Abstract
We characterize helix surfaces (constant angle surfaces) in the special linear group \(\mathrm {SL}(2,{\mathbb {R}})\). In particular, we give an explicit local description of these surfaces by means of a suitable curve and a 1-parameter family of isometries of \(\mathrm {SL}(2,{\mathbb {R}})\).
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1 Introduction
In recent years much work has been done to understand the geometry of surfaces whose unit normal vector field forms a constant angle with a fixed field of directions of the ambient space. These surfaces are called helix surfaces or constant angle surfaces and they have been studied in most of the 3-dimensional geometries. In [2] Cermelli and Di Scala analyzed the case of constant angle surfaces in \({\mathbb {R}}^3\) obtaining a remarkable relation with a Hamilton–Jacobi equation and showing their application to equilibrium configurations of liquid crystals. Later, Dillen–Fastenakels–Van der Veken–Vrancken [4], and Dillen–Munteanu [3], classified the surfaces making a constant angle with the \({\mathbb {R}}\)-direction in the product spaces \({\mathbb {S}}^2\times {\mathbb {R}}\) and \({\mathbb {H}}^2\times {\mathbb {R}}\), respectively. Moreover, helix submanifolds have been studied in higher dimensional euclidean spaces and product spaces in [5, 6, 10].
The spaces \({\mathbb {S}}^2\times {\mathbb {R}}\) and \({\mathbb {H}}^2\times {\mathbb {R}}\) can be seen as two particular cases of Bianchi–Cartan–Vranceanu spaces (BCV-spaces) which include all 3-dimensional homogeneous metrics whose group of isometries has dimension 4 or 6, except for those of constant negative sectional curvature. A crucial feature of BCV-spaces is that they admit a Riemannian submersion onto a surface of constant Gaussian curvature, called the Hopf fibration, that, in the cases of \({\mathbb {S}}^2\times {\mathbb {R}}\) and \({\mathbb {H}}^2\times {\mathbb {R}}\), it is the natural projection onto the first factor. Consequently, one can consider the angle \(\vartheta \) that the unit normal vector field of a surface in a BCV-space forms with the Hopf vector field, which is, by definition, the vector field tangent to the fibers of the Hopf fibration. This angle \(\vartheta \) has a crucial role in the study of surfaces in BCV-spaces as shown by Daniel, in [1], where he proved that the equations of Gauss and Codazzi are given in terms of the function \(\nu =\cos \vartheta \) and that this angle is one of the fundamental invariants for a surface in BCV-spaces. Consequently, in [7], the authors considered the surfaces in a BCV-space for which the angle \(\vartheta \) is constant, giving a complete local classification in the case that the BCV-space is the Heisenberg space \({\mathbb {H}}_3\).
Later, López–Munteanu, in [8], defined and classified two types of constant angle surfaces in the homogeneous 3-manifold \(\mathrm {Sol}_3\), whose isometry group has dimension 3. Also, Montaldo–Onnis, in [9], characterized helix surfaces in the 1-parameter family of Berger spheres \({\mathbb {S}}^3_\epsilon \), with \(\epsilon >0\), proving that, locally, a helix surface is determined by a suitable 1-parameter family of isometries of the Berger sphere and by a geodesic of a 2-torus in the 3-dimensional sphere.
This paper is a continuation of our work [9] and it is devoted to the study and characterization of helix surfaces in the homogeneous 3-manifold given by the special linear group \(\mathrm {SL} (2,{\mathbb {R}})\) endowed with a suitable 1-parameter family \(g_{\tau }\) of metrics that we shall describe in sect. 2. Our study of helix surfaces in \((\mathrm {SL}(2,{\mathbb {R}}),g_{\tau })\) will depend on a constant \(B:=(\tau ^2+1)\cos ^2\vartheta -1\), where \(\vartheta \) is the constant angle between the normal to the surface and the Hopf vector field of \(\mathrm {SL} (2,{\mathbb {R}})\). A similar constant appeared also in the study of helix surfaces in the Berger sphere (see [9]) but in that case the constant was always positive. Thus we shall divide our study according to the three possibilities: \(B>0,\, B=0\) and \(B<0\).
2 Preliminaries
Let \({\mathbb {R}}_2^4\) denote the 4-dimensional pseudo-Euclidean space endowed with the semi-definite inner product of signature (2,2) given by
We identify the special linear group with
and we shall use the Lorentz model of the hyperbolic plane with constant Gauss curvature \(-4\), that is
where \({\mathbb {R}}^3_1\) is the Minkowski 3-space. Then the Hopf map \(\psi :\mathrm {SL}(2,{\mathbb {R}})\rightarrow {\mathbb {H}}^2(-4)\) given by
is a submersion, with circular fibers, and if we put
we have that \(X_1\) is a vertical vector field while \(X_2,\, X_3\) are horizontal. The vector field \(X_1\) is called the Hopf vector field.
We shall endow \(\mathrm {SL}(2,{\mathbb {R}})\) with the 1-parameter family of metrics \(g_\tau ,\, \tau >0\), given by
which renders the Hopf map \(\psi :(\mathrm {SL}(2,{\mathbb {R}}),g_{\tau })\rightarrow {\mathbb {H}}^2(-4)\) a Riemannian submersion.
For those familiar with the notations of Daniel [1], we point out that \((\mathrm {SL}(2,{\mathbb {R}}),g_{\tau })\) corresponds to a model for a homogeneous space \(E(k, \tau )\) with curvature of the basis \(k=-4\) and bundle curvature \(\tau >0\).
With respect to the inner product in \({\mathbb {R}}_2^4\) the metric \(g_{\tau }\) is given by
From now on, we denote \((\mathrm {SL}(2,{\mathbb {R}}),g_\tau )\) with \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\). Obviously
is an orthonormal basis on \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\) and the Levi-Civita connection \({{\nabla }}^\tau \) of \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\) is given by (see, for example, [11]):
Finally, we recall that the isometry group of \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\) is the 4-dimensional indefinite unitary group \(\mathrm {U}_1(2)\) that can be identified with:
where \(J_1\) is the complex structure of \({\mathbb {R}}^4\) defined by
while
is the indefinite orthogonal group.
We observe that \(\mathrm {O}_2(4)\) is the group of \(4\times 4\) real matrices preserving the semi-definite inner product of \({\mathbb {R}}_2^4\).
Suppose now we are given a 1-parameter family \(A(v), v\in (a,b)\subset {\mathbb {R}}\), consisting of \(4\times 4\) indefinite orthogonal matrices commuting (anticommuting, respectively) with \(J_1\). In order to describe explicitly the family \(A(v)\), we shall use two product structures of \({\mathbb {R}}^4\), namely
Since \(A(v)\) is an indefinite orthogonal matrix, the first row must be a unit vector \({\mathbf {r}}_1(v)\) 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 \({\mathbf {r}}_2(v)=\pm J_{1}{\mathbf {r}}_1(v)\). Now, the four vectors \(\{{\mathbf {r}}_1, J_1{\mathbf {r}}_1, J_2{\mathbf {r}}_1,J_3{\mathbf {r}}_1\}\) form a pseudo-orthonormal basis of \({\mathbb {R}}^4_2\), thus the third row \({\mathbf {r}}_3(v)\) of \(A(v)\) must be a linear combination of them. Since \({\mathbf {r}}_3(v)\) is unit and it is orthogonal to both \({\mathbf {r}}_1(v)\) and \(J_1{\mathbf {r}}_1(v)\), there exists a function \(\xi (v)\) such that
Finally the fourth row of \(A(v)\) is \({\mathbf {r}}_4(v)=\pm J_1{\mathbf {r}}_3(v)=\mp \cos \xi (v) J_3 {\mathbf {r}}_1(v)\pm \sin \xi (v) J_2 {\mathbf {r}}_1(v)\). This means that any 1-parameter family \(A(v)\) of \(4\times 4\) indefinite orthogonal matrices commuting (anticommuting, respectively) with \(J_1\) can be described by four functions \(\xi _1,\xi _2,\xi _3\) and \(\xi \) as
3 Constant angle surfaces
We start this section giving the definition of constant angle surface in \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\).
Definition 3.1
We say that a surface in the special linear group \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\) is a helix surface or a constant angle surface if the angle \(\vartheta \in [0,\pi )\) between the unit normal vector field and the unit Killing vector field \(E_1\) (tangent to the fibers of the Hopf fibration) is constant at every point of the surface.
Let \(M^2\) be an oriented helix surface in \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\) and let \(N\) be a unit normal vector field. Then, by definition,
for fixed \(\vartheta \in [0,\pi /2]\). Note that \(\vartheta \ne 0\). In fact, if it were zero then the vector fields \(E_2\) and \(E_3\) would be tangent to the surface \(M^2\), which is absurd since the horizontal distribution of the Hopf map is not integrable. If \(\vartheta =\pi /2\), we have that \(E_1\) is always tangent to \(M\) and, therefore, \(M\) is a Hopf cylinder. Therefore, from now on we assume that the constant angle \(\vartheta \ne \pi /2,0\).
The Gauss and Weingarten formulas are
where with \(A\) we have indicated the shape operator of \(M\) in \(\mathrm {SL}(2,{\mathbb {R}})_\tau \), with \(\nabla \) the induced Levi-Civita connection on \(M\) and by \(\alpha \) the second fundamental form of \(M\) in \(\mathrm {SL}(2,{\mathbb {R}})_\tau \). Projecting \(E_1\) onto the tangent plane to \(M\) we have
where \(T\) is the tangent part which satisfies \(g_\tau (T,T)=\sin ^2\vartheta \).
For all \(X\in TM\), we have that
On the other hand, if \(X=\sum X_i E_i\),
where \(JX=N\wedge X\) denotes the rotation of angle \(\pi /2\) on \(TM\). Identifying the tangent and normal component of (6) and (7) respectively, we obtain
and
Lemma 3.2
Let \(M^2\) be an oriented helix surface in \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\) with constant angle \(\vartheta \). Then, we have the followings properties.
- (i):
-
With respect to the basis \(\{T,JT\}\), the matrix associates to the shape operator \(A\) takes the form
$$\begin{aligned} A=\left( \begin{array}{cc} 0 &{}\quad -\tau \\ -\tau &{}\quad \lambda \\ \end{array}\right) , \end{aligned}$$for some function \(\lambda \) on \(M\).
- (ii):
-
The Levi-Civita connection \(\nabla \) of \(M\) is given by
$$\begin{aligned} \nabla _T T=-2\tau \cos \vartheta \, JT,\qquad \nabla _{JT} T=\lambda \cos \vartheta \, JT, \\ \nabla _T JT=2\tau \cos \vartheta \, T,\qquad \nabla _{JT} JT=-\lambda \cos \vartheta \, T. \end{aligned}$$ - (iii):
-
The Gauss curvature of \(M\) is constant and satisfies
$$\begin{aligned} K=-4(1+\tau ^2)\,\cos ^2\vartheta . \end{aligned}$$ - (iv):
-
The function \(\lambda \) satisfies the equation
$$\begin{aligned} T \lambda +\lambda ^2\,\cos \vartheta +4B\,\cos \vartheta =0, \end{aligned}$$(10)where \(B:=(\tau ^2+1)\cos ^2\vartheta -1\).
Proof
Point (i) follows directly from (9). From (8) and using
we obtain (ii). From the Gauss equation in \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\) (we refer to the equation in Corollary 3.2 of [1] with \(\nu =\cos \theta \) and \(k=-4\)), and (i), we have that the Gauss curvature of \(M\) is given by
Finally, (10) follows from the Codazzi equation (see [1]):
putting \(X=T,\, Y=JT\) and using (ii). In fact, it is easy to check that
and
\(\square \)
As \(g_\tau (E_1,N)=\cos \vartheta \), there exists a smooth function \(\varphi \) on \(M\) such that
Therefore
and
Also
Comparing (12) with (i) of Lemma 3.2, it results that
We observe that, as
the compatibility condition of system (13):
is equivalent to (10).
We now choose local coordinates \((u,v)\) on \(M\) such that
Also, as \(\partial _v\) is tangent to \(M\), it can be written in the form
for certain functions \(a=a(u,v)\) and \(b=b(u,v)\). As
then
Moreover, the Eq. (10) of Lemma 3.2 can be written as
Depending on the value of \(B\), by integration of (17), we have the following three possibilities.
-
(i)
If \(B=0\)
$$\begin{aligned} \lambda (u,v)=\frac{1}{u\,\cos \vartheta +\eta (v)}, \end{aligned}$$for some smooth function \(\eta \) depending on \(v\). Thus the solution of system (16) is given by
$$\begin{aligned} \left\{ \begin{array}{l} a(u,v)=-\tau \,u\,\cos \vartheta \,(u\,\cos \vartheta +2\,\eta (v)),\\ b(u,v)=u\,\cos \vartheta +\eta (v). \end{array}\right. \end{aligned}$$ -
(ii)
If \(B>0\)
$$\begin{aligned} \lambda (u,v)=2\,\sqrt{B}\tan (\eta (v)-2\cos \vartheta \sqrt{B}\,u), \end{aligned}$$for some smooth function \(\eta \) depending on \(v\) and system (16) has the solution
$$\begin{aligned} \left\{ \begin{array}{l} a(u,v)=\frac{\tau }{\sqrt{B}}\sin (\eta (v)-2\cos \vartheta \sqrt{B}\,u),\\ b(u,v)=\cos (\eta (v)-2\cos \vartheta \sqrt{B}\,u). \end{array}\right. \end{aligned}$$ -
(iii)
If \(B<0\)
$$\begin{aligned} \lambda (u,v)=2\,\sqrt{-B}\tanh ({\eta }(v)+2\cos \vartheta \sqrt{-B}\,u), \end{aligned}$$for some smooth function \({\eta }\) depending on \(v\). Solving the system (16), we have
$$\begin{aligned} \left\{ \begin{array}{l} a(u,v)=-\frac{\tau }{\sqrt{-B}}\sinh ({\eta }(v)+2\cos \vartheta \sqrt{-B}\,u),\\ b(u,v)=\cosh ({\eta }(v)+2\cos \vartheta \sqrt{-B}\,u). \end{array}\right. \end{aligned}$$
Moreover, in the case (i) the system (13) becomes
and so \(\varphi (u,v)=v+c,\, c\in {\mathbb {R}}\). In the cases (ii) and (iii), the system (13) becomes
of which the general solution is given by
where \(c\) is a real constant.
With respect to the local coordinates \((u,v)\) chosen above, we have the following characterization of the position vector of a helix surface.
Proposition 3.3
Let \(M^2\) be a helix surface in \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\subset {\mathbb {R}}_2^4\) with constant angle \(\vartheta \). Then, with respect to the local coordinates \((u,v)\) on \(M\) defined in (14), the position vector \(F\) of \(M^2\) in \({\mathbb {R}}^4_2\) satisfies the following equation:
- (a):
-
if \(B=0\),
$$\begin{aligned} \frac{\partial ^2F}{\partial u^2}=0, \end{aligned}$$(20) - (b):
-
if \(B\ne 0\),
$$\begin{aligned} \frac{\partial ^4F}{\partial u^4} + (\tilde{b}^2-2\tilde{a})\,\frac{\partial ^2F}{\partial u^2}+\tilde{a}^2\,F=0, \end{aligned}$$(21)where
$$\begin{aligned} \tilde{a}=-\tau ^{-2}\sin ^2\vartheta \, B, \qquad \tilde{b}=-2 \tau ^{-1}\, B. \end{aligned}$$(22)
Proof
Let \(M^2\) be a helix surface and let \(F\) be the position vector of \(M^2\) in \({\mathbb {R}}_2^4\). Then, with respect to the local coordinates \((u,v)\) on \(M\) defined in (14), we can write \(F(u,v)=(F_1(u,v),\ldots ,F_4(u,v))\). By definition, taking into account (11), we have that
Using the expression of \(E_1,\, E_2\) and \(E_3\) with respect to the coordinates vector fields of \({\mathbb {R}}^4_2\), we obtain
Therefore, if \(B=0\), taking the derivative of (23) with respect to \(u\) and using (18), we obtain that \(F_{uu}=0\).
If \(B\ne 0\), taking the derivative of (23) with respect to \(u\) and using (19), we find two constants \(\tilde{a}\) and \(\tilde{b}\) such that
where
Finally, taking twice the derivative of (24) with respect to \(u\) and using (23) and (24) in the derivative we obtain the desired Eq. (21). \(\square \)
Remark 3.4
As \(\langle F,F\rangle =1\), using (21), (23) and (24), we find that the position vector \(F(u,v)\) and its derivatives must satisfy the relations:
where
In addition, as
using (21)–(25), we obtain the following identities
Using Remark 3.4 we can prove the following proposition that gives the conditions under which an immersion defines a helix surface.
Proposition 3.5
Let \(F:\Omega \rightarrow {{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\subset {\mathbb {R}}_2^4\) be an immersion from an open set \(\Omega \subset {\mathbb {R}}^2\), with local coordinates \((u,v)\), such that the projection of \(E_1=-\tau ^{-1}J_1F\) to the tangent space of \(F(\Omega )\subset {{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\) is \(F_u\). Then \(F(\Omega )\subset {{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\) defines a helix surface of constant angle \(\vartheta \) if and only if
and
Proof
Suppose that \(F\) is a helix surface of constant angle \(\vartheta \). Then
Similarly
Finally, using (15), we have
For the converse, put
Then, if we denote by \(N\) the unit normal vector field to the surface \(F(\Omega ),\, \{F_u,T_2,N\}\) is an orthogonal bases of the tangent space of \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\) along the surface \(F(\Omega )\). Now, using (28), we get \(g_{\tau }(E_1,T_2)=0\), thus \(E_1=a\, F_u+c\, N\). Moreover, using (27) and that \(g_{\tau }(E_1,F_u)=a\, g_{\tau }(F_u,F_u)\), we conclude that \(a=1\). Finally,
which implies that \(c^2=\cos ^2\vartheta \). Thus the angle between \(E_1\) and \(N\) is
\(\square \)
4 The case \(B=0\)
Theorem 4.1
Let \(M^2\) be a helix surface in the \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\subset {\mathbb {R}}_2^4\) with constant angle \(\vartheta \) such that \(B=0\). Then \(\cos \vartheta =\frac{1}{\sqrt{1+\tau ^2}}\) and, locally, the position vector of \(M^2\) in \({\mathbb {R}}^4_2\), with respect to the local coordinates \((u,v)\) on \(M\) defined in (14), is given by
where \(A(v)=A(\xi ,\xi _1,\xi _2,\xi _3)(v)\) is a 1-parameter family of \(4\times 4\) indefinite orthogonal matrices commuting with \(J_1\), as described in (4), with
Conversely, a parametrization
with \(A(v)\) as above, defines a helix surface in the special linear group with constant angle \(\vartheta =\arccos \frac{1}{\sqrt{1+\tau ^2}}\).
Proof
Since \(B=0\) we have immediately that \(\cos ^2\vartheta =1/({1+\tau ^2})\). Integrating (20), we obtain that
where \(h^i(v),\, i=1,2\), are vector fields in \({\mathbb {R}}^4_2\), depending only on \(v\).
Evaluating in \((0,v)\) the identities:
it results that
Moreover, using (23) in \((0,v)\), we have that
where \(h^3(v)\) is a vector field of \({\mathbb {R}}_2^4\) satisfying
Consequently, if we fix 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 \(A(v)\in \mathrm {O}_2(4)\), with \(J_1\,A(v)=A(v)\,J_1\), such that
Then (31) becomes
Finally, the 1-parameter family \(A(v)\) depends, according to (4), on four functions \(\xi _1(v),\, \xi _2(v),\, \xi _3(v)\) and \(\xi (v)\) and, in this case, condition (28) reduces to \(\langle F_u, F_v\rangle =0\) which is equivalent to (30).
For the converse, let
be a parametrization where \(A(v)=A(\xi (v),\xi _1(v),\xi _2(v),\xi _3(v))\) is a 1-parameter family of indefinite orthogonal matrices with functions \(\xi (v),\xi _1(v),\xi _2(v),\xi _3(v)\) satisfying (30). Since \(A(v)\) satisfies (30), then \(F\) satisfies (28), thus, in virtue of Proposition 3.5, we only have to show that (27) is satisfied for some constant angle \(\vartheta \). For this we put
Now, using (1) and taking into account that \(A(v)\) commutes with \(J_1\), we get
and we can choose \(\vartheta \) such that \(\tau ^2/(1+\tau ^2)=\sin ^2\vartheta \). Similarly,
\(\square \)
Example 4.2
If we take \(\xi _2=\xi _3={{\mathrm{constant}}}\), (30) becomes \(\xi '(1-\sinh ^2\xi _1)=0\). Thus, if also \(\xi ={{\mathrm{constant}}}\), we find, from (4), a 1-parameter family \(A(v)=A(\xi _1(v))\) of indefinite orthogonal matrices such that (29) defines a helix surface for any function \(\xi _1\).
5 The case \(B>0\)
Supposing \(B>0\), integrating (21) we have the following
Proposition 5.1
Let \(M^2\) be a helix surface in \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\) with constant angle \(\vartheta \) so that \(B>0\). Then, with respect to the local coordinates \((u,v)\) on \(M\) defined in (14), the position vector \(F\) of \(M^2\) in \({\mathbb {R}}^4_2\) is given by
where
are positive real constants, while the \(g^i(v),\, i\in \{1,\ldots ,4\}\), are mutually orthogonal vector fields in \({\mathbb {R}}^4_2\), depending only on \(v\), such that
Proof
First, a direct integration of (21), gives the solution
where
are two constants, while the \(g^i(v),\, i\in \{1,\ldots ,4\}\), are vector fields in \({\mathbb {R}}^4_2\) which depend only on \(v\). Now, taking into account the values of \(\tilde{a}\) and \(\tilde{b}\) given in (22), we get
Putting \(g_{ij}(v)=\langle g^i(v),g^j(v)\rangle \), and evaluating the relations (25) in \((0,v)\), we obtain:
From (38), (39), (43), (44), it follows that
Also, from (36), (40) and (41), we obtain
Finally, using (37), (42) and (45), we obtain
We observe that
Therefore, taking into account (35), we obtain the expressions (34). \(\square \)
We are now in the right position to state the main result of this section.
Theorem 5.2
Let \(M^2\) be a helix surface in the \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\subset {\mathbb {R}}_2^4\) with constant angle \(\vartheta \ne \pi /2\) so that \(B>0\). Then, locally, the position vector of \(M^2\) in \({\mathbb {R}}^4_2\), with respect to the local coordinates \((u,v)\) on \(M\) defined in (14), is
where
is a curve in \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau },\, g_{11},\, g_{33},\, \alpha _1,\, \alpha _2\) are the four constants given in Proposition 5.1, and \(A(v)=A(\xi ,\xi _1,\xi _2,\xi _3)(v)\) is a 1-parameter family of \(4\times 4\) indefinite orthogonal matrices commuting with \(J_1\), as described in (4), with \(\xi ={{\mathrm{constant}}}\) and
Conversely, a parametrization \(F(u,v)=A(v)\,\gamma (u)\), with \(\gamma (u)\) and \(A(v)\) as above, defines a constant angle surface in \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\) with constant angle \(\vartheta \ne \pi /2\).
Proof
With respect to the local coordinates \((u,v)\) on \(M\) defined in (14), Proposition 5.1 implies that the position vector of the helix surface in \({\mathbb {R}}^4_2\) is given by
where the vector fields \(\{g^i(v)\}_{i=1}^4\) are mutually orthogonal and
Thus, if we put \(e_i(v)=g^i(v)/||g^i(v)||,\, i\in \{1,\ldots ,4\}\), we can write:
Now, the identities (26), evaluated in \((0,v)\), become respectively:
We point out that to obtain the previous identities we have divided by \(\alpha _1^2-\alpha _2^2=4 \tau ^{-1} \sqrt{B^3} \cos \vartheta \) which is, by the assumption on \(\vartheta \), always different from zero. From (54) and (55), taking into account the \(\alpha _1^2-\alpha _2^2\ne 0\), it results that
Therefore
Substituting (56) in (50) and (52), we obtain the system
a solution of which is
Now, as
it results that
Moreover, as
it results that \(\langle J_1e_1,e_2\rangle <0\). Consequently, \(\langle J_1e_1,e_2\rangle =\langle J_1e_3,e_4\rangle =-1\) and \(J_1e_1=e_2,\, J_1e_3=-e_4\).
Then, if we fix the orthonormal basis of \({\mathbb {R}}^4_2\) given by
there must exists a 1-parameter family of \(4\times 4\) indefinite orthogonal matrices \(A(v)\in \mathrm {O}_2(4)\), with \(J_1A(v)=A(v)J_1\), such that \(e_i(v)=A(v)\tilde{E}_i\). Replacing \(e_i(v)=A(v)\tilde{E}_i\) in (49) we obtain
where
is a curve in \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\).
Let now examine the 1-parameter family \(A(v)\) that, according to (4), depends on four functions \(\xi _1(v),\xi _2(v),\xi _3(v)\) and \(\xi (v)\). From (15), it results that \(\langle F_v, F_v\rangle =-\sin ^2 \vartheta ={{\mathrm{constant}}}\). The latter implies that
Now, if we denote by \({\mathbf {c}_1},{\mathbf {c}_2},{\mathbf {c}_3},{\mathbf {c}_4}\) the four colons of \(A(v)\), (57) implies that
where with \(^\prime \) we means the derivative with respect to \(v\). Replacing in (58) the expressions of the \({\mathbf {c}_i}\)’s as functions of \(\xi _1(v),\xi _2(v),\xi _3(v)\) and \(\xi (v)\), we obtain
where \(h(v)\) and \(k(v)\) are two functions such that
From (59) we have two possibilities:
-
(i)
\(\xi ={{\mathrm{constant}}}\);
or
-
(ii)
\(4 (\xi _1^{\prime })^2+\sinh ^2(2\xi _1)\, (-\xi ^{\prime }+\xi _2^{\prime }+\xi _3^{\prime })^2=0\).
We will show that case (ii) cannot occur, more precisely we will show that if (ii) happens then the parametrization \(F(u,v)=A(v)\gamma (u)\) defines a Hopf tube, that is the Hopf vector field \(E_1\) is tangent to the surface. To this end, we write the unit normal vector field \(N\) as
A long but straightforward computation (that can be also made using a software of symbolic computations) gives
Now case (ii) occurs if and only if \(\xi _1={{\mathrm{constant}}}=0\), or if \(\xi _1={{\mathrm{constant}}}\ne 0\) and \(-\xi ^{\prime }+\xi _2^{\prime }+\xi _3^{\prime }=0\). In both cases \(N_1=0\) and this implies that \(g_{\tau }(N,J_1F)=-\tau g_{\tau }(N,E_1)=0\), i.e. the Hopf vector field is tangent to the surface. Thus we have proved that \(\xi ={{\mathrm{constant}}}\).
Finally, in this case, (28) is equivalent to
Since \(\vartheta \ne \pi /2\) we conclude that condition (48) is satisfied.
The converse follows immediately from Proposition 3.5 since a direct calculation shows that \(g_{\tau }(F_u,F_u)=g_{\tau }(E_1,F_u)=\sin ^2\vartheta \) which is (27), while (48) is equivalent to (28). \(\square \)
6 The case \(B<0\)
In this section we study the case \(B<0\). Integrating (21) we have the following result:
Proposition 6.1
Let \(M^2\) be a helix surface in \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\) with constant angle \(\vartheta \) and \(B<0\). Then, with respect to the local coordinates \((u,v)\) on \(M\) defined in (14), the position vector \(F\) of \(M^2\) in \({\mathbb {R}}^4_2\) is given by
where
is a real constant, \(\tilde{b}=-2 \tau ^{-1}\, B\), while the \(w^i(v),\, i\in \{1,\ldots ,4\}\), are vector fields in \({\mathbb {R}}^4_2\), depending only on \(v\), such that
Proof
A direct integration of (21), gives the solution
where
is a constant, while the \(w^i(v),\, i\in \{1,\ldots ,4\}\), are vector fields in \({\mathbb {R}}^4\) which depend only on \(v\). If \(w_{ij}(v):=\langle w^i(v),w^j(v)\rangle \), evaluating the relations (25) in \((0,v)\), we obtain
From (62), (66) and (67), it follows that
Also, from (64) and (70), we obtain
and, therefore, from (65) and (69),
Moreover, using (63), (68) and (71), we get
\(\square \)
Theorem 6.2
Let \(M^2\) be a helix surface in \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\) with constant angle \(\vartheta \ne \pi /2\) so that \(B<0\). Then, locally, the position vector of \(M^2\) in \({\mathbb {R}}^4_2\), with respect to the local coordinates \((u,v)\) on \(M\) defined in (14), is given by
where the curve \(\gamma (u)=(\gamma _1(u),\gamma _2(u),\gamma _3(u),\gamma _4(u))\) is given by
\(\beta =\sqrt{-B}\,\cos \vartheta ,\, \tilde{b}=-2 \tau ^{-1}\, B\) and \(A(v)=A(\xi ,\xi _1,\xi _2,\xi _3)(v)\) is a 1- parameter family of \(4\times 4\) indefinite orthogonal matrices anticommuting with \(J_1\), as described in (4), with \(\xi ={{\mathrm{constant}}}\) and
Conversely, a parametrization \(F(u,v)=A(v)\,\gamma (u)\), with \(\gamma (u)\) and \(A(v)\) as above, defines a helix surface in \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\) with constant angle \(\vartheta \ne \pi /2\).
Proof
From (61), 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 (26) in \((0,v)\), and taking into account that:
we conclude that
Then,
Therefore, we obtain that
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 \(A(v)\in \mathrm {O}_2(4)\), with \(J_1A(v)=-A(v)J_1\), such that \(e_i(v)=A(v)\hat{E}_i,\, i\in \{1,\ldots ,4\}\). As
computing \(\langle F,e_i\rangle \) and substituting \(e_i(v)=A(v)\hat{E}_i\), we obtain that \(F(u,v)=A(v)\,\gamma (u)\), where the curve \(\gamma (u)\) of \({{\mathrm {SL}}(2,{\mathbb {R}})_\tau }\) is given in (72).
Let now examine the 1-parameter family \(A(v)\) that, according to (4), depends on four functions \(\xi _1(v),\xi _2(v),\xi _3(v)\) and \(\xi (v)\). Similarly to what we have done in the proof of Theorem 5.2 we have that the condition
implies that the functions \(\xi _1(v),\xi _2(v),\xi _3(v)\) and \(\xi (v)\) satisfy the equation
Then we have two possibilities:
-
(i)
\(\xi ={{\mathrm{constant}}}\);
or
-
(ii)
\(2\sin (\xi _2-\xi _3)\,\xi _1^{\prime }-(\xi _2^{\prime }+\xi _3^{\prime }-\xi ^{\prime })\cos (\xi _2-\xi _3)\,\sinh (2\,\xi _1)=0\).
Also in this case, using the same argument as in Theorem 5.2, condition (ii) would implies that the surface is a Hopf tube, thus we can assume that \(\xi ={{\mathrm{constant}}}\).
Finally, a long but straightforward computation shows that, in the case \(\xi ={{\mathrm{constant}}}\), (28) is equivalent to (73).
The converse of the theorem follows immediately from Proposition 3.5 since a direct calculation shows that \(g_{\tau }(F_u,F_u)=g_{\tau }(E_1,F_u)=\sin ^2\vartheta \) which is (27) while (73) is equivalent to (28). \(\square \)
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Acknowledgments
S. Montaldo was supported by P.R.I.N. 2010/11—Varietà reali e complesse: geometria, topologia e analisi armonica—Italy and INdAM. A. Passos Passamani was supported by Capes—Brazil.
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Montaldo, S., Onnis, I.I. & Passos Passamani, A. Helix surfaces in the special linear group. Annali di Matematica 195, 59–77 (2016). https://doi.org/10.1007/s10231-014-0452-0
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DOI: https://doi.org/10.1007/s10231-014-0452-0