On Some Weingarten Surfaces in the Special Linear Group SL(2,$\mathbb{R}$) ^†

Abstract

We classify Weingarten conoids in the real special linear group $\mathrm{SL}(2,\mathbb{R})$. In particular, there is no linear Weingarten nontrivial conoids in $\mathrm{SL}(2,\mathbb{R})$. We also prove that the only conoids in $\mathrm{SL}(2,\mathbb{R})$ with constant Gaussian curvature are the flat ones. Finally, we show that any surface that is invariant under left translations of the subgroup $\mathcal{N}$ is a Weingarten surface.

1. Introduction

The geometry of the real special linear group $\mathrm{SL}(2,\mathbb{R})$ is very rich, and there are many important and interesting papers investigating its fundamental properties; see, e.g., [1,2,3,4]. We can define a canonical left-invariant Riemannian metric on $\mathrm{SL}(2,\mathbb{R})$ with the isometry group of dimension 4. It admits a structure of naturally reductive homogeneous space. On the other hand, it is possible to equip $\mathrm{SL}(2,\mathbb{R})$ with a left-invariant metric such that the isometry group is only three-dimensional; see, e.g., [5]. The Iwasawa decomposition allows to make use of global coordinates on $\mathrm{SL}(2,\mathbb{R})$ . A contact form can be defined in a canonical way, and it can be regarded as a connection form of the principal circle bundle $\mathrm{SL}(2,\mathbb{R})\to {\mathbb{H}}^2(-4)$ over the hyperbolic plane of constant curvature $-4$. The projection becomes a Riemannian submersion. A canonical homogeneous Sasakian structure of constant holomorphic $\phi$-sectional curvature $-7$ may be also defined; see, e.g., [2]. For a better understanding of $\mathrm{SL}(2,\mathbb{R})$ geometry, several investigations must be carried out to study its submanifolds. Over the last two decades, a large number of papers have investigated the geometry of curves and surfaces in $\mathrm{SL}(2,\mathbb{R})$. Among them, we mention only a few: [5,6,7,8].

The present paper is structured as follows. The next section is a detailed description of the geometry of $\mathrm{SL}(2,\mathbb{R})$. We collect several fundamental properties and put them together in order to obtain a self-contained paper. In Section 3, we are interested in some surfaces in $\mathrm{SL}(2,\mathbb{R})$ and recall rotational surfaces, parallel surfaces and conoids. Section 4 is devoted to Weingarten conoids in $\mathrm{SL}(2,\mathbb{R})$. As basic examples, we have minimal and flat conoids and conoids with constant Gaussian curvature, respectively, constant mean curvature conoids. All these are studied in detail. Theorem 1 gives the classification of Weingarten conoids in $\mathrm{SL}(2,\mathbb{R})$. In Section 5, we study surfaces that are invariant by the left action of the nilpotent group $\mathcal{N}$ given in the Iwasawa decomposition. After we study minimal (respectively flat) $\mathcal{N}$-surfaces in $\mathrm{SL}(2,\mathbb{R})$, we prove that every $\mathcal{N}$-surface in $\mathrm{SL}(2,\mathbb{R})$ is a Weingarten surface (Theorem 2).

2. Basics of the Geometry of $\mathrm{SL}(2,\mathbb{R})$

The group $\mathrm{SL}(2,\mathbb{R})$ is defined as the following subgroup of the group $\mathrm{GL}(2,\mathbb{R})$ with respect to the matrix multiplication law:

$$\mathrm{SL}(2,\mathbb{R})=\left\{\left(\begin{array}{cc}a& b\\ c& d\end{array}\right):a,b,c,d\in \mathbb{R},ad-bc=1\right\}.$$

(1)

2.1. $\mathrm{SL}(2,\mathbb{R})\sim \mathrm{SU}(1,1)$

The group $\mathrm{SU}(1,1)$ is a subgroup of the group $\mathrm{GL}(2,\mathbb{C})$, defined as follows:

$$\mathrm{SU}(1,1)=\left\{A\in {\mathcal{M}}_2\left(\mathbb{C}\right):A^{\ast }JA=J,detA=1\right\},$$

(2)

where $A^{\ast }$ denotes the transposed conjugation of the complex matrix A and $J=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$. It is straightforward that $\mathrm{SU}(1,1)$ can be express as

$$\mathrm{SU}(1,1)=\left\{\left(\begin{array}{cc}z& \overline{w}\\ w& \overline{z}\end{array}\right):z,w\in {\mathbb{C},\left|z\right|}^2-{\left|w\right|}^2=1\right\}.$$

(3)

The group $\mathrm{SL}(2,\mathbb{R})$ is isomorphic to the group $\mathrm{SU}(1,1)$, and we point out such an isomorphism. To achieve this, we fix the matrix $T=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& i\\ i& 1\end{array}\right)\in \mathrm{SU}\left(2\right)$. We can associate to any matrix $A\in \mathrm{SU}(1,1)$ the matrix $TA{T}^{\ast }\in \mathrm{SL}(2,\mathbb{R})$. Define

$$\begin{array}{cccc}\hfill f:\hfill & \mathrm{SU}(1,1)& \to & \mathrm{SL}(2,\mathbb{R})\\ \hfill \hfill & \left(\begin{array}{cc}z& \overline{w}\\ w& \overline{z}\end{array}\right)& \mapsto & \left(\begin{array}{cc}\mathrm{Re}\left(z\right)-\mathrm{Im}\left(w\right)& \mathrm{Im}\left(z\right)+\mathrm{Re}\left(w\right)\\ -\mathrm{Im}\left(z\right)+\mathrm{Re}\left(w\right)& \mathrm{Re}\left(z\right)+\mathrm{Im}\left(w\right)\end{array}\right).\\ \hfill \hfill & {\left|z\right|}^2-{\left|w\right|}^2=1& \end{array}$$

(4)

It is bijective and has the inverse given by:

$$\begin{array}{cccc}\hfill f^{-1}:\hfill & \mathrm{SL}(2,\mathbb{R})& \to & \mathrm{SU}(1,1)\\ \hfill \hfill & \left(\begin{array}{cc}a& b\\ c& d\end{array}\right)& \mapsto & \left(\begin{array}{cc}\frac{a+d}{2}+\frac{b-c}{2}i& \frac{b+c}{2}-\frac{d-a}{2}i\\ \frac{b+c}{2}+\frac{d-a}{2}i& \frac{a+d}{2}-\frac{b-c}{2}i\end{array}\right).\\ \hfill \hfill & ad-bc=1& \end{array}$$

(5)

It is worth to point out that f realizes an isomorphism between $\mathrm{SU}(1,1)$ and $\mathrm{SL}(2,\mathbb{R})$.

2.2. Iwasawa Decomposition

Consider the following subgroups of $\mathrm{SL}(2,\mathbb{R})$:

$$\begin{array}{cc}\mathcal{N}=\left\{\left(\begin{array}{cc}1& x\\ 0& 1\end{array}\right):x\in \mathbb{R}\right\}\hfill & \hfill \left(\text{Nilpotent part}\right);\\ \mathcal{A}=\left\{\left(\begin{array}{cc}r& 0\\ 0& 1/r\end{array}\right):r>0\right\}\hfill & \hfill \left(\text{Abelian part}\right);\\ \mathcal{K}=\left\{\left(\begin{array}{cc}cos\theta & sin\theta \\ -sin\theta & cos\theta \end{array}\right):0\le \theta <2\pi \right\}=\mathrm{SO}\left(2\right)\hfill & \hfill \left(\text{maximal torus}\right).\end{array}$$

As it is proved in [1], every matrix from $\mathrm{SL}(2,\mathbb{R})$ has a unique representation as a product $kan$, where $k\in \mathcal{K}$, $a\in \mathcal{A}$ and $n\in \mathcal{N}$. This decomposition is known as the Iwasawa decomposition. Nevertheless, we prefer to write an element of $\mathrm{SL}(2,\mathbb{R})$ as $nak$, and we call it the Iwasawa decomposition as well. So, when we write a matrix $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \mathrm{SL}(2,\mathbb{R})$ as $nak$, we obtain

$$A=\left(\begin{array}{cc}rcos\theta -\frac{x}{r}sin\theta & rsin\theta +\frac{x}{r}cos\theta \\ -\frac{sin\theta }{r}& \frac{cos\theta }{r}\end{array}\right),$$

(6)

where $\{x,r,\theta \}$ are uniquely given in terms of $a,b,c,d$ (with $ad-bc=1$) by

$$x=\frac{ac+bd}{c^2+d^2},r=\frac{1}{\sqrt{c^2+d^2}},e^{i\theta }=\frac{d-ic}{\sqrt{c^2+d^2}}.$$

(7)

Subsequently, as topological spaces, $\mathrm{SL}(2,\mathbb{R})$ is homeomorphic to the inside of a solid torus. Indeed, we have $\mathcal{N}$∼$\mathbb{R}$, $\mathcal{A}$∼$(0,\infty )$∼$\mathbb{R}$ and $\mathcal{K}$∼${\mathbb{S}}^1$. From the previous decomposition, we identify $\mathrm{SL}(2,\mathbb{R})$ with the product $\mathcal{N}\times \mathcal{A}\times \mathcal{K}$ via continuous maps. Since the plane ${\mathbb{R}}^2$ is homeomorphic to the open unit disk ${\mathbb{D}}^2$ (via the continuous map $•\mapsto \frac{•}{\sqrt{1+\left|\right|•{\left|\right|}^2}}$ and its inverse, which is also continuous), we immediately conclude that $\mathrm{SL}(2,\mathbb{R})$ is homeomorphic to ${\mathbb{D}}^2\times {\mathbb{S}}^1$, which is the inside of a solid torus. For more details, see, e.g., [1].

2.3. Hyperbolic Plane

Let ${\mathbb{H}}^2(-4)$ be the hyperbolic plane of constant curvature $-4$. There are several geometric models for ${\mathbb{H}}^2(-4)$, and we emphasize two of them here.

The first one is the hyperboloid model $\mathcal{H}$, also known as the Minkowski model of the hyperbolic plane. To be more precise, we denote by ${\mathbb{R}}_1^3$ the Minkowski 3-space with coordinates $x_1,x_2,x_3$, endowed with the Lorentzian metric

$$\langle ,\rangle =d{x}_1^2+d{x}_2^2-d{x}_3^2.$$

(8)

Then, the hyperbolic plane ${\mathbb{H}}^2(-4)$ can be considered as the upper sheet $(x_3>0)$ of the two-sheeted hyperboloid

$$\mathcal{H}=\left\{(x_1,x_2,x_3)\in {\mathbb{R}}_1^3:x_1^2+x_2^2-x_3^2=-\frac{1}{4}\right\}.$$

(9)

The metric on $\mathcal{H}$ is that induced from ${\mathbb{R}}_1^3$, and it is a Riemannian metric.

The second model of ${\mathbb{H}}^2(-4)$ that we use here, denoted by $H^{+}$, is the upper half-plane model equipped with the Poincaré metric, that is

$$H^{+}=\left(\{(u,v)\in {\mathbb{R}}^2,v>0\},g_{+}=\frac{d{u}^2+d{v}^2}{4v^2}\right).$$

(10)

Let us recall the Cayley transform between the two models of the hyperbolic plane

$$\mathsf{\Psi }:\mathcal{H}\to H^{+},\mathsf{\Psi }(x_1,x_2,x_3)=\left(\frac{x_1}{x_3-x_2},\frac{1}{2(x_3-x_2)}\right)$$

(11)

with its inverse

$${\mathsf{\Psi }}^{-1}(u,v)=\left(\frac{u}{2v},\frac{u^2+v^2-1}{4v},\frac{u^2+v^2+1}{4v}\right).$$

(12)

2.4. Hopf Map

Let us identify a matrix of the form $\left(\begin{array}{cc}z& \overline{w}\\ w& \overline{z}\end{array}\right)$ with the element $(z,w)\in {\mathbb{C}}^2$, as well as with an element $(v_1,v_2,v_3,v_4)$ in ${\mathbb{R}}_2^4$, where $z=v_1+i{v}_2$ and $w=v_3+i{v}_4$. The condition ${\left|z\right|}^2-{\left|w\right|}^2=1$ reads as $v_1^2+v_2^2-v_3^2-v_4^2=1$, and this justifies the index 2. To be more precise, we see later (in the Appendix A) that the signature of the scalar product in ${\mathbb{R}}_2^4$ is $(-,-,+,+)$.

The Hopf map is defined as a projection from $\mathrm{SU}(1,1)\sim {\mathbb{C}}^2$ to $\mathcal{H}$:

$$\begin{array}{c}\psi :\mathrm{SU}(1,1)\to \mathcal{H}\\ \psi (z,w)=(z\overline{w},\frac{{\left|z\right|}^2+{\left|w\right|}^2}{2})\end{array}{\left(\mathrm{where}\right|z|}^2-{\left|w\right|}^2=1).$$

(13)

It can also be expressed as

$$\psi (v_1,v_2,v_3,v_4)=\left(v_1{v}_3+v_2{v}_4,v_2{v}_3-v_1{v}_4,\frac{v_1^2+v_2^2+v_3^2+v_4^2}{2}\right).$$

(14)

Construct the map

$$\begin{array}{cccc}\hfill \mathsf{\Psi }\circ \psi :\hfill & \mathrm{SU}(1,1)& \to & H^{+}\\ \hfill \hfill & (v_1,v_2,v_3,v_4)& \mapsto & \left({\displaystyle \frac{2(v_1{v}_3+v_2{v}_4)}{{(v_1+v_4)}^2+{(v_2-v_3)}^2}},{\displaystyle \frac{1}{{(v_1+v_4)}^2+{(v_2-v_3)}^2}}\right).\end{array}$$

(15)

The isomorphism f can be rewritten as

$$\begin{array}{cccc}\hfill f:\hfill & \mathrm{SU}(1,1)& \to & \mathrm{SL}(2,\mathbb{R})\\ \hfill \hfill & (v_1,v_2,v_3,v_4)& \mapsto & \left(\begin{array}{cc}{v}_1-v_4& v_2+v_3\\ -v_2+v_3& v_1+v_4\end{array}\right).\end{array}$$

(16)

In terms of the coordinates $(x,r,\theta )$ obtained from the Iwasawa decomposition, one obtains

$$\begin{array}{ccc}& & \hfill x={\displaystyle \frac{2(v_1{v}_3+v_2{v}_4)}{{(v_1+v_4)}^2+{(v_2-v_3)}^2}},r={\displaystyle \frac{1}{\sqrt{{(v_1+v_4)}^2+{(v_2-v_3)}^2}}},\\ & & \hfill e^{i\theta }={\displaystyle \frac{(v_1+v_4)+i(v_2-v_3)}{\sqrt{{(v_1+v_4)}^2+{(v_2-v_3)}^2}}}.\end{array}$$

(17)

Set $y=r^2$ and consider $(x,y,\theta )$ as global coordinates on $\mathrm{SL}(2,\mathbb{R})$. They are more natural than $(x,r,\theta )$. To justify this, consider the projection

$$\pi :\mathrm{SL}(2,\mathbb{R})\to H^{+},(x,y,\theta )\mapsto (x,y).$$

(18)

In this settings, the diagram

commutes.

2.5. Action on the ${\mathbb{H}}^2(-4)$

The group $\mathrm{SL}(2,\mathbb{R})$ acts on $H^{+}$ as follows. Let $H^{+}$ be

$$H^{+}=\{z\in \mathbb{C}:Im\left(z\right)>0\}$$

and define the action, called the linear fractional transformation, by

$$Az=\frac{az+b}{cz+d},$$

(19)

where $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in \mathrm{SL}(2,\mathbb{R}).$ Indeed, we have $A_1\left(A_{2}z\right)=\left(A_1{A}_2\right)z$, for any two matrices $A_1,A_2\in \mathrm{SL}(2,\mathbb{R})$ and

$$Im\left(\frac{az+b}{cz+d}\right)=\frac{Im\left(z\right)}{{|cz+d|}^2}>0.$$

Recall that one can join $i=(0,1)\in H^{+}$ to any other element of $H^{+}$ using this action. More precisely, if $z=u+iv$, $v>0$, one needs, e.g., the matrix $A=\left(\begin{array}{cc}\sqrt{v}& u/\sqrt{v}\\ 0& 1/\sqrt{v}\end{array}\right)\in \mathrm{SL}(2,\mathbb{R})$ to map i to z. This matrix belongs to the $\mathcal{N}\mathcal{A}$ group.

The stabilizer of i (or the isotropy subgroup of $\mathrm{SL}(2,\mathbb{R})$ at the point $(0,1)$) can be computed as

$$\mathrm{Stab}\left(i\right)=\{A\in \mathrm{SL}(2,\mathbb{R}):Ai=i\}=\{A=\left(\begin{array}{cc}a& -c\\ c& a\end{array}\right):a^2+c^2=1\}=\mathrm{SO}\left(2\right),$$

which is the group $\mathcal{K}$.

The action of $\mathrm{SL}(2,\mathbb{R})$ on $H^{+}={\mathbb{H}}^2(-4)$ via linear fractional transformations is transitive and isometric (which we see later). The natural projection $\pi :\mathrm{SL}(2,\mathbb{R})\to {\mathbb{H}}^2(-4)$ is explicitly given by the formula

$$\pi (\left(\begin{array}{cc}a& b\\ c& d\end{array}\right))=\frac{1}{c^2+d^2}(ac+bd,1).$$

(20)

This is nothing but Equation (18), given before. A linear fractional transformation determined by $A\in \mathrm{SL}(2,\mathbb{R})$, $A\ne \pm I_2$ is said to be the following:

• elliptic if $\left|\mathsf{trace}A\right|<2$;
• parabolic if $\left|\mathsf{trace}A\right|=2$;
• hyperbolic if $\left|\mathsf{trace}A\right|>2$.

In terms of the isomorphism f, this classification is equivalent to the following one (see, e.g., [9]).

An element $\left(\begin{array}{cc}z& \overline{w}\\ w& \overline{z}\end{array}\right)\in \mathrm{SU}(1,1)$, with ${\left|z\right|}^2-{\left|w\right|}^2=1$, is called the following:

• elliptic if $\left|\mathrm{Re}\right(z\left)\right|<1$;
• parabolic if $\left|\mathrm{Re}\right(z\left)\right|=1$;
• hyperbolic if $\left|\mathrm{Re}\right(z\left)\right|>1$.

2.6. Lie Algebra $\mathfrak{sl}(2,\mathbb{R})$

The Lie algebra of $\mathrm{SL}(2,\mathbb{R})$ consists in all trace-free matrices of order 2. Consider the following basis of $\mathrm{SL}(2,\mathbb{R})$

$$E=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),F=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),H=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),$$

(21)

whose commutation relations may be written as

$$[E,F]=H,[F,H]=2F,[H,E]=2E.$$

In view of the Iwasawa decomposition, we remark that the Lie algebras $\mathfrak{n}$, $\mathfrak{a}$ and $\mathfrak{k}$ of $\mathcal{N}$, $\mathcal{A}$ and $\mathcal{K}$ are, respectively, given by

$$\mathfrak{n}=\mathbb{R}E,\mathfrak{a}=\mathbb{R}H,\mathfrak{k}=\mathbb{R}(E-F).$$

On $\mathfrak{sl}(2,\mathbb{R})$, one can define a scalar product by

$$\langle X,Y\rangle =\frac{1}{2}\mathrm{trace}\left({}^{t}XY\right),\forall X,Y\in \mathfrak{sl}(2,\mathbb{R}).$$

(22)

Remark that if the “transpose” operator is omitted, we obtain a pseudoscalar product on $\mathfrak{sl}(2,\mathbb{R})$; see Appendix A.

2.7. Riemannian Metrics on $\mathrm{SL}(2,\mathbb{R})$

We introduced the coordinates $x,y$ and $\theta$ from the $nak$ decomposition of $\mathrm{SL}(2,\mathbb{R})$. For a matrix $A\in \mathrm{SL}(2,\mathbb{R})$, expressed as

$$A=\left(\begin{array}{cc}\sqrt{y}cos\theta -\frac{x}{\sqrt{y}}sin\theta & \sqrt{y}sin\theta +\frac{x}{\sqrt{y}}cos\theta \\ -\frac{sin\theta }{\sqrt{y}}& \frac{cos\theta }{\sqrt{y}}\end{array}\right),$$

(23)

we consider the natural basis of the tangent space $T_A\mathrm{SL}(2,\mathbb{R})$

$${\partial }_x=\frac{\partial A}{\partial x},{\partial }_y=\frac{\partial A}{\partial y},{\partial }_{\theta }=\frac{\partial A}{\partial \theta }.$$

(24)

Then, the vectors $A^{-1}{\partial }_x$, $A^{-1}{\partial }_y$ and $A^{-1}{\partial }_{\theta }$ are tangent vectors at the identity matrix; hence, they may be considered as elements in the Lie algebra $\mathfrak{sl}(2,\mathbb{R})$. Remark that

$$\left\{\begin{array}{c}{\displaystyle A^{-1}{\partial }_x=\frac{{cos}^2\theta }{y}E-\frac{{sin}^2\theta }{y}F-\frac{sin\theta cos\theta }{y}H,}\hfill \\ {\displaystyle A^{-1}{\partial }_y=\frac{sin2\theta }{2y}E+\frac{sin2\theta }{2y}F+\frac{cos2\theta }{2y}H,}\hfill \\ {\displaystyle A^{-1}{\partial }_{\theta }=E-F.}\hfill \end{array}\right.$$

(25)

We introduce a frame field $\{e_1,e_2,e_3\}$ on $\mathrm{SL}(2,\mathbb{R})$ by

$$e_1=2y{\partial }_x-{\partial }_{\theta },e_2=2y{\partial }_y,e_3={\partial }_{\theta }.$$

(26)

It follows that

$$A^{-1}(cos2\theta e_1+sin2\theta e_2)=E+F,A^{-1}(-sin2\theta e_1+cos2\theta e_2)=H,A^{-1}{e}_3=E-F.$$

The dual coframe of $\{e_1,e_2,e_3\}$ is given by

$${\omega }^1=\frac{dx}{2y},{\omega }^2=\frac{dy}{2y},{\omega }^3=d\theta +\frac{dx}{2y}.$$

(27)

Let us equip $\mathrm{SL}(2,\mathbb{R})$ with a pseudo-Riemannian metric

$$g_{\alpha }=\frac{d{x}^2+d{y}^2}{4y^2}+\alpha {\left(d\theta +\frac{dx}{2y}\right)}^2,\alpha \in \mathbb{R}\setminus \left\{0\right\}.$$

(28)

A pseudoscalar product $(·,·)$ of $\mathfrak{sl}(2,\mathbb{R})$ induces a left-invariant pseudo-Riemannian metric on $\mathrm{SL}(2,\mathbb{R})$ by

$$g(AX,AY)=(X,Y),\forall X,Y\in \mathfrak{sl}(2,\mathbb{R}).$$

For example, if $(·,·)$ is the scalar product $\langle ,\rangle$ given by (22), then the metric is $g_1$. If $(X,Y)=\frac{1}{2}\mathrm{trace}\left(XY\right)$, then we obtain the metric $g_{-1}$.

Every metric $g_{\alpha }$ is left-invariant but not necessarily bi-invariant. For example, for $\alpha =1$, the metric is only a left-invariant Riemannian metric, while for $\alpha =-1$, the metric is a bi-invariant Lorentzian metric. Obviously, $g_{\alpha }$ is Riemannian for $\alpha >0$ and Lorentzian for $\alpha <0$.

The isometry group of $(\mathrm{SL}(2,\mathbb{R}),g_1)$ has dimension 4. Note that it is possible to construct Riemannian metrics on $\mathrm{SL}(2,\mathbb{R})$ such that the isometry group has dimension 3. See, for example, [5].

Remark 1.

For any $\lambda \in \mathbb{R}\setminus \left\{0\right\}$, define a pseudoscalar product ${(·,·)}_{\lambda }$ on $\mathfrak{sl}(2,\mathbb{R})$ by

$${(E,E)}_{\lambda }={(F,F)}_{\lambda }=\frac{1+\epsilon }{4},{(H,H)}_{\lambda }=\lambda ,{(E,F)}_{\lambda }=\frac{1-\epsilon }{4},{(E,H)}_{\lambda }={(F,H)}_{\lambda }=0,$$

where $\epsilon =\pm 1$. By left-translating this pseudoscalar product, we can endow $\mathrm{SL}(2,\mathbb{R})$ with a left-invariant pseudo-Riemannian metric ${(·,·)}_{\lambda }$. For the sake of simplicity, we kept the same notation. By using (25), it is straightforward to show that ${({\partial }_x,{\partial }_y)}_{\lambda }=\frac{(1-\lambda )sin2\theta cos2\theta }{4y^2}$. Therefore, x-coordinate curves and y-coordinate curves are orthogonal if and only if $\lambda =1$. We find ${(·,·)}_1$ is the metric $g_{\epsilon }$ from the Equation (28).

2.8. The Curvature of $\mathrm{SL}(2,\mathbb{R})$

On $\mathrm{SL}(2,\mathbb{R})$, we consider the Riemannian metric $g_1$ from (28). For the rest of this paper, we denote it by $d{s}^2$

$$d{s}^2=\frac{d{x}^2+d{y}^2}{4y^2}+{\left(d\theta +\frac{dx}{2y}\right)}^2.$$

(29)

We have seen that the isotropy subgroup of $\mathrm{SL}(2,\mathbb{R})$ at $i=(0,1)$ is the rotation group $SO\left(2\right)$. The natural projection (also defined before)

$$\pi :\left(\mathrm{SL}(2,\mathbb{R}),d{s}^2\right)\to \left(\mathrm{SL}(2,\mathbb{R})/SO\left(2\right)\equiv {\mathbb{H}}^2(-4)\equiv H^{+},g_{+}\right)$$

becomes a Riemannian submersion with totally geodesic fibers. This submersion is often called the hyperbolic Hopf fibration.

The frame $\{e_1,e_2,e_3\}$ given by (26) is orthonormal, and it is globally defined on $\mathrm{SL}(2,\mathbb{R})$. The coframe $\{{\omega }^1,{\omega }^2,{\omega }^3\}$ defined by the Equation (27) satisfies the following structure equations:

$$d{\omega }^1=2{\omega }^1\wedge {\omega }^2,d{\omega }^2=0,d{\omega }^3=2{\omega }^1\wedge {\omega }^2,$$

which can be written as the first structure equation:

$$d\left(\begin{array}{c}{\omega }^1\\ {\omega }^2\\ {\omega }^3\end{array}\right)=-\omega \wedge \left(\begin{array}{c}{\omega }^1\\ {\omega }^2\\ {\omega }^3\end{array}\right),$$

where $\omega$ is the connection matrix for the Levi-Civita connection of $(\mathrm{SL}(2,\mathbb{R}),d{s}^2)$, and it is given by

$$\omega =\left(\begin{array}{ccc}0& -(2{\omega }^1+{\omega }^3)& -{\omega }^2\\ 2{\omega }^1+{\omega }^3& 0& {\omega }^1\\ {\omega }^2& -{\omega }^1& 0\end{array}\right).$$

Hence, the essential components of the connection matrix $\omega$, called the connection 1-forms, are ${\omega }_2^1=-2{\omega }^1-{\omega }^3$, ${\omega }_3^1=-{\omega }^2$ and ${\omega }_3^2={\omega }^1$. Now, by using the formula ${\tilde{\nabla }}_X{e}_i={\omega }_i^j\left(X\right)e_j$, we immediately obtain the expression of the Levi-Civita connection of $d{s}^2$ :

$$\begin{array}{ccc}{\tilde{\nabla }}_{e_1}{e}_1=2e_2,\hfill & {\tilde{\nabla }}_{e_2}{e}_1=e_3,\hfill & {\tilde{\nabla }}_{e_3}{e}_1=e_2,\hfill \\ {\tilde{\nabla }}_{e_1}{e}_2=-2e_1-e_3,\hfill & {\tilde{\nabla }}_{e_2}{e}_2=0,\hfill & {\tilde{\nabla }}_{e_3}{e}_2=-e_1,\hfill \\ {\tilde{\nabla }}_{e_1}{e}_3=e_2,\hfill & {\tilde{\nabla }}_{e_2}{e}_3=-e_1,\hfill & {\tilde{\nabla }}_{e_3}{e}_3=0.\hfill \end{array}$$

(30)

We now compute

$$d\omega =\left(\begin{array}{ccc}0& -6{\omega }^1\wedge {\omega }^2& 0\\ 6{\omega }^1\wedge {\omega }^2& 0& 2{\omega }^1\wedge {\omega }^2\\ 0& -2{\omega }^1\wedge {\omega }^2& 0\end{array}\right)$$

and

$$\omega \wedge \omega =\left(\begin{array}{ccc}0& -{\omega }^1\wedge {\omega }^2& {\omega }^1\wedge {\omega }^3\\ {\omega }^1\wedge {\omega }^2& 0& -2{\omega }^1\wedge {\omega }^2+{\omega }^2\wedge {\omega }^3\\ -{\omega }^1\wedge {\omega }^3& 2{\omega }^1\wedge {\omega }^2-{\omega }^2\wedge {\omega }^3& 0\end{array}\right).$$

Let $\tilde{R}$ be the curvature tensor of $\tilde{\nabla }$ defined as $\tilde{R}(X,Y)Z={\tilde{\nabla }}_X{\tilde{\nabla }}_{Y}Z-{\tilde{\nabla }}_Y{\tilde{\nabla }}_{X}Z-{\tilde{\nabla }}_{[X,Y]}Z$. The curvature 2-forms are defined as usual by $R(X,Y)e_i={\mathsf{\Omega }}_i^j(X,Y)e_j$. The second structure equation may be written as

$$\mathsf{\Omega }=d\omega +\omega \wedge \omega ,$$

that is, ${\mathsf{\Omega }}_i^j=d{\omega }_i^j+{\omega }_k^j\wedge {\omega }_i^k$. Throughout this paper, we use the Det convention. Combining with the previous relations, we obtain

$$\mathsf{\Omega }=\left(\begin{array}{ccc}0& -7{\omega }^1\wedge {\omega }^2& {\omega }^1\wedge {\omega }^3\\ 7{\omega }^1\wedge {\omega }^2& 0& {\omega }^2\wedge {\omega }^3\\ -{\omega }^1\wedge {\omega }^3& -{\omega }^2\wedge {\omega }^3& 0\end{array}\right).$$

The Ricci tensor is defined as $\widetilde{Ric}(X,Y)=\mathrm{trace}\left(V\mapsto \tilde{R}(V,X)Y\right)$ and hence

$${\tilde{R}}_{ij}:=\widetilde{Ric}(e_i,e_j)={\mathsf{\Omega }}_j^k(e_k,e_i).$$

This means that

$$\begin{array}{cc}\hfill \widetilde{Ric}\hfill & =\left(e_1{e}_2{e}_3\right)·\mathsf{\Omega }·\left(\begin{array}{c}{\omega }^1\\ {\omega }^2\\ {\omega }^3\end{array}\right)=(-6{\omega }^1-6{\omega }^{2}2{\omega }^3)\left(\begin{array}{c}{\omega }^1\\ {\omega }^2\\ {\omega }^3\end{array}\right)\hfill \\ & =-6{\left({\omega }^1\right)}^2-6{\left({\omega }^2\right)}^2+2{\left({\omega }^3\right)}^2.\hfill \end{array}$$

The scalar curvature $\widetilde{\mathsf{scal}}=\mathrm{trace}\left(\widetilde{Ric}\right)=-10$.

The sectional curvature ${\tilde{K}}_{ij}$ defined by the plane $\mathrm{span}\{e_i,e_j\}$ is equal to

$${\tilde{K}}_{ij}=d{s}^2(e_i,\tilde{R}(e_i,e_j)e_j)={\mathsf{\Omega }}_j^i(e_i,e_j).$$

Thus, we have ${\tilde{K}}_{12}=-7$, ${\tilde{K}}_{23}=1$ and ${\tilde{K}}_{13}=1$.

3. Surfaces in $\mathrm{SL}(2,\mathbb{R})$

Let M be a surface in $\mathrm{SL}(2,\mathbb{R})$ having N as the unit normal vector field. Denote by g the induced metric and by ∇ its Levi-Civita connection. The Gauss and Weingarten formulas define the second fundamental form h and the shape operator S, respectively, on M:

$$\begin{array}{cc}{\tilde{\nabla }}_{X}Y={\nabla }_{X}Y+h(X,Y),\hfill & \hfill \left(\mathbf{G}\right)\\ {\tilde{\nabla }}_{X}N=-SX,\hfill & \hfill \left(\mathbf{W}\right)\end{array}$$

where $X,Y$ are tangent to M.

As a consequence of the previous two formulas, we have

$$\tilde{R}(X,Y)Z=R(X,Y)Z+S_{h(X,Z)}Y-S_{h(Y,Z)}X+\left({\overline{\nabla }}_{X}h\right)(Y,Z)-\left({\overline{\nabla }}_{Y}h\right)(X,Z),$$

(31)

for any $X,Y,Z\in \mathfrak{X}\left(M\right)$. Here, $\tilde{R}$ and R denote the curvature tensors of $\tilde{\nabla }$ and ∇, respectively, and the covariant derivative $\overline{\nabla }h$ is defined by

$$\left({\overline{\nabla }}_{X}h\right)(Y,Z)={\nabla }_X^{\perp }h(Y,Z)-h({\nabla }_{X}Y,Z)-h(Y,{\nabla }_{X}Z),$$

where ${\nabla }^{\perp }$ is the normal connection. The connection $\overline{\nabla }$ is often called the van der Waerden–Bortolotti connection. The tangential part in Equation (31) leads to the equation of Gauss, while the normal component is described by the Codazzi equation

$$\begin{array}{cc}{\left(\tilde{R}(X,Y)Z\right)}^{\perp }=\left({\overline{\nabla }}_{X}h\right)(Y,Z)-\left({\overline{\nabla }}_{Y}h\right)(X,Z).& \left(\mathbf{EC}\right)\end{array}$$

3.1. Rotational Surfaces

An immersed surface $\mathtt{f}:M\to \mathrm{SL}(2,\mathbb{R})$ is called rotational if it is invariant under the right translations of the subgroup $\mathcal{K}=SO\left(2\right)$ of $\mathrm{SL}(2,\mathbb{R})$, namely $\mathtt{f}\left(M\right)SO\left(2\right)\subseteq \mathtt{f}\left(M\right)$. See, e.g., [10]. It follows that such a surface can be parameterized as

$$(t,\theta )\stackrel{\mathtt{f}}{⟼}\left(\begin{array}{cc}1& x\left(t\right)\\ 0& 1\end{array}\right)\left(\begin{array}{cc}\sqrt{y\left(t\right)}& 0\\ 0& 1/\sqrt{y\left(t\right)}\end{array}\right)\left(\begin{array}{cc}cos\theta & sin\theta \\ -sin\theta & cos\theta \end{array}\right),$$

(32)

where $t\in I$, $\theta \in {\mathbb{S}}^1$ (I being an interval in $\mathbb{R}$). The curve $\left(x\right(t),y(t\left)\right)$ (that is, $\theta =0$) with $\dot{x}{\left(t\right)}^2+\dot{y}{\left(t\right)}^2\ne 0$ is called the generating curve for $\mathtt{f}$. We denote $\dot{x}\left(t\right)=\frac{dx}{dt}$, $\dot{y}\left(t\right)=\frac{dy}{dt}$ and so on.

Two important properties are proved in [10]:

(i)

The induced metric on a rotational surface is flat; that is, the Gaussian curvature is identically zero.

(ii)

A rotational surface $\mathtt{f}$ has constant mean curvature H if and only if the curve $\pi \circ \mathtt{f}$ (in $H^{+}$) has constant curvature H. See, for details, ref. [10] (Proposition 4.3).

A bit more generally, one can consider, for any constant $\sigma$, the following isometry of $(\mathrm{SL}(2,\mathbb{R}),d{s}^2)$:

$$\mathrm{SL}(2,\mathbb{R})\ni A\stackrel{{{\rm Y}}_t^{\sigma }}{⟼}\left(\begin{array}{cc}1& \sigma t\\ 0& 1\end{array}\right)A\left(\begin{array}{cc}cost& sint\\ -sint& cost\end{array}\right),$$

(33)

where $t\in \mathbb{R}$. We obtain a one-parameter group ${\left\{{{\rm Y}}_t^{\sigma }\right\}}_{t\in \mathbb{R}}$ of isometries on $\mathrm{SL}(2,\mathbb{R})$. Each element ${{\rm Y}}_t^{\sigma }$ is called a helicoidal motion of pitch $\sigma$.

3.2. Parallel Surfaces

A surface M is said to be parallel if $\overline{\nabla }h=0$. They are classified in [6] as follows: the only parallel surfaces in $\mathrm{SL}(2,\mathbb{R})$ are rotational surfaces of constant mean curvature H over Riemannian circles of curvature $2H$ in $H^{+}$. As a consequence, all these surfaces are flat.

This result says something more: there are no extrinsic spheres (i.e., totally umbilical surfaces with parallel mean curvature vector field) in $\mathrm{SL}(2,\mathbb{R})$. In particular, we have the following:

Proposition 1.

The space $(\mathrm{SL}(2,\mathbb{R}),d{s}^2)$ contains no totally geodesic surfaces.

Proof. 

Even though the result is a consequence of the main theorem in [6], we sketch the proof of this special statement.

Suppose that there exists a totally geodesic surface M in $(\mathrm{SL}(2,\mathbb{R}),d{s}^2)$, and let $N=A{e}_1+B{e}_2+C{e}_3$ be its unit normal. The functions $A,B,C$ satisfy $A^2+B^2+C^2=1$. We consider a local frame on M. A vector field on M is also expressed in terms of $\{e_1,e_2,e_3\}$. Let us distinguish two situations: $A=0$ and $A\ne 0$ (on a certain open set).

If $A=0$, then $X=e_1$ is tangent to M and takes $Y=C{e}_2-B{e}_3$, which is also tangent to M, since $d{s}^2(Y,N)=0$. The compatibility condition $[X,Y]\in \mathrm{span}\{X,Y\}$ must be fulfilled, and it leads to some differential equation. From the Gauss formula, since the second fundamental form h vanishes, we obtain that ${\tilde{\nabla }}_{e_1}{e}_1=2e_2$ is tangent to M. But $[e_1,e_2]=-2e_1-2e_3$ must also be tangent to M, and this is false.

Hence, $A\ne 0$. We consider a frame on M, defined by $X=C{e}_1-A{e}_3$, $Y=B{e}_1-A{e}_2$. Of course, it is not an orthonormal frame. The Equation (31) simplifies and yields that $\tilde{R}(X,Y)X$ and $\tilde{R}(X,Y)Y$ are tangent to M. Using the 3-linearity of the curvature we obtain

$$\begin{array}{c}\tilde{R}(X,Y)X=-A^{2}B{e}_1+A(A^2-7C^2)e_2-ABC{e}_3,\hfill \\ \tilde{R}(X,Y)Y=-7A^{2}C{e}_1-7ABC{e}_2-A(A^2+B^2)e_3.\hfill \end{array}$$

We obtain

$$0=d{s}^2(N,\tilde{R}(X,Y)X)=-8AB{C}^2\mathrm{and}0=d{s}^2(N,\tilde{R}(X,Y)Y)=-8AC(A^2+B^2).$$

We obtain $C=0$, and this implies $X=-A{e}_3$. Again, from the vanishing of h, we infer that ${\tilde{\nabla }}_{Y}X=\frac{dA\left(Y\right)}{A}X-AN$ must also be tangent to M, and this cannot be true. □

3.3. Hopf Cylinders

Let us only recall the notion of Hopf cylinder introduced by Pinkall (see [11]). Let $\pi :{\mathbb{S}}^3\to {\mathbb{S}}^2\left(4\right)$ be the classical Hopf fibration from the unit three-dimensional sphere onto the two-dimensional sphere of radius $1/2$. Take a curve $\overline{\gamma }$ on the base and consider the preimage $M={\pi }^{-1}\left(\overline{\gamma }\right)$. Then, M is a flat surface in ${\mathbb{S}}^3$, which is called a Hopf cylinder over $\overline{\gamma }$. This construction is valid for other Hopf fiberings (e.g., from the de Sitter space to the hyperbolic plane), and it can also be defined for our projection

$$\pi :\left(\mathrm{SL}(2,\mathbb{R}),d{s}^2\right)\to \left(\mathrm{SL}(2,\mathbb{R})/SO\left(2\right)\equiv H^{+},g_{+}\right).$$

So, a surface M in $\mathrm{SL}(2,\mathbb{R})$ is a Hopf cylinder over a curve in $H^{+}$ if and only if it is a rotational surface. See, e.g., [7].

3.4. Conoids

In ${\mathbb{R}}^3$, the right conoid is defined as a surface that can be parameterized by $(u,v)\mapsto (ucosv,usinv,\varphi (v\left)\right)$. A known result says that the right helicoid is the only minimal surface among the right conoids. A similar result is proved by Kokubu in [10], when the ambient space is the real special linear group $\mathrm{SL}(2,\mathbb{R})$. An immersed surface M in $\mathrm{SL}(2,\mathbb{R})$ is called conoid if it can be parameterized as

$$(u,v)\stackrel{\mathtt{f}}{⟼}\left(\begin{array}{cc}1& \varphi \left(v\right)\\ 0& 1\end{array}\right)\left(\begin{array}{cc}\sqrt{u}& 0\\ 0& 1/\sqrt{u}\end{array}\right)\left(\begin{array}{cc}cosv& sinv\\ -sinv& cosv\end{array}\right),$$

(34)

where $u\in (0,+\infty )$ and $\varphi$ is a smooth function on a certain interval of $\mathbb{R}$. A right helicoid is obtained when the above function $\varphi$ is an affine function.

For later use, we emphasize some aspects of the geometry of conoids in $\mathrm{SL}(2,\mathbb{R})$. See, e.g., [10].

We consider the induced metric $g={\mathtt{f}}^{\ast }d{s}^2$ on M, which can be expressed as

$$g=\frac{d{u}^2}{4u^2}+{\mathsf{\Phi }}^{2}d{v}^2,$$

(35)

where

$$\mathsf{\Phi }=\mathsf{\Phi }(u,v)=\sqrt{\frac{\dot{\varphi }{\left(v\right)}^2}{2u^2}+\frac{\dot{\varphi }\left(v\right)}{u}+1}.$$

(36)

Let us define the orthonormal coframe $\{{\rho }^1,{\rho }^2\}$ and its dual frame $\{{ϵ}_1,{ϵ}_2\}$ by

$${\rho }^1=\mathsf{\Phi }dv,{\rho }^2=\frac{du}{2u},{ϵ}_1=\frac{1}{\mathsf{\Phi }}\frac{\partial }{\partial v},{ϵ}_2=2u\frac{\partial }{\partial u}.$$

The first structure equation on M may be written as:

$$d\left(\begin{array}{c}{\rho }^1\\ {\rho }^2\end{array}\right)=-\rho \wedge \left(\begin{array}{c}{\rho }^1\\ {\rho }^2\end{array}\right),$$

(37)

where

$$\rho =\left(\begin{array}{cc}0& {\displaystyle \frac{2u{\mathsf{\Phi }}_u}{\mathsf{\Phi }}{\rho }^1}\\ {\displaystyle -\frac{2u{\mathsf{\Phi }}_u}{\mathsf{\Phi }}{\rho }^1}& 0\end{array}\right)$$

(38)

is the connection matrix on M. Here, ${\displaystyle {\mathsf{\Phi }}_u=\frac{\partial \mathsf{\Phi }}{\partial u}=-\frac{\dot{\varphi }}{2u^2\mathsf{\Phi }}\left(1+\frac{\dot{\varphi }}{u}\right)}$.

The second structure equation is written as $\varrho =d\rho +\rho \wedge \rho =d\rho$. More precisely, we have

$$\varrho =\left(\begin{array}{cc}0& {\displaystyle -\frac{4u({\mathsf{\Phi }}_u+u{\mathsf{\Phi }}_{uu})}{\mathsf{\Phi }}{\rho }^1\wedge {\rho }^2}\\ {\displaystyle \frac{4u({\mathsf{\Phi }}_u+u{\mathsf{\Phi }}_{uu})}{\mathsf{\Phi }}{\rho }^1\wedge {\rho }^2}& 0\end{array}\right).$$

It follows that the Ricci tensor and the scalar curvature of M are given by

$$Ric=-\frac{4u({\mathsf{\Phi }}_u+u{\mathsf{\Phi }}_{uu})}{\mathsf{\Phi }}\left({\left({\rho }^1\right)}^2+{\left({\rho }^2\right)}^2\right)\mathrm{and}\mathsf{scal}=-\frac{8u({\mathsf{\Phi }}_u+u{\mathsf{\Phi }}_{uu})}{\mathsf{\Phi }}.$$

Therefore, the Gaussian curvature of M is

$$K=-\frac{4u({\mathsf{\Phi }}_u+u{\mathsf{\Phi }}_{uu})}{\mathsf{\Phi }}.$$

(39)

We know that ${\nabla }_X{ϵ}_i={\rho }_i^j\left(X\right){ϵ}_j$, where ${\rho }_1^1={\rho }_2^2=0$ and ${\rho }_2^1=-{\rho }_1^2=\frac{2u{\mathsf{\Phi }}_u}{\mathsf{\Phi }}{\rho }^1$.

Thus, the Equation (38) is equivalent to

$$\left\{\begin{array}{c}{\nabla }_{{\partial }_u}{\partial }_u=-\frac{1}{u}{\partial }_u,{\nabla }_{{\partial }_u}{\partial }_v={\nabla }_{{\partial }_v}{\partial }_u=\frac{{\mathsf{\Phi }}_u}{\mathsf{\Phi }}{\partial }_v,\hfill \\ {\nabla }_{{\partial }_v}{\partial }_v=-4u^2\mathsf{\Phi }{\mathsf{\Phi }}_u{\partial }_u+\frac{{\mathsf{\Phi }}_v}{\mathsf{\Phi }}{\partial }_v.\hfill \end{array}\right.$$

(40)

The next step is to compute the second fundamental form of the immersion $\mathsf{f}$. We map $\{{ϵ}_1,{ϵ}_2\}$ via the differential ${\mathsf{f}}_{\ast }$ to obtain

$${\mathsf{f}}_{\ast }{ϵ}_1=\frac{\dot{\varphi }}{2u\mathsf{\Phi }}{e}_1+\frac{1}{\mathsf{\Phi }}\left(\frac{\dot{\varphi }}{2u}+1\right)e_3,{\mathsf{f}}_{\ast }{ϵ}_2=e_2.$$

Equivalently, we have

$${\mathsf{f}}_{\ast }{\partial }_u=\frac{1}{2u}{e}_2,{\mathsf{f}}_{\ast }{\partial }_v=\frac{\dot{\varphi }}{2u}{e}_1+\left(\frac{\dot{\varphi }}{2u}+1\right)e_3.$$

(41)

The unit normal to M is (up to sign)

$$N=-\frac{1}{\mathsf{\Phi }}\left(\frac{\dot{\varphi }}{2u}+1\right)e_1+\frac{\dot{\varphi }}{2u\mathsf{\Phi }}{e}_3.$$

Note that with this choice of orientation (for N), the frames $\{{\mathsf{f}}_{\ast }{ϵ}_1,{\mathsf{f}}_{\ast }{ϵ}_2,N\}$ and $\{e_1,e_2,e_3\}$ have the same orientation.

Combining (30), (40) and (41), we immediately obtain

$$h({\partial }_u,{\partial }_u)=0,h({\partial }_u,{\partial }_v)=\frac{1}{2u\mathsf{\Phi }}\left(1+\frac{2\dot{\varphi }}{u}+\frac{{\dot{\varphi }}^2}{2u^2}\right),h({\partial }_v,{\partial }_v)=-\frac{\ddot{\varphi }}{2u\mathsf{\Phi }}.$$

(42)

It follows that the mean curvature $H=\frac{1}{2}\left(h({ϵ}_1,{ϵ}_1)+h({ϵ}_2,{ϵ}_2)\right)$ of M is equal to

$$H=-\frac{\ddot{\varphi }\left(v\right)}{4u{\mathsf{\Phi }}^3}.$$

(43)

4. Weingarten Conoids

A surface M immersed in a three-dimensional Riemannian manifold is called a Weingarten surface if there exists some (smooth) relation $W(H,K)=0$ between its mean curvature H and its Gaussian curvature K. Obviously, minimal surfaces, CMC surfaces, flat surfaces and constant Gaussian curvature surfaces are typical examples of Weingarten surfaces.

The existence of a Weingarten relation $W(H,K)=0$ means that curvatures H and K (as functions of parameters u and v) are functionally related, and this is equivalent to the Jacobian condition

$$\frac{\partial H}{\partial u}\frac{\partial K}{\partial v}-\frac{\partial H}{\partial v}\frac{\partial K}{\partial u}=0,$$

(44)

at any $(u,v)$. The Jacobian condition characterizes Weingarten surfaces, and it is used to identify them when an explicit Weingarten relation cannot be immediately found.

In this section, we study conoids in $\mathrm{SL}(2,\mathbb{R})$ that are Weingarten surfaces, and we call them Weingarten conoids.

1. The first example is obtained from minimal conoids, when $W(H,K)=H$. They are studied in [10], and the following result is obtained:

Proposition 2.

A surface of the form

$$(0,\infty )\times \mathbb{R}\ni (u,v)\stackrel{\mathsf{f}}{\mapsto }\left(\begin{array}{cc}1& \zeta \\ 0& 1\end{array}\right)\left(\begin{array}{cc}1& \sigma v\\ 0& 1\end{array}\right)\left(\begin{array}{cc}\sqrt{u}& 0\\ 0& 1/\sqrt{u}\end{array}\right)\left(\begin{array}{cc}cosv& sinv\\ -sinv& cosv\end{array}\right),$$

where $\sigma ,\zeta \in \mathbb{R}$ is the only minimal conoid. Moreover, this surface is helicoidal, namely, it is invariant under any helicoidal motion in $\mathrm{SL}(2,\mathbb{R})$.

2. The second example is given by flat conoids; they are Weingarten surfaces with $W(H,K)=K$. We obtain the following result.

Proposition 3.

The only flat conoids in $\mathrm{SL}(2,\mathbb{R})$ are rotational surfaces parameterized by

$$(0,\infty )\times {\mathbb{S}}^1\ni (u,v)\stackrel{\mathsf{f}}{\mapsto }\left(\begin{array}{cc}1& {\varphi }_0\\ 0& 1\end{array}\right)\left(\begin{array}{cc}\sqrt{u}& 0\\ 0& 1/\sqrt{u}\end{array}\right)\left(\begin{array}{cc}cosv& sinv\\ -sinv& cosv\end{array}\right),$$

(45)

for which the generating curve is a vertical geodesic in $H^{+}$. Here, ${\varphi }_0$ is a real constant.

Proof. 

The Gaussian curvature K is computed in (39). Then, the surface is flat if and only if

$$u{\mathsf{\Phi }}_u+{\mathsf{\Phi }}_{uu}=0.$$

This equation is equivalent to $u{\mathsf{\Phi }}_u=\mu \left(v\right)$, where $\mu$ is a smooth function depending on v. Thus, we obtain ${\displaystyle \frac{p(p+1)}{2\sqrt{1+p+\frac{p^2}{2}}}+\mu \left(v\right)=0,\forall (u,v),}$ where $p=p(u,v)=\frac{\dot{\varphi }\left(v\right)}{u}$.

After we take the derivative with respect to u, we find

$$\frac{p^3+3p^2+5p+2}{\sqrt{2}{(2+2p+p^2)}^{3/2}}·\frac{\partial p}{\partial u}=0.$$

If $p_u$ is different from zero in a point, then it is different from zero on an open set. Therefore, on that open set, we need to have $p^3+3p^2+5p+2=0$. This equation has one (real) solution, call it $p_0\in (-1,0)$. It follows that $\frac{\dot{\varphi }\left(v\right)}{u}=p_0$ on that open set. This can only be possible if $\varphi \left(v\right)$ is a constant ${\varphi }_0$, which implies $p_0=0$, which is false. So, $p_u=0$ on a certain open set. We deduce that $\frac{\dot{\varphi }\left(v\right)}{u}$ depends only on v. Again, we need to have $\varphi \left(v\right)={\varphi }_0$, and hence $\mu \left(v\right)=0$.

For the last part of the statement, we just notice that the generating curve is parameterized by $({\varphi }_0,u)$, $u>0$, and this represents a vertical geodesic in $H^{+}$. □

3. For the third example, we consider conoids with constant Gaussian curvature; that is, the Weingarten function is $W(H,K)=K-K_0$, where $K_0\in \mathbb{R}$. The following statement is true:

Proposition 4.

The only conoids in $\mathrm{SL}(2,\mathbb{R})$ with constant Gaussian curvature are the flat ones.

Proof. 

Using the expression of the Gaussian curvature from (39), we obtain the following partial differential equation $4u({\mathsf{\Phi }}_u+u{\mathsf{\Phi }}_{uu})+K_0\mathsf{\Phi }=0$. Denote by $s=logu$. Since $u{\partial }_u={\partial }_s$, the previous equation may be rewritten as

$$p_{ss}+\frac{K_0}{4}p=0,$$

where we set $p(s,v)=\mathsf{\Phi }(e^s,v)$. The solution of this equation depends on the sign of $K_0$, so we have to distinguish three situations.

Case 1: $K_0=0$

The solution is an affine function in s, equivalently $\mathsf{\Phi }(u,v)=A\left(v\right)+B\left(v\right)logu$, where $A,B$ are two smooth functions depending on v. Squaring and then taking the derivative with respect to u yields

$$-\frac{\dot{\varphi }\left(v\right)}{u^2}-\frac{{\dot{\varphi }}^2\left(v\right)}{u^3}=\frac{2}{u}A\left(v\right)B\left(v\right)+2B^2\left(v\right)\frac{logu}{u}.$$

Multiply by u and take (again) the derivative with respect to u to obtain

$$\frac{\dot{\varphi }\left(v\right)}{u^2}+\frac{2{\dot{\varphi }}^2\left(v\right)}{u^3}=2B^2\left(v\right)\frac{1}{u}.$$

Because this relation is valid for arbitrary $(u,v)$ in an open set, we must have $B\left(v\right)=0$ and $\dot{\varphi }\left(v\right)=0$ for any v. Then, $A\left(v\right)=1$ and $\varphi \left(v\right)={\varphi }_0$, a real constant. The result from Proposition 3 is obtained.

Case 2: $K_0>0$

The solution of the differential equation is given by

$$\mathsf{\Phi }(u,v)=A\left(v\right)cos\left(\frac{\sqrt{K_0}}{2}logu\right)+B\left(v\right)sin\left(\frac{\sqrt{K_0}}{2}logu\right),$$

where $A,B$ are two smooth functions depending on v.

Similarly as before, we square and take the derivative with respect to u. We obtain

$$\begin{array}{cc}\hfill {\displaystyle -\frac{\dot{\varphi }\left(v\right)}{u^2}-\frac{{\dot{\varphi }}^2\left(v\right)}{u^3}=}\hfill & \hfill {\displaystyle \frac{B{\left(v\right)}^2-A{\left(v\right)}^2}{2}\frac{\sqrt{K_0}}{u}sin\left(\frac{\sqrt{K_0}}{2}logu\right)}\hfill \\ \hfill \hfill & \hfill {\displaystyle +A\left(v\right)B\left(v\right)\frac{\sqrt{K_0}}{u}cos\left(\frac{\sqrt{K_0}}{2}logu\right).}\hfill \end{array}$$

(46)

Multiply by u and, again, take the derivative with respect to u twice; we obtain

$$\begin{array}{cc}\hfill {\displaystyle -\frac{\dot{\varphi }\left(v\right)}{u^2}-\frac{4{\dot{\varphi }}^2\left(v\right)}{u^3}=}\hfill & \hfill {\displaystyle \frac{A{\left(v\right)}^2-B{\left(v\right)}^2}{2}\frac{{\sqrt{K_0}}^3}{u}sin\left(\frac{\sqrt{K_0}}{2}logu\right)}\hfill \\ \hfill \hfill & \hfill {\displaystyle -A\left(v\right)B\left(v\right)\frac{{\sqrt{K_0}}^3}{u}cos\left(\frac{\sqrt{K_0}}{2}logu\right).}\hfill \end{array}$$

(47)

Adding the Equation (46) multiplied by $K_0$ with the Equation (47), we obtain

$$-(K_0+1)\frac{\dot{\varphi }\left(v\right)}{u^2}-(K_0+4)\frac{\dot{\varphi }\left(v\right)}{u^3}=0.$$

This equation must be fulfilled for arbitrary $(u,v)$; hence, $\dot{\varphi }\left(v\right)=0$. This leads to $K_0=0$, which is a contradiction.

Case 3: $K_0<0$

The solution of the differential equation is given by

$$\mathsf{\Phi }(u,v)=A\left(v\right)cosh\left(\frac{\sqrt{-K_0}}{2}logu\right)+B\left(v\right)sinh\left(\frac{\sqrt{-K_0}}{2}logu\right),$$

where $A,B$ are two smooth functions depending on v. A similar technique as above yields a nonexistent result. □

4. A fourth example consists in CMC conoids, and they are obtained when the Weingarten function $W(H,K)=H-H_0$, where $H_0\in \mathbb{R}$. Since minimal conoids were discussed before, we consider $H_0\ne 0$. The mean curvature of a conoid is given in (43). Consider the function $p(u,v)=\frac{\dot{\varphi }\left(v\right)}{u}$. We have to solve the partial differential equation

$${\displaystyle \frac{p_v}{{\left(1+{(1+p)}^2\right)}^{\frac{3}{2}}}=\sqrt{2}{H}_0.}$$

Integrating with respect to v, we find $\frac{1+p}{\sqrt{1+{(1+p)}^2}}=\sqrt{2}{H}_{0}v+A\left(u\right)$, where A is a smooth function on u. Take the derivative with respect to u to obtain ${\displaystyle \frac{p_u}{{\left(1+{(1+p)}^2\right)}^{\frac{3}{2}}}=A^{\prime }\left(u\right)}$. As $p_u=-\frac{\dot{\varphi }\left(v\right)}{u^2}$, after some elementary computations, we obtain the following equation

$$-\frac{\dot{\varphi }\left(v\right)}{u^2}=\frac{A^{\prime }\left(u\right)}{{\left(1-{(\sqrt{2}{H}_{0}v+A\left(u\right))}^2\right)}^{\frac{3}{2}}}.$$

Squaring, we obtain

$$\dot{\varphi }{\left(v\right)}^2{\left(1-{(\sqrt{2}{H}_{0}v+A\left(u\right))}^2\right)}^3=u^4{A}^{\prime }{\left(u\right)}^4.$$

Take the derivative with respect to v. Recall that we excluded the minimal conoids; hence, $\dot{\varphi }$ does not vanish on a certain open interval. Moreover the expression $1-{(\sqrt{2}{H}_{0}v+A\left(u\right))}^2$ cannot be identically zero. After simplifications, we find

$$\ddot{\varphi }\left(v\right)\left(1-{(\sqrt{2}{H}_{0}v+A\left(u\right))}^2\right)=3\sqrt{2}\dot{\varphi }\left(v\right)H_0(\sqrt{2}{H}_{0}v+A\left(u\right))$$

The function A cannot be constant, otherwise $\dot{\varphi }$ is trivially zero. So, after we take two more derivatives with respect to u and perform the necessary simplifications, we obtain a contradiction.

The conclusion is given by the following:

Proposition 5.

Any CMC conoid in $\mathrm{SL}(2,\mathbb{R})$ is minimal. Moreover, it is a rotational surface parameterized by (45).

We have seen that all these four “classical” types of Weingarten conoids in $\mathrm{SL}(2,\mathbb{R})$ are helicoidal surfaces (with the special case when they are also rotational surfaces). Let us study the general case to understand if the function $\varphi$ is affine for any Weingarten conoid.

The key is to write the Jacobian condition that characterizes a Weingarten surface. With K and H obtained in (39) and (43), respectively, we develop the equation $\frac{\partial H}{\partial u}\frac{\partial K}{\partial v}-\frac{\partial H}{\partial v}\frac{\partial K}{\partial u}=0$. Therefore, the following holds:

(i)

either ${\varphi }^{\prime }\left(v\right){\varphi }^{‴}\left(v\right)-{\varphi }^{″}{\left(v\right)}^2=0$,

(ii)

or  ${\varphi }^{\prime }{\left(v\right)}^4+4u{\varphi }^{\prime }{\left(v\right)}^3+12u^2{\varphi }^{\prime }{\left(v\right)}^2+16u^3{\varphi }^{\prime }\left(v\right)+4u^4=0$, $\forall (u,v)$.

Obviously, the second situation cannot occur. We now focus on the first differential equation. The already known solution is $\varphi \left(v\right)=\sigma v+\zeta$ (with $\sigma ,\zeta \in \mathbb{R}$). We are looking for other solutions. Equation (i) is equivalent to ${\left(\frac{{\varphi }^{″}\left(v\right)}{{\varphi }^{\prime }\left(v\right)}\right)}^{\prime }=0$. The general solution is

$$\varphi \left(v\right)={\zeta }_1{e}^{\sigma v}+{\zeta }_0,$$

where $\sigma ,{\zeta }_0,{\zeta }_1\in \mathbb{R}$.

We formulate the following:

Theorem 1.

Let M be a conoid in $\mathrm{SL}(2,\mathbb{R})$, parameterized as in (34). Then, M is a Weingarten surface if and only if either the function ϕ is affine or $\varphi \left(v\right)={\zeta }_1{e}^{\sigma v}+{\zeta }_0,$ where $\sigma \in \mathbb{R}\setminus \left\{0\right\}$, ${\zeta }_0,{\zeta }_1\in \mathbb{R}$.

A Weingarten surface is called linear Weingarten if there exists a linear relation W between H and K, namely, the functional relation between H and K can be written as $W(H,K):=c_{1}H+c_{2}K+c_0=0$ for some $c_1,c_2,c_0\in \mathbb{R}$ with $c_1^2+c_2^2\ne 0$.

Corollary 1.

Linear Weingarten nontrivial conoids in $\mathrm{SL}(2,\mathbb{R})$ do not exist.

Proof. 

By nontrivial Weingarten conoids, we understand the conoids obtained in the second case of the previous theorem. With $\varphi \left(v\right)={\zeta }_1{e}^{\sigma v}+{\zeta }_0$, we can compute

$$K=-\frac{4(2{\mathsf{\Sigma }}^3+5{\mathsf{\Sigma }}^2+3\mathsf{\Sigma }+1)}{{(2{\mathsf{\Sigma }}^2+2\mathsf{\Sigma }+1)}^2},H=\pm \frac{\sigma {\mathsf{\Sigma }}^2}{\sqrt{2}{(2{\mathsf{\Sigma }}^2+2\mathsf{\Sigma }+1)}^{\frac{3}{2}}},$$

(48)

where $\mathsf{\Sigma }=\frac{u{e}^{-\sigma v}}{{\zeta }_1\sigma }$.

Suppose that we have the linear relation $c_{1}H+c_{2}K+c_0=0$ between H and K. Starting from $c_1^2{H}^2-{(c_{2}K+c_0)}^2=0$, after straightforward computations, we obtain a polynomial in $\mathsf{\Sigma }$ of degree 8. The leading coefficient is $-16c_0$. Now, analyzing the other coefficients (under the hyprthesis $c_0=0$), we conclude $c_2=0$ and $c_1=0$, which leads to a contradiction. □

5. $\mathcal{N}$-Surfaces

An immersed surface $\mathsf{f}:M\to \mathrm{SL}(2,\mathbb{R})$ is called an $\mathcal{N}$-surface if it is invariant under the left translations of the subgroup $\mathcal{N}$ of $\mathrm{SL}(2,\mathbb{R})$. It follows that such a surface can be parameterized as

$$(u,v)\stackrel{\mathsf{f}}{⟼}\left(\begin{array}{cc}1& u\\ 0& 1\end{array}\right)\left(\begin{array}{cc}\sqrt{\varphi \left(v\right)}& 0\\ 0& 1/\sqrt{\varphi \left(v\right)}\end{array}\right)\left(\begin{array}{cc}cosv& sinv\\ -sinv& cosv\end{array}\right),$$

(49)

with $u\in \mathbb{R}$ and $\varphi :{\mathbb{S}}^1\to (0,\infty )$. See, e.g., [12].

Let us briefly sketch the geometry of these parameterized surfaces.

The pull-back of the metric $d{s}^2$ on M is

$$g={\mathsf{f}}^{\ast }d{s}^2=\frac{1}{2}{\left(dv+\frac{du}{\varphi \left(v\right)}\right)}^2+\frac{1}{2}\left(1+\frac{\dot{\varphi }{\left(v\right)}^2}{2\varphi {\left(v\right)}^2}\right)d{v}^2.$$

(50)

Consider the orthonormal frame on M

$$\left\{{ϵ}_1=\sqrt{2}\varphi \left(v\right)\frac{\partial }{\partial u},{ϵ}_2=\frac{\sqrt{2}}{\mathsf{\Phi }}\left(-\varphi \left(v\right)\frac{\partial }{\partial u}+\frac{\partial }{\partial v}\right)\right\},$$

where $\mathsf{\Phi }=\mathsf{\Phi }\left(v\right)=\sqrt{1+\frac{\dot{\varphi }{\left(v\right)}^2}{2\varphi {\left(v\right)}^2}}$. Its orthonormal coframe is given by

$$\left\{{\rho }^1=\frac{1}{\sqrt{2}}\left(dv+\frac{du}{\varphi \left(v\right)}\right),{\rho }^2=\frac{\mathsf{\Phi }}{\sqrt{2}}dv\right\}.$$

We have $d{\rho }^1=\frac{\sqrt{2}\dot{\varphi }\left(v\right)}{\varphi \left(v\right)\mathsf{\Phi }}{\rho }^1\wedge {\rho }^2$ and $d{\rho }^2=0$; hence,

$$d\left(\begin{array}{c}{\rho }^1\\ {\rho }^2\end{array}\right)=-\rho \wedge \left(\begin{array}{c}{\rho }^1\\ {\rho }^2\end{array}\right),$$

(51)

where

$$\rho =\left(\begin{array}{cc}{\rho }_1^1& {\rho }_2^1\\ {\rho }_1^2& {\rho }_2^2\end{array}\right)=\left(\begin{array}{cc}0& -\frac{\sqrt{2}\dot{\varphi }\left(v\right)}{\varphi \left(v\right)\mathsf{\Phi }}{\rho }^1\\ \frac{\sqrt{2}\dot{\varphi }\left(v\right)}{\varphi \left(v\right)\mathsf{\Phi }}{\rho }^1& 0\end{array}\right)$$

is the connection matrix on M.

The second structure equation is written as $\varrho =d\rho +\rho \wedge \rho =d\rho$. More precisely, we have

$$\varrho =\left(\begin{array}{ccc}0& {\displaystyle \frac{2}{1+{\beta }^2}\left(\frac{\sqrt{2}{\beta }^{\prime }}{1+{\beta }^2}-2{\beta }^2\right){\rho }^1\wedge {\rho }^2}& \\ {\displaystyle -\frac{2}{1+{\beta }^2}\left(\frac{\sqrt{2}{\beta }^{\prime }}{1+{\beta }^2}-2{\beta }^2\right){\rho }^1\wedge {\rho }^2}& 0\end{array}\right),$$

where $\beta =\beta \left(v\right)=\frac{\dot{\varphi }\left(v\right)}{\sqrt{2}\varphi \left(v\right)}$.

It follows that the Ricci tensor and the scalar curvature of M are given by

$$Ric=K\left({\left({\rho }^1\right)}^2+{\left({\rho }^1\right)}^2\right)\mathrm{and}\mathsf{scal}=2K,$$

where the Gaussian curvature K of M is given by

$$K=\frac{2}{1+{\beta }^2}\left(\frac{\sqrt{2}{\beta }^{\prime }}{1+{\beta }^2}-2{\beta }^2\right).$$

(52)

We now easily compute

$${\mathsf{f}}_{\ast }{ϵ}_1=\frac{e_1+e_3}{\sqrt{2}},{\mathsf{f}}_{\ast }{ϵ}_2=\frac{1}{\mathsf{\Phi }}\left(\frac{e_3-e_1}{\sqrt{2}}+\beta \left(v\right)e_2\right).$$

(53)

It follows that the unit normal to M is

$$N=\frac{1}{\mathsf{\Phi }}\left(\beta \left(v\right)\frac{e_3-e_1}{\sqrt{2}}-e_2\right).$$

(54)

Remark that $\{{\mathsf{f}}_{\ast }{ϵ}_1,{\mathsf{f}}_{\ast }{ϵ}_2,n\}$ has the same orientation as $\{e_1,e_2,e_3\}$.

Let us write the expressions of the Levi-Civita connection ∇ on M and the (scalar) second fundamental form h of the immersion $\mathsf{f}$, respectively:

$${\nabla }_{{ϵ}_1}{ϵ}_1=\frac{2\beta }{\mathsf{\Phi }}{ϵ}_2,{\nabla }_{{ϵ}_1}{ϵ}_2=-\frac{2\beta }{\mathsf{\Phi }}{ϵ}_1,{\nabla }_{{ϵ}_2}{ϵ}_1=0,{\nabla }_{{ϵ}_2}{ϵ}_2=0,$$

(55)

$$h({ϵ}_1,{ϵ}_1)=-\frac{2}{\mathsf{\Phi }},h({ϵ}_1,{ϵ}_2)=h({ϵ}_2,{ϵ}_1)=1,h({ϵ}_1,{ϵ}_2)=-\frac{\sqrt{2}{\beta }^{\prime }}{{\mathsf{\Phi }}^3}.$$

(56)

Notice, as it is also pointed out in [12], that the surface M has no geodesic points.

The mean curvature of M is

$$H=-\frac{{\beta }^{\prime }+\sqrt{2}(1+{\beta }^2)}{\sqrt{2}{\mathsf{\Phi }}^3}.$$

(57)

Weingarten $\mathcal{N}$-Surfaces

The first problems we wish to investigate are the minimal $\mathcal{N}$-surfaces, respectively, the flat $\mathcal{N}$-surfaces.

Proposition 6.

An $\mathcal{N}$-surface defined by Equation (49) is minimal if and only if, under the initial conditions $\varphi \left(0\right)=1$ and ${\varphi }^{\prime }\left(0\right)=0$, the function ϕ is

$$\varphi \left(v\right)=cos\left(\sqrt{2}v\right).$$

Proof. 

The formula (57) is equivalent to that found in [7] for $\nu =1$. The minimality condition $H=0$ leads to the differential equation

$${\beta }^{\prime }+\sqrt{2}(1+{\beta }^2)=0.$$

With the initial condition $\beta \left(0\right)=0$, we obtain $\beta \left(v\right)=-tan\left(\sqrt{2}v\right)$. Then, the conclusion holds immediately.

Remark that the minimality condition is also equivalently to $\ddot{\varphi }+2\varphi =0$. □

Remark 2.

In [12], the authors provide examples of constant mean curvature $\mathcal{N}$-surfaces by considering a certain height function to be constant.

Proposition 7.

An $\mathcal{N}$-surface defined by Equation (49) is flat if and only if, under the initial conditions $\varphi \left(0\right)=1$ and ${\varphi }^{\prime }\left(0\right)=\sqrt{2}$, the function ϕ is

$$\varphi \left(v\right)=\frac{\sqrt{2}\tilde{p}(v_0-v\sqrt{2})}{\sqrt{1+{\tilde{p}}^2(v_0-v\sqrt{2})}},$$

(58)

where $\tilde{p}$ is is the inverse function of the function $p:I\subset (0,\infty )\to p\left(I\right)\subset \mathbb{R}$, $p\left(t\right)=arctant+\frac{1}{t}$ and $v_0=1+\frac{\pi }{4}$.

Proof. 

The flatness condition $K=0$ is equivalent to the differential equation

$${\beta }^{\prime }-\sqrt{2}{\beta }^2(1+{\beta }^2)=0.$$

Remark that $\frac{d}{dv}\left(arctan\beta \left(\frac{v}{\sqrt{2}}\right)+\frac{1}{\beta \left(\frac{v}{\sqrt{2}}\right)}\right)=-1$ for any v in an open interval where all the expressions make sense.

Consider the function $p:\mathbb{R}\setminus \left\{0\right\}\to \mathbb{R}$, $p\left(t\right)=arctant+\frac{1}{t}$, which satisfies the following properties:

• p is a strictly decreasing function, hence it is invertible.
• p is an odd function, i.e., $p(-t)=-p\left(t\right)$, $\forall t\in \mathbb{R}\setminus \left\{0\right\}$.
• $\underset{t\to \infty }{lim}p\left(t\right)=\frac{\pi }{2}$, $\underset{t\to 0_{+}}{lim}p\left(t\right)=\infty$.

We have seen that $\frac{d}{dv}p\left(\beta (\frac{v}{\sqrt{2}})\right)=-1$; hence, $p\left(\beta (\frac{v}{\sqrt{2}})\right)=-v+v_0$, where the integration constant is obtained from the initial conditions: $v_0=p\left(\beta \left(0\right)\right)=p\left(1\right)=1+\frac{\pi }{4}$. It follows that

$$\beta \left(v\right)=p^{-1}\left(v_0-v\sqrt{2}\right).$$

Since ${\beta }^{\prime }\ge 0$, the function $\beta$ is increasing, and because $\beta \left(0\right)=1$ (due to the initial conditions), the function $\beta \left(v\right)$ is positive (for $v\ge 0$). It follows that $-v+v_0\in (\frac{\pi }{2},\infty )$. So, the definition domain of v restricts to $(0,1-\frac{\pi }{4})$.

For the sake of simplicity, let us write $\tilde{p}=p^{-1}$. We compute

$${\tilde{p}}^{\prime }\left(\zeta \right)=-{\tilde{p}}^2\left(\zeta \right)\left(1+{\tilde{p}}^2\left(\zeta \right)\right),\forall \zeta \in \mathrm{Image}\left(\mathrm{p}\right).$$

At this point, with $\varphi$ given by (58), straightforward computations yield $\frac{\dot{\varphi }\left(v\right)}{\sqrt{2}\varphi \left(v\right)}=\beta \left(v\right)$. □

Remark 3.

The flatness equation can be written in terms of the function ϕ as follows:

$$2\ddot{\varphi }{\varphi }^3-4{\dot{\varphi }}^2{\varphi }^2-{\dot{\varphi }}^4=0.$$

We conclude this section with the following statement:

Theorem 2.

Every $\mathcal{N}$-surface is a Weingarten surface.

Proof. 

The proof is elementary, because we have concrete expressions for the mean curvature H and for the Gaussian curvature K. We remark that they only depend on the variable v, so, the Jacobi condition is automatically satisfied. □

Remark 4.

It is well known that $\mathrm{SL}(2,\mathbb{R})$ with the metric $g_{-1}$ (see Appendix A) is identified with the anti-de Sitter space ${\mathbb{H}}_1^3$. The Lie group $\mathrm{SL}(2,\mathbb{R})$ acts on $\mathfrak{sl}(2,\mathbb{R})$ by the Ad-action. See, e.g., [7]. The $Ad$-orbit of a space-like vector (in $\mathfrak{sl}(2,\mathbb{R})$ endowed with the pseudoscalar product) is the hyperbolic plane ${\mathbb{H}}^2(-4)$, the $Ad$-orbit of a time-like vector is the Lorentz sphere ${\mathbb{S}}_1^2\left(4\right)$ and the $Ad$-orbit of a null vector is the light-cone. Inoguchi noticed in [7] that rotational surfaces, as well as conoids, are Hopf cylinders over the hyperbolic plane and over Lorentz sphere, respecitvely. Of course, different metrics are considered on $\mathrm{SL}(2,\mathbb{R})$. In the same article, Inoguchi studied surfaces obtained over the $Ad$-orbit of a null vector, i.e., the inverse image in ${\mathbb{H}}_1^3$ of a curve in the light-cone of $\mathfrak{sl}(2,\mathbb{R})$. This surface is precisely the $\mathcal{N}$-surface defined in this section.

6. Conclusions

A Weingarten surface is a surface for which the mean curvature H is connected with its Gaussian curvature K by a functional relation. These surfaces were introduced by J. Weingarten [13,14] in the context of the problem of finding all surfaces isometric to a given surface of revolution. Several geometric problems that involve Weingarten surfaces have been formulated and solved, especially in the Euclidean 3-space. See, e.g., [15,16,17,18]. Later on, generalized Weingarten surfaces in the Euclidean space of dimension 3 were studied. See, e.g., [19,20,21]. The study of Weingarten surfaces may be extended to other ambient spaces (e.g., [22,23]). In this paper, we consider surfaces in the real special linear group $\mathrm{SL}(2,\mathbb{R})$. This group is one of the most important Lie groups, not only because it consists in all linear transformations of ${\mathbb{R}}^2$ that preserve the oriented area but also because its universal cover is one of the eight Thurston geometries.

Along this article, we briefly describe some aspects of the geometry of the real special linear group $\mathrm{SL}(2,\mathbb{R})$ and study some of its surfaces. We classify Weingarten conoids and show that there is no linear Weingarten nontrivial conoids in $\mathrm{SL}(2,\mathbb{R})$. We also prove that the only conoids in $\mathrm{SL}(2,\mathbb{R})$ with constant Gaussian curvature are the flat ones. Finally, we show that any surface that is invariant under left translations of the subgroup $\mathcal{N}$ (and we call them $\mathcal{N}$-surfaces, is a Weingarten surface.

There are several types of surfaces in $\mathrm{SL}(2,\mathbb{R})$ to be studied, and we plan to investigate some of them in future works.