1 , ξ 2 , ξ 3 )(y) is a 1-parameter family of 4 × 4 pseudo-orthogonal matrices commuting with J 1 and γ(x) is a curve on H 3 1,τ .Therefore, as F x = A(y)γ ′ , from (6.26) we get g τ (γ ′ , γ ′ ) = g τ (F x , F x ) = −λ (λ + ν 2 ), thus we conclude that if M is a spacelike (respectively, timelike) surface, then γ is a spacelike (respectively, timelike) curve.In both cases, the above equation yields γ ′ τ = λ + ν 2 .Moreover, as J 1 A(y) = A(y)J 1 , we have A(y)J 1 γ and then, from (6.26), we obtain Therefore, the angle function that γ 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, γ is a general helix, forming the same constant angle with its axis E 1 .
Remark 7.1.As we observed in Remark 5.2, when τ = 1 we get flat helix surfaces in H 3 1 (κ/4) equipped with its standard metric.The results we obtained are consistent with the ones deduced in [11], under the requirement of constant angle between N and E 1 .In this case, B = −λ and so: • the case B > 0 corresponds to Lorentzian helix surfaces considered in [11], as λ = −1; • the case B = 0 cannot occur; • the case B < 0 corresponds to Riemannian helix surfaces considered in [11], as λ = 1.