Taut contact hyperbolas on three-manifolds

Dedicated to my grandson Davide and my grand-daughter Elisa

Abstract

In this paper, we introduce the notion of taut contact hyperbola on three-manifolds. It is the hyperbolic analogue of the taut contact circle notion introduced by Geiges and Gonzalo (Invent. Math., 121: 147–209, 1995), (J. Differ. Geom., 46: 236–286, 1997). Then, we characterize and study this notion, exhibiting several examples, and emphasizing differences and analogies between taut contact hyperbolas and taut contact circles. Moreover, we show that taut contact hyperbolas are related to some classic notions existing in the literature. In particular, it is related to the notion of conformally Anosov flow, to the critical point condition for the Chern–Hamilton energy functional and to the generalized Finsler structures introduced by R. Bryant. Moreover, taut contact hyperbolas are related to the bi-contact metric structures introduced in D. Perrone (Ann. Global Anal. Geom., 52: 213–235, 2017).

1 Introduction

Following [13, 15], a pair $(\eta _1 ,\eta _2 )$ of contact 1-forms on a three-manifold M is called a contact circle if the linear combination $\eta _a = a_1 \eta _1 +a_2 \eta _2$ is also a contact form for every $a=(a_1 ,a_2 )\in {\mathbb {S}}^1$, the unit circle in $\mathbb {R}^2$. If in addition the volume forms $\eta _a\wedge (d\eta _a)$ are equal for every $a\in {\mathbb {S}}^1$, then $(\eta _1 ,\eta _2 )$ is said to be a taut contact circle.

In the paper [24], we studied taut contact circles on a three-manifold from point of view of the Riemannian geometry and introduced the notion of bi-contact metric structure $(\eta _1,\eta _2, g)$, that is, $(\eta _1 ,\eta _2 )$ is a pair of contact 1-forms and g is a Riemannian metric associated to both the contact forms $\eta _1$, $\eta _2$ such that the corresponding Reeb vector fields are orthogonal.

The main purpose of this paper is to start the study of the hyperbolic analogue, in dimension three, of taut contact circles study. The unit circle ${\mathbb {S}}^1$ of the Euclidean plane has its counterpart in the pseudo-Euclidean plane, that is, in the Minkowski plane, in the four arms of the unit equilateral hyperbolas ${\mathbb {H}}^1_r: x^2 - y^2 = r, r=\pm 1$. Then, we call contact hyperbola a pair of contact 1-forms $(\eta _1 ,\eta _2 )$ such that, for every $a=(a_1 ,a_2 )\in {\mathbb {H}}^1_r$, the linear combination $\eta _a = a_1 \eta _1 +a_2 \eta _2$ is also a contact form. If in addition the volume forms $\eta _a\wedge (d\eta _a)=r \eta _1\wedge (d\eta _1)$ for every $a\in {\mathbb {H}}^1_r$, then $(\eta _1 ,\eta _2 )$ is said to be a taut contact hyperbola. In particular, if $(\eta _1,\eta _2, g)$ is a bi-contact metric structure on a three-manifold, then $(\eta _1 ,\eta _2 )$ is either a taut contact circle or a taut contact hyperbola.

We note that the notion of taut contact hyperbola introduced in this paper is very natural because is related to some classic notions existing in the literature: the notion of conformally Anosov flow introduced by Mitsumatsu [21] and Eliashberg-Thurston [12], the critical point condition for the Chern–Hamilton energy functional [9, 26] and the generalized Finsler structures introduced by Bryant [5, 6], are related to the taut contact hyperbolas. So we believe that this study is worthy of subsequent insights.

The present paper, where in particular we emphasize differences and analogies between taut contact hyperbolas and taut contact circles, is organized in the following way.

In Sect. 2, we collect some basic facts about contact Riemannian geometry.

In Sect. 3, we introduce the notion of taut contact hyperbola on a three-manifold. In particular, in the compact case, a taut contact hyperbola defines a conformally Anosov flow in the sense of Mitsumatsu [21] and Eliashberg-Thurston [12]. Then, we study left invariant taut contact hyperbolas on 3D Lie groups. The Lie groups ${\widetilde{SL}}(2,R)$ and $Sol^3$ are the only unimodular Lie groups which admit left invariant taut contact hyperbolas; thus, we determine the left invariant taut contact hyperbolas on these Lie groups. Then, we study taut contact hyperbolas on non-unimodular Lie groups, in particular there are non-unimodular Lie groups with the Milnor’s invariant ${\mathcal {D}} =0$ which admit a taut contact hyperbola with the corresponding Reeb vector fields dependent.

Section 4 contains some characterization of taut contact hyperbolas (cf. Theorem 4.2), and a remark about a difference between a taut contact circle and a taut contact hyperbola in terms of symplectic structures.

In Sect. 5, in analogy with the notion of taut contact 2-sphere, we introduce the notion of taut contact 2-hyperboloid, in particular we get that the Lie group ${\widetilde{SL}}(2,R)$ is the only simply connected three-manifold which admits a taut contact 2-hyperboloid $(\eta _1,\eta _2,\eta _3)$ with the corresponding Reeb vector fields $(\xi _1,\xi _2,\xi _3)$ that constitute the frame dual of the coframe $(\eta _1,\eta _2,\eta _3)$.

In Sect. 6, we show that the critical point condition for the Chern–Hamilton energy functional ( [9, 26]) is a sufficient condition for the existence of a taut contact hyperbola on a non-Sasakian contact metric three-manifold $(M,\eta ,g)$ (cf. Theorem 6.1). In particular, in the compact case, $(M,\ker \eta )$ is universally tight. Thus, we exhibit an example related to this Theorem.

In Sect. 7 we study the geometry of a three-manifold M determined by the existence of a bi-contact metric structure. We characterize the existence of a bi-contact metric structure $(\eta _1,\eta _2,g)$ on M by the condition that $(\eta _1,\eta _2)$ is a $(-\varepsilon )$-Cartan structure (cf. Theorem 7.1). In particular, there are a 1-form $\eta _3$ and a function $\kappa$, that we call the Webster function (cf. Remark 7.4), uniquely determined by this structure. Then, (cf. Theorem 7.5) $\eta _3$ is Killing (resp. a contact form) if and only if $(\eta _1,\eta _2)$ is taut contact circle (resp. the Webster function $\kappa \ne 0$ everywhere). Besides, we study the geometry of M when the 1-form $\eta _3$ is a contact form and in particular when the Webster function $\kappa =\pm 1$ (cf. Theorem 7.5 and Corollary 7.6).

Finally, in Sect. 8, we see how bi-contact metric structures are related to the generalized Finsler structures (introduced by Bryant [5, 6]) and construct an explicit example of bi-contact metric structures $(\eta _1,\eta _2,g)$ where $(\eta _1,\eta _2)$ is a taut contact hyperbola with the Webster function $\kappa$ non-constant (in particular, this example gives a positive answer to a question posed in [24]).

2 Riemannian geometry of contact manifolds

In this section, we collect some basic facts about contact Riemannian geometry and refer to the two monographs [3, 4] for more information. All manifolds are supposed to be connected and smooth. Moreover, in what follows, for a Riemannian manifold (M, g), we shall denote by $\nabla$ the Levi-Civita connection of the Riemannian metric g, by R the corresponding Riemannian curvature tensor and by Ric the Ricci tensor.

A contact manifold is a $(2n+1)$-dimensional manifold M equipped with a global 1-form $\eta$ such that $\eta \wedge (d \eta ) ^n \ne 0$ everywhere on M. It has an underlying almost contact structure $(\xi ,\eta ,\varphi )$ where $\xi$ is a global vector field (called the Reeb vector field, or the characteristic vector field) and $\varphi$ is a global tensor of type (1,1) such that   $\eta (\xi ) =1 \,, \, \varphi \xi =0 \,, \, \varphi ^2 = -I+ \eta \otimes \xi$. A Riemannian metric g can be found such that

$$\begin{aligned} \eta = g(\xi , \cdot )\,, \quad d \eta =g(\cdot ,\varphi \cdot ) \,. \end{aligned}$$

In such a case, g is called an associated metric, and we refer to $(M, \eta ,g)$, or $(M,\xi ,\eta ,\varphi ,g)$, as a contact metric (or contact Riemannian) manifold. The tensor $h=\frac{1}{2}{\mathcal {L}}_{\xi } \varphi$, where ${\mathcal {L}}$ denotes the Lie derivative, plays a fundamental role in contact Riemannian geometry, it is symmetric and satisfies:  $h\varphi =- \varphi h$,   $h \xi = 0$   and

$$\begin{aligned} \nabla \xi = -\varphi -\varphi h . \end{aligned}$$

(2.1)

In particular, the Reeb vector field $\xi$ is a geodesic vector field: $\nabla _\xi \xi =0$.

More in general, given an almost contact structure $(\xi ,\eta ,\varphi )$, a Riemannian metric g can be found such that $g(\varphi X, \varphi Y) =g(X,Y) -\eta (X)\eta (Y)$, and in this case $(\xi ,\eta ,\varphi ,g)$ is called almost contact metric structure. An almost contact structure $(\xi ,\eta ,\varphi )$ is said to be normal if the almost complex structure J on $M\times {\mathbb {R}}$ defined by $J(X,f\mathrm {d}/\mathrm {d}t) = (\phi X - f\xi ,\eta (X)\mathrm {d}/\mathrm {d}t)$ is integrable, where f is a real-valued function. A contact metric manifold is said to be a K-contact manifold if the Reeb vector field $\xi$ is a Killing vector field with respect to the associated metric g. Since the torsion $\tau ={\mathcal {L}}_\xi g$ satisfies $\tau = 2g(\cdot ,h\varphi \cdot )$ and $Ric(\xi ,\xi )=2n-tr h^2$, a contact metric manifold M is K-contact if and only if the tensor $h=0$ or, equivalently, $Ric(\xi ,\xi )=2n$. A contact metric manifold is said to be a Sasakian manifold if the almost contact structure $(\eta ,\xi ,\varphi )$ is normal. Any Sasakian manifold is K-contact and the converse also holds in dimension three. A contact metric manifold $(M,\eta ,g)$ is said to be an H-contact manifold if Reeb vector field $\xi$ is a harmonic vector field, that is, $\xi$ satisfies the critical point condition for the energy functional defined on the space of all unit vector fields; moreover, a contact metric manifold $(M,\eta ,g)$ is H-contact if and only if $\xi$ is an eigenvector of the Ricci operator Q, that is, $Q \xi = (2n-\mathrm{tr} h^2) \xi$ [23]. Sasakian manifolds and K-contact manifolds are H-contact manifolds, but the converse, in general, is not true.

Recently, we have considered a Riemannian metric g as an associated metric for two contact forms. More precisely, we have

Definition 2.1

( [24]) Let M be a three-manifold. A bi-contact metric structure on M is a triple $(\eta _1,\eta _2,g)$ where $(\eta _1,\eta _2)$ is a pair of contact forms and g is an associated metric for both the contact forms $\eta _1,\eta _2$, such that the corresponding Reeb vector fields satisfy $g(\xi _1,\xi _2)=0$, equivalently the corresponding almost contact structures $(\xi _i,\eta _i,\varphi _i)$, $i=1,2$, satisfy the condition:

$$\begin{aligned} \varphi _1\varphi _2 +\varepsilon \eta _1\otimes \xi _2 = - (\varphi _2\varphi _1 +\varepsilon \eta _2\otimes \xi _1),\quad \varepsilon =\pm 1, \end{aligned}$$

(2.2)

where $\varepsilon$ is defined by $\varphi _2\xi _1=\varepsilon \,\varphi _1\xi _2$.

Then, in [24] we gave a complete classification of simply connected three-manifolds which admit a bi-H-contact metric structure $(\eta _1,\eta _2,g)$, i.e., $(\eta _1,g)$ and $(\eta _2,g)$are both H-contact.

We note that in the classical definition of contact metric 3-structure (see, for example, [4] Chapter 13 and [3] Chapter 14) we have three contact metric structures $(\xi _i,\eta _i,\varphi _i,g)$, $i=1,2,3$, such that:

$$\begin{aligned} \varphi _i\varphi _j - \eta _j\otimes \xi _i = \varphi _k= -(\varphi _j\varphi _i - \eta _i\otimes \xi _j) \end{aligned}$$

(2.3)

for any cyclic permutation (i, j, k) of (1, 2, 3). A contact metric 3-structure is called Sasakian 3-structure if the three contact metric structures are Sasakian. The condition (2.3) implies, in particular, the orthogonality of the three Reeb vector fields with respect to g. So, if $(\eta _1,\eta _2,\eta _3, g)$ is a contact metric 3-structure then $(\eta _i,\eta _j,g)$ are bi-contact metric structures for any $i,j=1,2,3, i\ne j$. However, this is only a necessary condition for a contact metric 3-structure. In fact, the Lie group ${\widetilde{SL}}(2,{\mathbb {R}})$ admits three bi-contact metric structures which do not define a contact metric 3-structure (cf. Remark 7.7).

3 Taut contact hyperbolas: first properties and examples

3.1 Definitions and first properties

We begin this Subsection recalling the definitions of contact circle, contact sphere and taut contact circle introduced by H. Geiges and J. Gonzalo on a manifold of dimension three (see, for example, [13, 15]). In all this section, by M we will denote always a three-manifold.

Let $(\eta _1 ,\eta _2 )$ be a pair of contact 1-forms on M. The pair $(\eta _1 ,\eta _2 )$ is called a contact circle if for every $a=(a_1 ,a_2 )\in {\mathbb {S}}^1$, the unit circle in $\mathbb {R}^2$, the linear combination $\eta _a = a_1 \eta _1 +a_2 \eta _2$ is also a contact form. A contact circle $(\eta _1 ,\eta _2 )$ is said to be a taut contact circle if the volume forms $\eta _a\wedge (d\eta _a)$ are equal for every $a\in {\mathbb {S}}^1$. Equivalently, a pair of contact forms $(\eta _1 ,\eta _2 )$ is a taut contact circle if and only if

$$\begin{aligned} \eta _1 \wedge d\eta _1 = \eta _2 \wedge d\eta _2 \quad \text {and} \quad \eta _1 \wedge d\eta _2 =-\eta _2 \wedge d\eta _1 . \end{aligned}$$

(3.1)

In the case of closed three-manifolds, taut contact circles exist only on compact left quotients of the Lie groups: ${\mathbb {S}}^3=SU(2)$, ${\widetilde{SL}}(2, R)$, ${{\widetilde{E}}}(2)$ (cf. [13], Theorem 1.2 ).

The unit circle ${\mathbb {S}}^1$ of the Euclidean plane has its counterpart, in the pseudo-Euclidean plane, that is, in the Minkowski plane, in the four arms of the unit equilateral hyperbolas

$$\begin{aligned} {\mathbb {H}}^1_r: x^2 - y^2 = r, \, \ r=\pm 1. \end{aligned}$$

Indeed, the equilateral hyperbolas have many of the properties of circles in the Euclidean plane (cf., for example, [8]). So we give the following definitions.

Definition 3.1

A pair $(\eta _1 ,\eta _2 )$ of contact forms on M is called a contact hyperbola if for every $a=(a_1 ,a_2 ) \in {\mathbb {H}}^1_r$, the linear combination $\eta _a = a_1 \eta _1 +a_2 \eta _2$ is also a contact form.

This definition implies that any non-trivial linear combination $\eta _a=a_1 \eta _1 +a_2 \eta _2$ with constant coefficients $(a_1 ,a_2 ), a^2_1-a_2^2\ne 0$, is again a contact form.

Definition 3.2

A contact hyperbola $(\eta _1 ,\eta _2 )$ is said to be a taut contact hyperbola if the volume forms $\eta _a\wedge (d\eta _a)$ satisfy

$$\begin{aligned} \eta _a\wedge (d\eta _a) = r \, \eta _1\wedge (d\eta _1) \, \ { for \, all} \, \ a\in {\mathbb {H}}^1_r . \end{aligned}$$

(3.2)

(equivalently, $\eta _a\wedge (d\eta _a) =- r\, \eta _2\wedge (d\eta _2)$).

In [21], Mitsumatsu introduced a bi-contact structure $(\eta _1,\eta _2)$ on a three-manifold, that is, $\eta _1$ and $\eta _2$ are mutually transverse contact 1-forms which induce opposite orientations. Anosov flow naturally induces a bi-contact structure whose intersection as a pair of plane fields is tangent to the flow. In general, the intersection of a bi-contact structure does not define an Anosov flow. In fact, he showed that if $(\eta _1,\eta _2)$ is a bi-contact structure on a compact three-manifold, then the vector field directing the intersection of the two contact subbundles is a conformally Anosov flow (that they called projectively Anosov flow) which is a generalization of an Anosov flow. Eliashberg and Thurston [12] studied bi-contact structures and conformally Anosov flows from the viewpoint of confoliation theory.

The next proposition shows that the notion of taut contact hyperbola is related to that of conformally Anosov flow.

Proposition 3.3

Let $(\eta _1,\eta _2 )$ be a pair of contact forms on M. Then, $(\eta _1 ,\eta _2 )$ is a taut contact hyperbola if and only if

$$\begin{aligned} \quad \eta _2 \wedge d\eta _2 = -\eta _1 \wedge d\eta _1 \quad \text {and} \quad \quad \eta _1 \wedge d\eta _2 =-\eta _2 \wedge d\eta _1 . \end{aligned}$$

(3.3)

In particular, when M is compact, a taut contact hyperbola defines a conformally Anosov flow. But, a pair of contact forms that defines a conformally Anosov flow, in general, does not define a taut contact hyperbola.

Proof

Let $(\eta _1 ,\eta _2 )$ be a taut contact hyperbola. If we take $a=(0,1)\in H^{1}_{r}$, $r=-1$, then $\eta _a=\eta _2$ and from (3.2) we get the condition $\eta _2 \wedge d\eta _2 = -\eta _1 \wedge d\eta _1$. Moreover, for all $a\in {\mathbb {H}}^{1}_{r}$, $r=\pm 1$, (3.2) and the above condition imply

$$\begin{aligned} r\eta _1 \wedge d\eta _1=\eta _a\wedge d\eta _a =(a_1^2-a_2^2)\eta _1 \wedge d\eta _1 +a_1a_2(\eta _1 \wedge d\eta _2+\eta _2\wedge d\eta _1), \end{aligned}$$

and thus we get $\eta _1 \wedge d\eta _2 =-\eta _2 \wedge d\eta _1$.

Vice versa, if we assume (3.3), from

$$\begin{aligned} \eta _a\wedge d\eta _a = a_1^2 \eta _1 \wedge d\eta _1 + a_2^2 \eta _2\wedge d\eta _2 +a_1a_2(\eta _1 \wedge d\eta _2+\eta _2\wedge d\eta _1), \end{aligned}$$

we obtain (3.2). In particular, a taut contact hyperbola is a bi-contact structure (in the sense of Mitsumatsu) and thus, in the compact case, it defines a conformally Anosov flow. The last part is a consequence of Remark 3.10. $\square$

Remark 3.4

By using notations of complex/hyperbolic numbers, the conditions (3.1) and (3.3) can be write, respectively, in the simple forms

$$\begin{aligned} \eta ^c\wedge d\eta ^c= 0, \, where \, \eta ^c=\eta _1+i\eta _2, \; i^2=-1, \quad \eta ^h\wedge d\eta ^h= 0, \, \hbox {where}\, \, \eta ^h=\eta _1+j\eta _2, j^2=1, j\ne \pm 1. \end{aligned}$$

Since the sphere ${\mathbb {S}}^3$ does not admit a conformally Anosov flow (cf. [21] , p.1420), from Proposition 3.3, follows that the sphere ${\mathbb {S}}^3$ does not admit a taut contact hyperbola (cf. also [24], Corollary 3.7).

Corollary 3.5

The torus ${\mathbb {T}}^3$ admits a taut contact hyperbola (and so a conformally Anosov flow).

Proof

Consider on ${\mathbb {R}}^3$ the volume form $\Omega = dx\wedge dy\wedge dz$ and the contact 1-forms

$$\begin{aligned} \eta _1= \cos z dx - \sin z dy \,\; \hbox {and} \;\, \eta _2= \cos z dx + \sin z dy. \end{aligned}$$

Then,

$$\begin{aligned} \eta _1\wedge d \eta _1= \Omega = -\eta _2\wedge d \eta _2 \quad \hbox {and} \quad \eta _2\wedge d \eta _1=(\cos ^2z-\sin ^2 z)\Omega = -\eta _1\wedge d \eta _2. \end{aligned}$$

Thus, by Proposition 3.3, $(\eta _1,\eta _2)$ defines a taut contact hyperbola on ${\mathbb {R}}^3$. On the other hand, $\eta _1$ and $\eta _2$ are invariant under translation by $2\pi$; therefore, $(\eta _1,\eta _2)$ defines a taut contact hyperbola on the torus ${\mathbb {T}}^3$. $\square$

Remark 3.6

The torus ${\mathbb {T}}^3$ has many conformally Anosov flows, while it has no Anosov flows because its fundamental group does not grow exponentially.

Now, recall that a taut contact circle with $\eta _1\wedge d\eta _2 =\eta _2 \wedge d\eta _1=0$ is said to be a Cartan structure (see [13]). On the other hand, if (3.3) holds with $\eta _1 \wedge d\eta _2 =\eta _2 \wedge d\eta _1=0$, we have a taut contact hyperbola like-Cartan structure. So it is natural to give the following

Definition 3.7

A pair of contact 1-forms $(\eta _1,\eta _2)$ is said to be a $(-\varepsilon )$-Cartan structure on the three-manifold M if    $\eta _2 \wedge d\eta _2 = -\varepsilon \eta _1 \wedge d\eta _1 \quad \text {and} \quad \eta _1 \wedge d\eta _2 =0=\eta _2 \wedge d\eta _1$, $\varepsilon =\pm 1$.

Of course, a 1-Cartan structure is a Cartan structure and $(-1)$-Cartan structure is a taut contact hyperbola with $\eta _1 \wedge d\eta _2=0$.

3.2 Taut contact hyperbolas on 3D Lie groups

3.3 Unimodular case

Let G be a simply connected unimodular 3D Lie group. Then, G contains a discrete subgroup $\Gamma$ such that the space of right cosets $\Gamma \backslash G$ is a differentiable manifold. Note that a three-dimensional Lie group G admits a discrete subgroup $\Gamma$ such that $\Gamma \backslash G$ is compact if and only if G is unimodular ( [20]). Moreover, each left-invariant tensor field on G descends to $\Gamma \backslash G$.

In particular, if we have a left-invariant taut contact hyperbola on G, then it descends to $\Gamma \backslash G$ and thus defines a conformally Anosov flow.

Now, we determine the simply connected unimodular 3D Lie groups G which admit taut contact hyperbolas. Since G is unimodular, there exist a basis of left invariant vector fields $(e_1,e_2,e_3)$ such that ( [20]) :

$$\begin{aligned} {}[e_2,e_3]=\lambda _1 e_1, \quad [e_3,e_1]=\lambda _2 e_2, \quad [e_1,e_2]=\lambda _3 e_3 , \end{aligned}$$

(3.4)

where $(\lambda _1,\lambda _2,\lambda _3)$ are constant. If $(\lambda _1,\lambda _2,\lambda _3)=(0,0,0)$, then G is the Abelian Lie group ${\mathbb {R}}^3$. The signature of $(\lambda _1,\lambda _2,\lambda _3)\ne (0,0,0)$ can be one of the following type:

• $(+,+,+)$ and in this case G is the three-sphere group SU(2);

•   $(+,+,0)$ and in this case $G={{\widetilde{E}}}(2)$ (the universal cover of the group of orientation-preserving isometries of the Euclidean plane);

•   $(+,0,0)$ and in this case G is the Heisenberg group ${\mathcal {H}}^3=Nil^3$;

•   $(+,-,-)$ and in this case $G= {\widetilde{SL}}(2,R)$;

•   $(+,-,0)$ and in this case $G=Sol^3$ (also known as the group E(1, 1) of orientation-preserving isometries of the Minkowski plane).

Denote by $\eta _i$ the dual 1-forms : $\eta _i(e_j)=\delta _{ij}$. Then

$$\begin{aligned} d\eta _1 = -\lambda _1 \eta _2 \wedge \eta _3, \quad d\eta _2 = - \lambda _2 \eta _3 \wedge \eta _1, \quad d\eta _3= - \lambda _1 \eta _1\wedge \eta _2, \end{aligned}$$

and thus

$$\begin{aligned} \eta _i\wedge d\eta _i = - \lambda _i \eta _1 \wedge \eta _2 \wedge \eta _3 \quad \hbox {and} \quad \eta _i\wedge d\eta _j=0 \quad \hbox { for any } i\ne j. \end{aligned}$$

In general, if $(\eta _1,\eta _2,\eta _3)$ is a coframe on a three-manifold, the non-trivial 1-forms $\eta _a=a_1 \eta _1 +a_2 \eta _2 + a_3\eta _3$, $\eta _b=b_1 \eta _1 +b_2 \eta _2 + b_3\eta _3$ with constant coefficients $(a_i), (b_i)$, satisfy

$$\begin{aligned} \eta _a\wedge d\eta _b&= a_1b_1 \eta _1\wedge d\eta _1 +a_2b_2 \eta _2\wedge d\eta _2+a_3b_3 \eta _3\wedge d\eta _3 + a_1 b_2 \eta _1\wedge d\eta _2+ a_2 b_1\eta _2\wedge d\eta _1 \nonumber \\&\quad + a_1b_3 \eta _1\wedge d\eta _3+ a_3 b_1\eta _3\wedge d\eta _1 +a_2b_3 \eta _2\wedge d\eta _3+ a_3 b_2\eta _3\wedge d\eta _2. \end{aligned}$$

(3.5)

Thus, two arbitrary left invariant 1-forms $\eta _a,\eta _b$ on the unimodular Lie group G satisfy

$$\begin{aligned} \eta _a\wedge d\eta _b = - {\mathcal {L}}(a,b)\, \eta _1 \wedge \eta _2 \wedge \eta _3, \quad \text {where we put} \quad {\mathcal {L}}(a,b) = \lambda _1 a_1 b_1 + \lambda _2 a_2 b_2 +\lambda _3 a_3 b_3 . \end{aligned}$$

Consequently, by Proposition 3.3, the left invariant 1-forms $(\eta _a,\eta _b)$ define a taut contact hyperbola if and only if the symmetric bilinear map ${\mathcal {L}}$ satisfies

$$\begin{aligned} {\mathcal {L}}(a,a) = -{\mathcal {L}}(b,b)\ne 0, \quad {\mathcal {L}}(a,b) = 0. \end{aligned}$$

(3.6)

Therefore, the constant $(\lambda _1,\lambda _2,\lambda _3)$ that define the unimodular Lie groups SU(2), ${{\tilde{E}}}(2)$, ${\mathcal {H}}^3$ and ${\mathbb {R}}^3$ do not satisfy the condition (3.6). Only the constant $(\lambda _1,\lambda _2,\lambda _3)$ that define the unimodular Lie groups $Sol^3$ and ${\widetilde{SL}}(2,{\mathbb {R}})$ satisfy the condition (3.6). Next, we determine the left invariant taut contact hyperbolas on these unimodular Lie groups.

Example 3.8

On the Lie group $Sol^3$ we can consider a basis of left invariant vector fields $(e_1,e_2,e_3)$ such that

$$\begin{aligned}{}[e_2,e_3]=2 e_1, \quad [e_3,e_1]=-2 e_2, \quad [e_1,e_2]=0. \end{aligned}$$

(3.7)

In Example 8.2, we will give an explicit presentation of left invariant vector fields satisfying (3.7). The dual 1-forms $\eta _i$ satisfy

$$\begin{aligned} d\eta _1 = -2\eta _2 \wedge \eta _3, \quad d\eta _2 = 2\eta _3 \wedge \eta _1, \quad d\eta _3=0, \end{aligned}$$

and thus

$$\begin{aligned} \eta _1 \wedge d\eta _1 = -2 \eta _1 \wedge \eta _2 \wedge \eta _3= -\eta _2 \wedge d\eta _2,\qquad \eta _1 \wedge d\eta _2=0=\eta _2\wedge d\eta _1. \end{aligned}$$

Therefore, by (3.6) $(\eta _1,\eta _2)$ is a left invariant taut contact hyperbola on the unimodular Lie group $Sol^3$ and this structure descends to any compact left-quotient.

Moreover, by using (3.6), two arbitrary left invariant 1-forms $(\eta _a,\eta _b)$ on the Lie group $Sol^3$ define a taut contact hyperbola if and only if

$$\begin{aligned} (a_1^2 -a_2^2)\ne 0 \, \text { and } \, (b_1,b_2)=\pm (a_2,a_1). \end{aligned}$$

Example 3.9

On the Lie group ${\widetilde{SL}}(2,{\mathbb {R}})$, we can consider a basis of left invariant vector fields $(e_1,e_2,e_3)$ such that

$$\begin{aligned} {}[e_2,e_3]=2 e_1, \quad [e_3,e_1]=-2 e_2, \quad [e_1,e_2]=-2 e_3. \end{aligned}$$

(3.8)

In Example 8.2, we will give an explicit presentation of left invariant vector fields satisfying (3.8). The dual 1-forms $\eta _i$ satisfy

$$\begin{aligned} {d\eta _1 = -2\eta _2 \wedge \eta _3, \quad d\eta _2 = 2\eta _3 \wedge \eta _1, \quad d\eta _3=2\eta _1 \wedge \eta _2, } \end{aligned}$$

and thus

$$\begin{aligned} - \eta _1 \wedge d\eta _1 = 2 \eta _1 \wedge \eta _2 \wedge \eta _3 = \eta _2 \wedge d\eta _2= \eta _3 \wedge d\eta _3,\qquad \eta _1 \wedge d\eta _j=0 \text { for any } i\ne j. \end{aligned}$$

Therefore, $(\eta _1,\eta _2)$ and $(\eta _1,\eta _3)$ satisfy (3.6), i.e., they are left invariant taut contact hyperbolas on the unimodular Lie group ${\widetilde{SL}}(2,{\mathbb {R}})$ and these structures descend to any compact left-quotient.

Moreover, we can classify all left invariant taut contact hyperbolas on the Lie group ${\widetilde{SL}}(2,{\mathbb {R}})$. In this case, the bilinear map ${\mathcal {L}}$ defines on ${\mathbb {R}}^3$ the Lorentzian metric

$$\begin{aligned} g_0(a,b)= {\mathcal {L}}(a,b)= a_1b_1 - a_2b_2- a_3b_3, \, \, \text { for any } a,b\in {\mathbb {R}}^3. \end{aligned}$$

Then, by (3.6), two arbitrary left invariant contact 1-forms $\eta _a$,$\eta _b$ on ${\widetilde{SL}}(2,{\mathbb {R}})$, define a taut contact hyperbola if and only if   $g_0(b,b)=- g_0(a,a)\ne 0$ and $g_0(a,b)=0$, i.e., the vectors a and b are two orthogonal vectors, one of which is timelike and so the other is spacelike.

Remark 3.10

We can consider two left invariant 1-forms $\eta _a,\eta _b$ on the Lie group $Sol^3$ with

$$\begin{aligned} {\mathcal {L}}(b,b)=b_1^2 -b_2^2 = -(a_1^2 -a_2^2)=-{\mathcal {L}}(a,a)\ne 0 \text { and } {\mathcal {L}}(a,b)=a_1b_1 -a_2b_2\ne 0, \end{aligned}$$

then $\eta _a,\eta _b$ satisfy $\eta _b\wedge d\eta _b= -\eta _a\wedge d\eta _a$ and $\eta _a\wedge d\eta _b= \eta _b\wedge d\eta _a \ne 0$. Analogously, we can consider $\eta '_a,\eta '_b$ on the Lie group ${\widetilde{SL}}(2,{\mathbb {R}})$ with

$$\begin{aligned} g_0(b,b) =-g_0(a,a) \ne 0 \text { and }\ g_0(a,b) \ne 0. \end{aligned}$$

In both the cases, $(\eta _a,\eta _b)$ and $(\eta '_a,\eta '_b)$ do not define a taut contact hyperbola, but they define a bi-contact structure in the sense of Mitsumatsu [21], and so define a conformally Anosov on any compact quotient of $Sol^3$ and ${\widetilde{SL}}(2,{\mathbb {R}})$, respectively. Therefore, the notion of taut contact hyperbola is stronger than the notion of conformally Anosov.

3.4 Non-unimodular case

Let G be a non-unimodular 3D Lie group. Then G admits a basis of left invariant vector fields $e_1, e_2, e_3$ such that (cf. [20])

$$\begin{aligned} {}[e_1,e_2]=\alpha e_2 +\beta e_3, \quad [e_1,e_3]= \gamma e_2 +\delta e_3, \quad [e_2,e_3]=0, \quad \alpha +\delta \ne 0. \end{aligned}$$

(3.9)

This Lie group G can be presented as a semi-direct product Lie group ${\mathbb {R}}^2\rtimes _A{\mathbb {R}}$, where $A = \left( \begin{array}{cc} \alpha &{} \gamma \\ \beta &{} \delta \end{array}\right)$, tr$A=\alpha +\delta \ne 0$. Denote by

$$\begin{aligned} {\mathcal {D}}= 4 {\det A}/{(\mathrm{tr} A)^2} \end{aligned}$$

the invariant introduced by Milnor [20, p. 321], which, unless A is a multiple of the identity matrix, completely determines the non-unimodular Lie algebra (and so the Lie group) up to isomorphisms.

Let $(\vartheta ^1,\vartheta ^2,\vartheta ^3)$ be the basis of 1-forms dual of $(e_1,e_2,e_3)$. Then,

$$\begin{aligned} d \vartheta ^1=0, \quad d\vartheta ^2= -\alpha \vartheta ^1\wedge \vartheta ^2- \gamma \vartheta ^1\wedge \vartheta ^3,\quad d \vartheta ^3= -\beta \vartheta ^1\wedge \vartheta ^2- \delta \vartheta ^1\wedge \vartheta ^3, \end{aligned}$$

and thus

$$\begin{aligned} \vartheta ^2\wedge d\vartheta ^2= \gamma \ \Omega , \quad \vartheta ^3\wedge d\vartheta ^3= -\beta \ \Omega , \quad \vartheta ^2\wedge d\vartheta ^3= \delta \ \Omega \quad \text { and } \quad \vartheta ^3\wedge d\vartheta ^2= -\alpha \ \Omega , \end{aligned}$$

where $\Omega =\vartheta ^1\wedge \vartheta ^2\wedge \vartheta ^3$. Then, by using (3.5), two arbitrary left invariant 1-forms $\eta _a= \sum ^{3}_{i=1}a_i \vartheta ^i$ and $\eta _b= \sum ^{3}_{i=1}b_i \vartheta ^i$ satisfy

$$\begin{aligned} \eta _a\wedge d\eta _b&= (\gamma a_2b_2 +\delta a_2b_3 -\alpha a_3b_2 -\beta a_3b_3)\Omega ,\\ \eta _a\wedge d\eta _a&= (\gamma a_2^2 +(\delta -\alpha ) a_2a_3 -\beta a_3^3)\Omega . \end{aligned}$$

Therefore, $(\eta _a,\eta _b)$ is a taut contact hyperbola on the non-unimodular Lie group G if, and only if, are satisfied the following:

$$\begin{aligned} \left\{ \begin{array}{cc} &{}(\gamma a_2^2 +(\delta -\alpha ) a_2 a_3 -\beta a_3^2)=- (\gamma b_2^2 +(\delta -\alpha ) b_2 b_3 -\beta b_3^2)\ne 0, \\ &{} 2\gamma a_2b_2 +(\delta -\alpha ) (a_2 b_3 + a_3 b_2) -2\beta a_3b_3 = 0. \end{array} \right. \end{aligned}$$

(3.10)

In particular,

$$\begin{aligned} (\vartheta ^2,\vartheta ^3) \text { is a taut contact hyperbola } \, \Longleftrightarrow \, \beta =\gamma \ne 0 \text { and } \alpha =\delta \ne 0. \end{aligned}$$

In this case, the non-unimodular Lie group is defined by the matrix $A = \left( \begin{array}{cc} \alpha &{} \beta \\ \beta &{} \alpha \end{array}\right)$, $\alpha ,\beta \ne 0$, and the Reeb vector fields of $\vartheta ^2,\vartheta ^3$ are given by   $\xi _2= e_2- ({\alpha }/{\beta }) e_3$,   $\xi _3= -({\alpha }/{\beta }) e_2 +e_3$. Then, $\xi _2,\xi _3$ are linearly independent if and only if the Milnor’s invariant ${\mathcal {D}} =4 \det A/ (tr A)^2\ne 0$. Moreover, ${\mathcal {D}}=0$ if and only if $\xi _2=\pm \xi _3$. Thus, we get the following

Proposition 3.11

The non-unimodular Lie group $G={\mathbb {R}}^2\rtimes _A{\mathbb {R}}$, where $A = \left( \begin{array}{cc} \alpha &{} \pm \alpha \\ \pm \alpha &{} \alpha \end{array}\right)$, $\alpha \ne 0$, admits a taut contact hyperbola $(\eta _1,\eta _2)$ with the corresponding Reeb vector fields satisfying $\xi _2=\mp \xi _1$.

Remark 3.12

The result of the Proposition 3.11 gives an interesting difference with respect to the case of a taut contact circle. In fact, the Reeb vector fields of any taut contact circle are linearly independent (cf. Theorem 4.1).

Remark 3.13

Not all non-unimodular Lie groups admit a left invariant taut contact hyperbola. In fact, for $\alpha =\delta \ne 0$ and $\beta \gamma <0$, the system (3.10) does not admit solution.

Now, we give an explicit example of non-unimodular Lie group satisfying Proposition 3.11.

Example 3.14

Consider the hyperbolic plane ${\mathbb {H}}^2 = \left\{ (x_1,x_2)\in {\mathbb {R}}^2: x_2>0\right\}$ equipped with standard Lie group structure. For any $\alpha \ne 0$, the vector fields

$$\begin{aligned} E_1= 2\alpha \,x_2 \partial _1, \quad E_2= 2\alpha \,x_2 \partial _2 \end{aligned}$$

define a basis of left invariant vector fields on ${\mathbb {H}}^2$. Now, consider the direct product Lie group

$$\begin{aligned} {\mathcal {G}}_{{\mathcal {H}}}= {\mathbb {H}}^2\times {\mathbb {R}}. \end{aligned}$$

Then $(E_1, E_2, E_3= \partial _t)$ is a basis of left invariant vector fields on ${\mathcal {G}}_{{\mathcal {H}}}$. We note that with respect to the basis

$$\begin{aligned} e_1=E_2, \quad e_2= (E_1+E_3),\quad e_3= (E_1-E_3), \end{aligned}$$

the Lie algebra of ${\mathcal {G}}_{{\mathcal {H}}}$ is defined by   $[e_2,e_3]=0, \quad [e_1,e_2]=[e_1,e_3]= \alpha (e_2 +e_3)$. So, ${\mathcal {G}}_{{\mathcal {H}}}$ is the non-unimodular Lie group ${\mathbb {R}}^2\rtimes _{A}{\mathbb {R}}$, where $A=\alpha I_2$, with the Milnor’s invariant ${\mathcal {D}}=0$.

4 Some characterization of taut contact hyperbolas

We start this section recalling the following characterizations of taut contact circles.

Theorem 4.1

(Zessin [27]) Let $(\eta _1,\eta _2 )$ be a contact circle on a three-manifold M, and let $\xi _1$, $\xi _2$ be the corresponding Reeb vector fields. Then, $\xi _1$, $\xi _2$ are everywhere linearly independent and $d\eta _1(\xi _2, \cdot ),\,d\eta _2(\xi _1, \cdot )$ never vanish. Moreover, the following properties are equivalent

(i):

  $(\eta _1 ,\eta _2 )$ is a taut contact circle;

(ii):

  $\xi _a= a_1 \xi _1 +a_2 \xi _2$ is the Reeb vector field of $\eta _a= a_1 \eta _1 +a_2 \eta _2$ for any $a\in {\mathbb {S}}^1$;

(iii):

  $\eta _2(\xi _1)=-\eta _1(\xi _2)$ and $d\eta _1(\xi _2, \cdot )= - d\eta _2(\xi _1, \cdot )$.

In Proposition 3.11, we showed the existence of taut contact hyperbolas $(\eta _1,\eta _2)$ with the corresponding Reeb vector fields $(\xi _1,\xi _2=\pm \xi _1)$. Next, we give some characterization of taut contact hyperbolas with $\xi _2 \ne \pm \xi _1$. More precisely, we show the following theorem.

Theorem 4.2

Let $(\eta _1,\eta _2)$ be a pair of contact forms on a three-manifold M with $\xi _2 \ne \pm \xi _1$ in any point $p\in M$. Then, the following are equivalent:

(a):

  $(\eta _1,\eta _2)$ is a taut contact hyperbola ;

(b):

  $\eta _1(\xi _2)=\eta _2(\xi _1)$ and $d\eta _1(\xi _2,\cdot ) =d\eta _2(\xi _1,\cdot )\ne 0$ in any point $p\in M$;

(c):

  $\eta _a=a_1\eta _1+a_2\eta _2$is a contact form with Reeb vector field$\xi _a= r(a_1 \xi _1-a_2 \xi _2)$for any $a\in {\mathbb {H}}^1_r$.

In particular, in a such case, $\xi _1, \xi _2$ are linearly independent.

We first give the following lemmas.

Lemma 4.3

Let $(\eta _1,\eta _2 )$ be a contact hyperbola on M. Then,

$\xi _1$, $\xi _2$ are pointwise linearly independent $\Longleftrightarrow$ $d\eta _1(\xi _2, \cdot )$ and $d\eta _2(\xi _1, \cdot )$ are $\ne 0$ in any point.

Moreover,

$\xi _a= r(a_1 \xi _1 - a_2 \xi _2)$is the Reeb vector field of $\eta _a= a_1 \eta _1 +a_2 \eta _2$ for all $a\in {\mathbb {H}}^1_r$   if and only if   $\eta _2(\xi _1)= \eta _1(\xi _2)$ and $d\eta _1(\xi _2, \cdot )= d\eta _2(\xi _1, \cdot )$.

Proof

($\Rightarrow$) Suppose that there exists a point $p\in M$ such that $d\eta _2(\xi _1, \cdot )=0$ at p, and so $d\eta _a(\xi _1, \cdot )=a_1 d\eta _1(\xi _1, \cdot )+ a_2 d\eta _2(\xi _1, \cdot )=0$ at p for any $a\in {\mathbb {H}}^1_r$. Then, since $(\eta _a\wedge d\eta _a)(\xi _1,\cdot ,\cdot ) \ne 0$, we get $\eta _a(\xi _1)_p\ne 0$ for any $a\in {\mathbb {H}}^1_r$, and in particular $\eta _2(\xi _1)_p\ne 0$. Now, we put $\xi _1'=(1/\varrho )\xi _1$, where $\varrho =\eta _2(\xi _1)_p \ne 0$. Then,   $\eta _2(\xi _1')_p=1$   and   $(d\eta _2)(\xi _1',\cdot )=0$

give the contradiction $(\xi _2)_p=(1/\varrho )(\xi _1)_p$, that is, $\xi _1,\xi _2$ linearly dependent at p.

($\Leftarrow$) If we suppose $\xi _1,\xi _2$ linearly dependent in some point p, i.e., $(\xi _2)_p=\varrho (\xi _1)_p$ for some constant $\varrho \ne 0$, then we have the contradiction $(d\eta _1)(\xi _2,\cdot )_p= \varrho (d\eta _1)(\xi _1,\cdot )_p=0$. For the second part, it is sufficient to remark that

  $\eta _a(\xi _a)= 1+ r a_1 a_2\big (\eta _2(\xi _1)-\eta _1(\xi _2)\big )$   and   $d\eta _a(\xi _a,\cdot )= r a_1 a_2\big (d\eta _2(\xi _1,\cdot )-d\eta _1(\xi _2,\cdot )\big )$. $\square$

Lemma 4.4

Let $(\eta _1,\eta _2)$ be a contact hyperbola on M with $\xi _2 \ne \pm \xi _1$ in any point $p\in M$. Then, $d\eta _1(\xi _2,\cdot )$ and $d\eta _2(\xi _1,\cdot )$ are $\ne 0$ in any point $p\in M$.

In particular, if $\xi _2 \ne \pm \xi _1$ in any point $p\in M$, then $\xi _1, \xi _2$ are linearly independent in any point $p\in M$.

Proof

Assume that $\xi _2 \ne \pm \xi _1$ in any point, and suppose that there exists a point $p\in M$ such that $d\eta _2(\xi _1,\cdot )_p=0$. Then, for all $a\in {\mathbb {H}}^1_r$

$$\begin{aligned} d\eta _a(\xi _1,\cdot ) = a_1d\eta _1(\xi _1,\cdot ) +a_2 d\eta _2(\xi _1,\cdot )=0 \, \text { at } p. \end{aligned}$$

Consequently, since $\eta _a$ is a contact form, i.e., $\eta _a\wedge d\eta _a(\xi _1,\cdot ,\cdot )\ne 0$, we obtain

$$\begin{aligned} \eta _a(\xi _1)_p\ne 0 \, \text { for all }\, a\in {\mathbb {H}}^1_r. \end{aligned}$$

In particular, taking $a=(0,1)$, we have that   $\lambda :=\eta _2(\xi _1)_p\ne 0$.

Now, we show that $\lambda \ne 0$ gives a contradiction. Consider the function $f : {\mathbb {H}}^1_r \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} f(a)=\eta _a(\xi _1)_p=a_1\eta _1(\xi _1)_p + a_2\eta _2(\xi _1)_p= a_1 + \lambda a_2. \end{aligned}$$

Consider the cases:   $(I)\ \, \lambda >0$   and   $(II)\ \, \lambda <0$.

For the case (I), we distinguish the following subcases:

$$\begin{aligned} (I_1)\ \, \lambda >1, \quad (I_2) \ \, 0<\lambda <1, \quad (I_3)\,\ \lambda =1. \end{aligned}$$

For the subcase $(I_1)$ consider the function f defined on the connected subset $C_{1}=\{a\in {\mathbb {R}}^2: a_1^2-a_2^2=1, a_1>0\}$. Put $\mu = (\lambda /\sqrt{\lambda ^2-1})>1$ and take

$$\begin{aligned} a=(a_1,a_2), {\bar{a}}=(a_1,-a_2)\in C_{1}, \text { with } a_1>\mu>1 \text { and } a_2=\sqrt{a_1^2-1}>0. \end{aligned}$$

Then, $f(a)= a_1 + \lambda a_2 >0$ and $f({\bar{a}})= a_1 -\lambda a_2 <0$. Thus, it should exist $b\in C_1$ such that $f(b)=0$, and this gives a contradiction.

For the subcase $(I_2)$ consider the function f defined on the connected subset $C_{2}=\{a\in {\mathbb {R}}^2: a_1^2-a_2^2=-1, a_2>0\}$. Put $\mu = (1/\lambda )>1$ and take

$$\begin{aligned} a=(a_1,a_2), {\bar{a}}=(-a_1,a_2)\in C_2, \text { with } a_1>0 \text { and } a_2> \mu /\sqrt{\mu ^2-1}>0. \end{aligned}$$

Then, $f(a)= a_1 + \lambda a_2 >0$ and $f({\bar{a}})= -a_1 +\lambda a_2 <0$. Thus, it should exist $b\in C_2$ such that $f(b)=0$, and this gives a contradiction.

For the subcase $(I_3)$, $\eta _2(\xi _1)_p=1$ and $d\eta _2(\xi _1,\cdot )_p=0$ give the contradiction $\xi _2=\xi _1$ at p.

For the case (II), we distinguish the following subcases:

$$\begin{aligned} (II_1) \, \lambda<-1, \quad (II_2) \, -1<\lambda <0, \quad (II_3)\, \lambda =-1. \end{aligned}$$

For the subcase $(II_1)$ consider the function f defined on the connected subset $C_3=\{a\in {\mathbb {R}}^2: a_1^2-a_2^2=1, a_1<0\}$. Put $\mu = (\lambda /\sqrt{\lambda ^2-1})<-1$ and take

$$\begin{aligned} a=(a_1,a_2), {\bar{a}}=(a_1,-a_2)\in C_3, \text { with } a_1<\mu <-1 \text { and } a_2=\sqrt{a_1^2-1}>0. \end{aligned}$$

Then, $f(a)= a_1 + \lambda a_2 < 0$ and $f({\bar{a}})= a_1 -\lambda a_2 >0$. Thus, it should exist $b\in C_3$ such that $f(b)=0$, and this gives a contradiction.

For the subcase $(II_2)$ consider the function f defined on the connected subset $C_4=\{a\in {\mathbb {R}}^2: a_1^2-a_2^2=-1, a_2<0\}$. Put $\mu = (1/\lambda )<-1$ and take

$$\begin{aligned} a=(a_1,a_2), {\bar{a}}=(-a_1,a_2)\in C_4, \text { with } a_1 =\sqrt{a_2^2-1}>0 \text { and } a_2< \mu /\sqrt{\mu ^2-1}<-1. \end{aligned}$$

Then, $f(a)= a_1 + \lambda a_2 >0$ and $f({\bar{a}})= -a_1 +\lambda a_2 <0$. Thus, it should exist $b\in C_4$ such that $f(b)=0$, and this gives a contradiction.

For the subcase $(II_3)$, $\eta _2(-\xi _1)_p=1$ and $d\eta _2(-\xi _1,\cdot )_p=0$ give the contradiction $\xi _2=-\xi _1$ at p. We proceed analogously if suppose $d\eta _1(\xi _2,\cdot )_p=0$. The second part of the Lemma follows from Lemma 4.3. $\square$

Proof of Theorem 4.2

$(a) \Longleftrightarrow (b)$

If we suppose (a) (respectively (b)), by Lemma 4.4 (respectively Lemma 4.3), we have that the Reeb vector fields $\xi _1$, $\xi _2$ are linearly independent. Moreover, we have

$$\begin{aligned} \eta _1\wedge d\eta _1(\xi _1,\xi _2,\cdot )= & {} d\eta _1(\xi _2,\cdot ), \quad \eta _2\wedge d\eta _2(\xi _1,\xi _2,\cdot ) = -d\eta _2(\xi _1,\cdot ),\\ \eta _1\wedge d\eta _2(\xi _1,\xi _2,\cdot )= & {} -\eta _1(\xi _2) d\eta _2(\xi _1,\cdot ), \quad \eta _2\wedge d\eta _1(\xi _1,\xi _2,\cdot ) = \eta _2(\xi _1) d\eta _1(\xi _2,\cdot ). \end{aligned}$$

Then, by using Proposition 3.3, it is not difficult to see that $(a)\Leftrightarrow (b)$.

$(c) \Longleftrightarrow (a)$

Suppose (c), that is, $\eta _a=a_1\eta _1+a_2\eta _2$ is a contact form with Reeb vector field $\xi _a= r(a_1 \xi _1-a_2 \xi _2)$ for any $a\in {\mathbb {H}}^1_r$. Then, $(\eta _1,\eta _2)$ is a contact hyperbola and, from Lemma 4.4, $\xi _1,\xi _2$ are linearly independent and $d\eta _1(\xi _2,\cdot ), d\eta _2(\xi _1,\cdot )\ne 0$ in any point $p\in M$. Then, Lemma 4.3 gives $\eta _1(\xi _2)= \eta _2(\xi _1)$ and $d\eta _1(\xi _2,\cdot )= d\eta _2(\xi _1,\cdot )\ne 0$. So, we get (b) and thus (a).

Conversely, suppose (a), that is, $(\eta _1,\eta _2)$ is a taut contact hyperbola and thus $\eta _a=a_1\eta _1+a_2\eta _2$ is a contact form for any $a\in {\mathbb {H}}^1_r$. Since (a) is equivalent to (b), by Lemma 4.3 we get (c).$\square$

We close this subsection with remarking that the taut contact hyperbolas are related to the symplectic pair.

Remark 4.5

(taut contact hyperbola/circle and symplectic structures)

Let $(M, \eta ,\xi ,\varphi )$ be an almost contact manifold. We denote by $C(M)= {\mathbb {R}}_+\times M$ the cone on M, for more information about the geometry of the cone C(M) we refer, for example, to [4] Section 6.5. Consider the (1, 1)-tensor J on C(M) defined by

$$\begin{aligned} J X=\varphi X \, \text { for } \, X\in \ker \eta , \quad J \xi =\zeta , \quad J \zeta =-\xi , \end{aligned}$$

where $\zeta = t \frac{\partial }{\partial t}$ is Liouville (or Euler) vector field. Then, J is an almost complex structure invariant under the flow of $\zeta$ : ${\mathcal {L}}_\zeta J=0$. Moreover, it well-known that $\eta$ is a contact form on M if and only if the 2-form $\Omega = d(t^2\eta )$ is a symplectic form on C(M).

Now, let $(\eta _1,\eta _2)$ be a pair of contact forms on a three-manifold M and $(\eta _i,\xi _i,\varphi _i)$, $i=1,2$, underlying almost contact structures. Then the corresponding symplectic forms $\Omega _i= d(t^2 \eta _i)= 2t d t \wedge \eta _i + t^2 d\eta _i$, $i=1,2$, satisfy:

$$\begin{aligned} \Omega _1 \wedge \Omega _1&= =4t^3dt \wedge \eta _1 \wedge d\eta _1 ,\\ \Omega _2 \wedge \Omega _2&= 4t^3dt \wedge \eta _2 \wedge d\eta _2 ,\\ \Omega _1 \wedge \Omega _2&=2t^3dt \wedge (\eta _1 \wedge d\eta _2 + \eta _2 \wedge d\eta _1 )= \Omega _2 \wedge \Omega _1,\\ (a_1\Omega _1+ a_2 \Omega _2)\wedge&(a_1\Omega _1+ a_2 \Omega _2) = 4t^3dt \wedge \eta _a \wedge d \eta _a . \end{aligned}$$

So, $(\eta _1,\eta _2)$ is a contact hyperbola (resp. circle) if and only if $\Omega _a:=(a_1\Omega _1+ a_2 \Omega _2)$ is a symplectic 2-form on the four-dimensional cone C(M) for any $(a_1,a_2)\in {\mathbb {H}}^1_r$ (resp. for any $(a_1,a_2)\in {\mathbb {S}}^1$).

On the other hand, on a four-manifold, following Bande and Kotschick [2]: a symplectic pair is defined by two symplectic forms $(\omega _1,\omega _2)$ that satisfy

$$\begin{aligned} \omega _1\wedge \omega _2= 0 \, \ \text {and} \,\ \omega _1\wedge \omega _1=-\omega _2\wedge \omega _2 , \end{aligned}$$

and following Geiges [14]: $(\omega _1,\omega _2)$ is said to be a conformal symplectic couple if

$$\begin{aligned} \omega _1\wedge \omega _2= 0 \, \ \text { and } \, \ \omega _1\wedge \omega _1=\omega _2\wedge \omega _2. \end{aligned}$$

Therefore (cf. also Section 6 of [24] where we studied the metric cone of a bi-contact metric manifold) we get:

$(\eta _1,\eta _2)$ is a taut contact hyperbola (resp. circle) if and only if the corresponding symplectic 2-forms $(\Omega _1,\Omega _2)$ define a symplectic pair (resp. a conformal symplectic couple) .

Moreover, given a pair of contact forms $(\eta _1,\eta _2)$ on M with $\xi _2 \ne \pm \xi _1$ in any point $p\in M$, the corresponding symplectic 2-forms $\Omega _1$, $\Omega _2$ satisfy:

$$\begin{aligned} \Omega _1(\xi _2,X)= & {} 2t (dt\wedge \eta _1)(\xi _2,X) + t^2 d\eta _1(\xi _2,X) = t^2 (d\eta _1)(\xi _2,X) \text { for } X \text { tangent to } M;\\ \Omega _1(\xi _2,\partial t)= & {} 2t (dt\wedge \eta _1)(\xi _2,\partial t) + t^2 d\eta _1(\xi _2,\partial t) = -t \eta _1 (\xi _2) ;\\ \Omega _2(\xi _1,X)= & {} 2t (dt\wedge \eta _2)(\xi _1,X) + t^2 d\eta _2(\xi _1,X) = t^2 (d\eta _2)(\xi _1,X) \text { for } X \text { tangent to } M;\\ \Omega _2(\xi _1,\partial t)= & {} 2t (dt\wedge \eta _2)(\xi _1,\partial t) + t^2 d\eta _2(\xi _1,\partial t) = -t \eta _2 (\xi _1) . \end{aligned}$$

Then, by Theorems 4.2 and 4.1, we get

$(\eta _1,\eta _2)$ is a taut contact hyperbola (resp. circle) if and only if the corresponding symplectic 2-forms $(\Omega _1,\Omega _2)$ satisfy $\Omega _2(\xi _1,\cdot )=\Omega _1(\xi _2,\cdot )\ne 0$ (resp. $\Omega _2(\xi _1,\cdot )=-\Omega _1(\xi _2,\cdot )\ne 0$) in any point.

5 Taut contact 2-hyperboloid

Recall (cf. [13]) that a contact sphere on a three-manifold M is a triple of contact 1-forms $(\eta _1,\eta _2,\eta _3)$ such that any linear combination $(a_1\eta _1 +a_2\eta _2+a_3\eta _3)$, $a\in {\mathbb {S}}^2$, is a contact form; moreover, it is taut if the volume forms $\eta _a\wedge (d\eta _a)$ on M are equal for every $a\in {\mathbb {S}}^2$; moreover in this case the 1-forms $(\eta _1,\eta _2,\eta _3)$ parallelize the three-manifold M.

Now, consider the surface

$$\begin{aligned} H^2_r: a_1^2 -a_2^2 -a_3^2 = r, r\pm 1, \end{aligned}$$

that is, $H^2_{-1}$ is an one-sheeted hyperboloid and $H^2_{1}$ is a two-sheeted hyperboloid. In analogy with the definition of (taut) contact 2-sphere we give the following definition. We say that a triple of contact 1-forms $(\eta _1,\eta _2,\eta _3)$ on a three-manifold M, is a contact 2-hyperboloid if the 1-form

$$\begin{aligned} {{\eta _a= a_1\eta _1 + a_2\eta _2 + a_3\eta _3\, is\, a \,contact \,form\, for\, any\, a\in H^2_r}.} \end{aligned}$$

This definition implies that any non-trivial linear combination $\eta _a=a_1 \eta _1 +a_2 \eta _2 + a_3\eta _3$ with constant coefficients $(a_1,a_2,a_3), a_1^2 -a_2^2 -a_3^2 \ne 0$, is again a contact form. We call the triple $(\eta _1,\eta _2,\eta _3)$ a taut contact 2-hyperboloid if the volume forms $r(\eta _a\wedge d\eta _a)$ are equal for every $a\in H^2_r$, that is,

$$\begin{aligned} \eta _a\wedge d\eta _a = r \, \eta _1\wedge d\eta _1 \quad \text {for every}\ \, a\in H^2_r. \end{aligned}$$

Besides, we note that

$$\begin{aligned} \eta _a\wedge d\eta _a&= a_1^2 \eta _1\wedge d\eta _1 +a_2^2 \eta _2\wedge d\eta _2+a_3^2 \eta _3\wedge d\eta _3 + a_1a_2(\eta _1\wedge d\eta _2+\eta _2\wedge d\eta _1) \\&\quad + a_1a_3(\eta _1\wedge d\eta _3+\eta _3\wedge d\eta _1) +a_2a_3(\eta _3\wedge d\eta _2+\eta _2\wedge d\eta _3). \end{aligned}$$

Consequently, we get

Proposition 5.1

A triple of contact 1-forms $(\eta _1,\eta _2,\eta _3)$ on a three-manifold M is a taut contact 2-hyperboloid if and only if $(\eta _1,\eta _2)$ and $(\eta _1,\eta _3)$ are taut contact hyperbola and the other pair $(\eta _2,\eta _3)$ is a taut contact circle.

Remark 5.2

We note that a triple of contact 1-forms $(\eta _1,\eta _2,\eta _3)$ on a three-manifold M defines a taut contact 2-sphere if and only if $(\eta _1,\eta _2)$, $(\eta _1,\eta _3)$ and $(\eta _2,\eta _3)$ are taut contact circles.

A difference, with respect to the taut contact 2-spheres, is that: in general the 1-forms $\eta _1,\eta _2,\eta _3$ that define a taut contact 2-hyperboloid are not linearly independent. In fact, we have the following.

Example 5.3

On the torus ${\mathbb {T}}^3$ the contact 1-forms

$$\begin{aligned} \eta _1= (\cos z)dx - (\sin z) dy, \quad \eta _2= (\cos z)dx + (\sin z) dy \, \text { and } \, \eta _3= (\sin z) dx - (\cos z)dy, \end{aligned}$$

are linearly dependent and define a taut contact 2-hyperboloid. More precisely: $(\eta _1,\eta _2)$, $(\eta _1,\eta _3)$ are taut contact hyperbola, and $(\eta _2,\eta _3)$ is a taut contact circle.

Now, we give the following

Proposition 5.4

Let $(\eta _1,\eta _2,\eta _3)$ be a coframe of contact 1-forms on a three-manifold M. Denote by $(e_1,e_2,e_3)$ the frame dual of $(\eta _1,\eta _2,\eta _3)$. Then, $(\eta _1,\eta _2,\eta _3)$ is a taut contact 2-hyperboloid if and only if there exist three 1-forms $\beta _1,\beta _2,\beta _3$ and a nonzero smooth function $\lambda$ such that

$$\begin{aligned} \left\{ \begin{array}{cc} d\eta _1 = \beta _1\wedge \eta _1 +\lambda \eta _2\wedge \eta _3,\\ d\eta _2 = \beta _2\wedge \eta _2 -\lambda \eta _3\wedge \eta _1,\\ d\eta _3 = \beta _3\wedge \eta _3 -\lambda \eta _1\wedge \eta _2, \end{array} \right. \end{aligned}$$

(5.1)

where the 1-forms $\beta _i$ satisfy

$$\begin{aligned} (*) \quad \quad \quad \quad \quad \quad \quad \beta _i(e_i)=0 \, \text { and } \, \beta _i(e_j)=\beta _k(e_j) \, \text { for any } (i,j,k) \text { permutation of } (1,2,3). \end{aligned}$$

In particular, $\beta _1=\beta _2=\beta _3$ if and only if $\beta _i=0$ for any $i=1,2,3$. If this is the case, then the function $\lambda$ is a constant $\ne 0$.

Proof

Since $(\eta _1,\eta _2,\eta _3)$ is a coframe, we put

$$\begin{aligned} \left\{ \begin{array}{cc} d\eta _1 = f_1 \eta _1\wedge \eta _2 +f_2 \eta _2\wedge \eta _3 + f_3 \eta _3\wedge \eta _1,\\ d\eta _2 = g_1 \eta _1\wedge \eta _2 +g_2 \eta _2\wedge \eta _3 + g_3 \eta _3\wedge \eta _1,\\ d\eta _3 = h_1 \eta _1\wedge \eta _2 +h_2 \eta _2\wedge \eta _3 + h_3 \eta _3\wedge \eta _1, \end{array} \right. \end{aligned}$$

where $f_i,g_i,h_i$ are smooth functions. Then, the conditions

$$\begin{aligned} \eta _1\wedge d\eta _1=-\eta _2\wedge d\eta _2=-\eta _3\wedge d\eta _3 \end{aligned}$$

are equivalent to the conditions $f_2=-g_3=-h_1=\lambda$, where $\lambda$ is a nowhere zero function. Moreover, the conditions

$$\begin{aligned} 0=\eta _1\wedge d\eta _2+\eta _2\wedge d\eta _1= \eta _1\wedge d\eta _3+\eta _3\wedge d\eta _1=\eta _2\wedge d\eta _3+\eta _3\wedge d\eta _2 \end{aligned}$$

are equivalent to the conditions $g_2=-f_3=-\lambda _3, \, h_2=-f_1= \lambda _2, h_3= - g_1=-\lambda _1$. Thus, $(\eta _1,\eta _2,\eta _3)$ is a taut contact 2-hyperboloid if and only if

$$\begin{aligned} \left\{ \begin{array}{cc} d\eta _1 = \beta _1 \wedge \eta _1 + \lambda \eta _2\wedge \eta _3,\\ d\eta _2 = \beta _2\wedge \eta _2 - \lambda \eta _3\wedge \eta _1,\\ d\eta _3 = \beta _3\wedge \eta _3 - \lambda \eta _1\wedge \eta _2, \end{array} \right. \end{aligned}$$

where

$$\begin{aligned} \beta _1= (\lambda _2 \eta _2+\lambda _3 \eta _3),\quad \beta _2=(\lambda _1\eta _1 + \lambda _3 \eta _3) \, \text { and } \, \beta _3=(\lambda _1 \eta _1 + \lambda _2 \eta _2). \end{aligned}$$

Consequently, the 1-forms $\beta _i$ satisfy $(*)$; moreover, $\beta _1=\beta _2=\beta _3$ if and only if $\beta _1=\beta _2=\beta _3=0$. In this case, we have

$$\begin{aligned} 0= d^2\eta _1= d\lambda \wedge \eta _2\wedge \eta _3 + \lambda (d\eta _2)\wedge \eta _3 - \lambda \eta _2\wedge d\eta _3 =d\lambda \wedge \eta _2\wedge \eta _3 \Rightarrow e_1(\lambda )=0. \end{aligned}$$

Analogously, $e_2(\lambda )=e_3(\lambda )=0$, and so $\lambda$ is a constant. $\square$

Corollary 5.5

A simply connected three-manifold M admits a taut contact 2-hyperboloid $(\eta _1,\eta _2,\eta _3)$, with the corresponding Reeb vector fields $(\xi _1,\xi _2,\xi _3)$ that constitute the frame dual of the coframe $(\eta _1,\eta _2,\eta _3)$, if and only if M is the Lie group ${\widetilde{SL}}(2,{\mathbb {R}})$.

Proof

In the Proposition 5.4, the condition $\beta _1=\beta _2=\beta _3$, i.e., $\beta _i=0$ for any $i=1,2,3$, is equivalent to the condition that the vector fields $e_i$ are the Reeb vector fields of the contact forms $\eta _i, i=1,2,3$. Then, if $(\eta _1,\eta _2,\eta _3)$ is a taut contact 2-hyperboloid with the corresponding Reeb vector fields $(\xi _1,\xi _2,\xi _3)$ that constitute the frame dual of the coframe $(\eta _1,\eta _2,\eta _3)$, from (5.1) we get that the 1-forms $\beta _i$ vanish and so the Reeb vector fields satisfy

$$\begin{aligned} {}[\xi _1,\xi _2]= \lambda \xi _3, \quad [\xi _2,\xi _3]= -\lambda \xi _1, \quad [\xi _3,\xi _1]= \lambda \xi _2, \, \lambda =\mathrm{const.}\ne 0. \end{aligned}$$

Therefore, M admits a Lie group structure isomorphic to ${\widetilde{SL}}(2,{\mathbb {R}})$.

Conversely, if we consider the Lie group ${\widetilde{SL}}(2,{\mathbb {R}})$, by using the notations of the Example 3.9, we have the 1-forms $(\eta _1,\eta _2)$ and $(\eta _1,\eta _3)$ are left invariant contact hyperbolas and $(\eta _2,\eta _3)$ is a left invariant taut contact circle. Then, by Proposition 5.1, we get that the 1-forms $(\eta _1,\eta _2,\eta _3)$ define a left invariant taut contact 2-hyperboloid. Moreover, the corresponding Reeb vector fields $(\xi _1,\xi _2,\xi _3)$ constitute the frame dual of the coframe $(\eta _1,\eta _2,\eta _3)$. $\square$

Remark 5.6

In the compact case, taut contact 2-spheres exist only on left quotients of the three-sphere group ${\mathbb {S}}^3=SU(2)$ ( [13], Theorem 1.10). In particular, the torus ${\mathbb {T}}^3$ does not admit a taut contact 2-sphere, however it admits a taut contact 2-hyperboloid.

6 Taut contact hyperbolas and the Chern–Hamilton energy functional

Let $(M,\eta )$ be an oriented compact contact manifold. Denote by ${\mathcal {M}}(\eta )$ the set of all Riemannian metrics associated to the contact form $\eta$ and by ${\mathcal {A}}(\eta )$ the set of all almost CR structures J for which the Levi form is positive definite. The sets ${\mathcal {M}}(\eta )$ and ${\mathcal {A}}(\eta )$ can be identified (cf., for example, Proposition 8 of [25]). Tanno [26] considered the Dirichlet energy

$$\begin{aligned} E(g)=\int _M \Vert \tau \Vert ^2 dv, \quad \tau ={\mathcal {L}}_\xi g, \end{aligned}$$

(6.1)

defined for any $g\in {\mathcal {M}}(\eta )$. Then, he found the critical point condition ( [26], Theorem 5.1)

$$\begin{aligned} \nabla _\xi \tau = 2\tau \varphi , \, \text {equivalently} \ \nabla _\xi h = -2\varphi h. \end{aligned}$$

(6.2)

The Dirichlet energy (6.1) was first studied by Chern and Hamilton [9] for compact contact three-manifolds as a functional defined on the set ${\mathcal {A}}(\eta )$ (there was an error in their calculation of the critical point condition, as was pointed out by Tanno). This functional is known in literature also with the name of Chern–Hamilton energy functional. Moreover, since $Ric(\xi ,\xi ) = 2n - tr h^2= 2n- \Vert \tau \Vert ^2/4$, the functional (6.1) is equivalent to the functional $L(g) = \int _M Ric(\xi ,\xi ) dv$ studied in general dimension, for compact regular contact manifold, by Blair ( [3], Section 10.3). We note that K-contact metrics and Sasakian metrics are trivial critical metrics, besides we note that the critical point condition (6.2) has a tensorial character, so it holds also in the non-compact case. On the other hand, the sphere ${\mathbb {S}}^3$ admits a Sasakian structure, therefore: in general a Sasakian three-manifold fails to admit a taut contact hyperbola.

Next, we show that the critical point condition (6.2) is a sufficient condition for the existence of a taut contact hyperbola on a non-Sasakian contact metric three-manifold. In fact, we have the following.

Theorem 6.1

Let $(M,\eta ,g,\varphi ,\xi )$ be a non-Sasakian contact metric three-manifold, that is, the torsion $\tau \ne 0$ at any point. If the metric g satisfies the critical point condition for the Dirichlet energy functional (6.1), then M admits a taut contact hyperbola.

Proof

Let $(M,\eta ,g,\varphi ,\xi )$ be a non-Sasakian contact metric three-manifold. Let $\{e_1,e_2 =\varphi e_1 ,\xi \}$ be an orthonormal basis of smooth eigenvectors for h with $h\xi =0$, $he_1=\lambda e_1$, $he_2=-\lambda e_2$, $\lambda$ being the positive eigenvalue. Since the three eigenvalues $0,\lambda ,-\lambda$ of h are everywhere distinct, the corresponding line fields are global and by the orientability the basis can be taken to be global. Let $\eta _1$, $\eta _2$ be the 1-forms g-dual to $e_1$ and $e_2$, respectively, and hence $(\eta _1,\eta _2,\eta )$ is a global basis of 1-forms. Using (2.1), we have

$$\begin{aligned} \nabla _{e_1}\xi = -\varphi e_1 -\varphi he_1=-(1+\lambda )e_2 \quad \text {and} \quad \nabla _{e_2}\xi =(1-\lambda )e_1 . \end{aligned}$$

(6.3)

By straightforward computation and using $\nabla _\xi \xi =0$, we get

$$\begin{aligned} \nabla _{\xi }e_1 = a\,e_2 \qquad \text {and}\qquad \nabla _{\xi }e_2 = -a\,e_1 , \end{aligned}$$

(6.4)

where $a=g(\nabla _\xi e_1,e_2)$ is a smooth function. Moreover, $(\nabla _{\xi }h)\xi = 0$ and by using (6.4) we obtain

$$\begin{aligned} (\nabla _{\xi }h)e_1 = \nabla _{\xi }he_1 -h(\nabla _{\xi }e_1) = \xi (\lambda )e_1 +2 a \lambda \,e_2 \end{aligned}$$

and

$$\begin{aligned} (\nabla _{\xi }h) e_2 = (\nabla _{\xi }h)\varphi e_1= -\varphi (\nabla _{\xi }h)e_1= -\xi (\lambda )e_2 +2 a \lambda \, e_1. \end{aligned}$$

Thus,

$$\begin{aligned} (\nabla _{\xi }h)=-2a\,h\varphi +\big (\xi (\lambda )/\lambda \big ) h. \end{aligned}$$

Consequently, since g satisfies the critical point condition (6.2), we have $a =-1$ and $\xi (\lambda )=0$. Thus, (6.4) becomes

$$\begin{aligned} \nabla _{\xi }e_1 = -e_2 \qquad \text {and} \qquad \nabla _{\xi }e_2 = e_1 . \end{aligned}$$

(6.5)

Then, by using (6.3) and (6.5) we have

$$\begin{aligned} \eta _1\wedge d\eta _1(\xi ,e_1,e_2 )= & {} -(d\eta _1)(\xi ,e_2)= {\frac{1}{2}}g(e_1 , \nabla _\xi e_2 -\nabla _{e_2}\xi ) = {\frac{\lambda }{2}}>0, \\ \eta _2\wedge d\eta _2(\xi ,e_1,e_2 )= & {} (d\eta _2)(\xi ,e_1)= -{\frac{1}{2}}g(e_2 , \nabla _\xi e_1 -\nabla _{e_1}\xi ) =-{\frac{\lambda }{2}}<0, \\ \eta _1\wedge d\eta _2(\xi ,e_1,e_2 )= & {} -(d\eta _2)(\xi ,e_2)= {\frac{1}{2}}g(e_2 , \nabla _\xi e_2 -\nabla _{e_2}\xi ) = 0,\\ \eta _2\wedge d\eta _1(\xi ,e_1,e_2 )= & {} (d\eta _1)(\xi ,e_1)= -{\frac{1}{2}}g(e_1 , \nabla _\xi e_1 -\nabla _{e_1}\xi ) = 0, \end{aligned}$$

Therefore, by using Proposition 3.3, the 1-forms $(\eta _1,\eta _2)$ define a taut contact hyperbola. $\square$

Following Y. Eliashberg [11], a contact manifold $(M, \eta )$ is called overtwisted if there exists an embedded disk D in M such that $T_pD = \ker \eta _p$ for all $p\in \partial D$. It is called tight if it is not overtwisted. Moreover, the contact distribution is called universally tight if even its lift to the universal cover of M is tight. Recently, S. Hozoori ( [18], Theorem 1.4) proved, in the compact case, that a conformally Anosov contact three-manifold is universally tight. On the other hand, by proof of Theorem 6.1 we note that the intersection of the bi-contact structure $(\eta _1,\eta _2)$ is given by ${\mathbb {R}}\xi$ and thus, by [21], $(M,\ker \eta )$ is a conformally Anosov contact three-manifold. Therefore, we have the following

Corollary 6.2

Let $(M,\eta ,g,\varphi ,\xi )$ be a compact non-Sasakian contact metric three-manifold. If the metric g is a critical metric for the Dirichlet energy functional, then $(M,\ker \eta )$ is universally tight.

Next, we exhibit an example of compact non-Sasakian contact metric three-manifold with the contact Riemannian metric g critical for the Dirichlet energy (6.1).

Example 6.3

Let (M, g) be a compact 2-dimensional Riemannian manifold of constant sectional curvature $k<0$. By Theorem 7 and Corollary 3 of the paper [1], we get that the unit tangent sphere bundle $T_1M$ admits a family of non-Sasakian contact metric structures $({\tilde{\eta }}_a,{{\tilde{G}}}_a)$, depending on one parameter $a>0$, satisfying the critical point condition (6.2), where the critical metric ${{\tilde{G}}}_a$ is a Riemannian g-natural metric. In particular, for $a=1/4$ and $k=-1$, $({\tilde{\eta }}_a,{{\tilde{G}}}_a)$ is the standard (non-Sasakian) contact Riemannian structure on $T_1M$ satisfying the critical point condition (6.2) ( [3], Th. 10.13, p.208), where ${{\tilde{G}}}_a$ is the classic Sasaki metric ${{\tilde{G}}}_S$. In general, for $a>0$, ${{\tilde{G}}}_a$ is a metric of Kaluza–Klein type, i.e., horizontal and tangential lifts are mutually orthogonal with respect to ${{\tilde{G}}}_a$. To note that the Sasaki metric on $T_1M$, in general, is not Sasakian in the sense of the contact Riemannian geometry.

7 Geometry of bi-contact metric structures

In this section, we study the geometry of a three-manifold determined by the existence of a bi-contact metric structure. The following theorem, which can be considered as a more complete presentation of Theorem 3.6 of [24], will be very useful for this study.

Theorem 7.1

Let $(\eta _1,\eta _2)$ be a pair of contact forms on a three-manifold M, with Reeb vector fields $(\xi _1,\xi _2)$, $\xi _2\ne \pm \xi _1$. Then, the following are equivalent.

(I):

$(\eta _1,\eta _2)$ defines a bi-contact metric structure, i.e., there exists a Riemannian metric g for which $(\eta _1,\eta _2,g)$ is a bi-contact metric structure.

(II):

$(\eta _1,\eta _2)$ is a $(-\varepsilon )$-Cartan structure, i.e.,

$$\begin{aligned} \eta _1\wedge d\eta _2= \eta _2\wedge d\eta _1=0, \quad \eta _2\wedge d\eta _2= -\varepsilon \eta _1\wedge d\eta _1 , \, \varepsilon =\pm 1. \end{aligned}$$

(III):

There exists a unique 1-form $\eta _3$ such that

$$\begin{aligned} d\eta _1=-2 \eta _2\wedge \eta _3, \quad d\eta _2=-2 \varepsilon \eta _1\wedge \eta _3, \quad d\eta _3= 2\kappa \eta _2\wedge \eta _1, \end{aligned}$$

where the smooth function $\kappa =(d\eta _3)(\xi _2,\xi _1)$ satisfies $d\kappa \wedge \eta _1\wedge \eta _2=0$.

Proof

$(I)\Rightarrow (II)$.

Let $(\eta _1,\eta _2,g)$ be a bi-contact metric structure and $(\eta _1,\xi _1,\varphi _1,g)$, $(\eta _2,\xi _2,\varphi _2,g)$ the corresponding contact metric structure with $g(\xi _1,\xi _2)=0$. Consider the vector field $\xi _3=\varphi _1\xi _2=\varepsilon \varphi _2\xi _1$, $\varepsilon =\pm 1$ (cf. Definition 2.1). Then $(\xi _1,\xi _2,\xi _3)$ is a global orthonormal basis and $\eta _i(\xi _j)=\delta _{ij}$ for $i=1,2$ and $j=1,2,3$. Consequently, $\eta _1\wedge d\eta _2 =0=\eta _2\wedge d\eta _1$ and

$$\begin{aligned} (\eta _2\wedge d\eta _2) (\xi _1,\xi _2,\xi _3)=-\eta _2(\xi _2) d\eta _2 (\xi _1,\xi _3)=-g(\xi _1,\varphi _2\xi _3)=\varepsilon ,\, (\eta _1\wedge d\eta _1) (\xi _1,\xi _2,\xi _3)=...=1 \end{aligned}$$

that is,   $\eta _2\wedge d\eta _2 =-\varepsilon \eta _1\wedge d\eta _1$.

$(II)\Rightarrow (III)$.

By using (II), (3.1) and (3.3), we get that $(\eta _1,\eta _2)$ ia taut contact hyperbola (resp. circle) if $\varepsilon =1$ (resp. $\varepsilon =-1$). From Theorem 4.2 (if $\varepsilon =1$) and Theorem 4.1 (if $\varepsilon =-1$), we get that $\xi _1,\xi _2$ are linearly independent, $\eta _1(\xi _2)= \varepsilon \eta _2(\xi _1)$ and $d\eta _1(\xi _2,\cdot ) = \varepsilon d\eta _2(\xi _1,\cdot )\ne 0$ everywhere. Then, $\eta _1\wedge d\eta _2 =0=\eta _2\wedge d\eta _1$ implies $\eta _1(\xi _2)= \eta _2(\xi _1)=0$. Now, consider the 1-form

$$\begin{aligned} \eta _3= d\eta _1(\cdot ,\xi _2) = \varepsilon d\eta _2(\cdot , \xi _1)\ne 0 \, \text { everywhere}. \end{aligned}$$

Then, $\eta _3(\xi _2)= \eta _3(\xi _1)=0$ and there exists a vector field $\xi _3$ such that $\eta _3(\xi _3)=1$, $(\xi _1,\xi _2,\xi _3)$ is a basis and thus $\eta _1\wedge \eta _2\wedge \eta _3$ is a volume form and ($\eta _1\wedge \eta _2,\eta _1\wedge \eta _3,\eta _2\wedge \eta _3$) is a basis of 2-forms. Since

$$\begin{aligned} d\eta _1(\xi _1,\cdot ) =d\eta _2(\xi _2,\cdot )=0, d\eta _1(\xi _2,\xi _3)=-\eta _3(\xi _3)=-1 \text { and } d\eta _2(\xi _1,\xi _3)=-\varepsilon \eta _3(\xi _3)=-\varepsilon , \end{aligned}$$

we get

$$\begin{aligned} d\eta _1=-2 \eta _2\wedge \eta _3, \quad d\eta _2=-2 \varepsilon \eta _1\wedge \eta _3 . \end{aligned}$$

(7.1)

Finally, by using (7.1), we have

$$\begin{aligned} 0=d\eta _2\wedge \eta _3= \eta _2\wedge d\eta _3 \quad \text { and } \quad 0=d\eta _1\wedge \eta _3= \eta _1\wedge d\eta _3 \end{aligned}$$

and hence  $d\eta _3= 2\kappa \, \eta _2\wedge \eta _3$ where $\kappa$ is a smooth function satisfying $d\kappa \wedge \eta _1\wedge \eta _2=0$, that is, $\kappa _3=0$ where we put $d \kappa =\sum _i \kappa _i\eta _i$. The 1-form $\eta _3$ satisfying (III) is unique because from (7.1) one gets that $\eta _3=- d\eta _1(\xi _2,\cdot )= -\varepsilon d\eta _2(\xi _1,\cdot )$.

$(III)\Rightarrow (I)$.

Let $(\xi _1,\xi _2,\xi _3)$ be the triple of vector fields dual to the basis $(\eta _1,\eta _2,\eta _3)$ of 1-forms. We note that $\xi _1,\xi _2$ are necessarily the Reeb vector fields of $\eta _1,\eta _2$, respectively. Moreover by the usual formulae, the dual of the equations of (III) are

$$\begin{aligned} {}[\xi _1,\xi _2]= 2\kappa \,\xi _3, \quad [\xi _2,\xi _3]=2\, \xi _1, \quad [\xi _3,\xi _1]= -2\varepsilon \,\xi _2, \end{aligned}$$

(7.2)

where $\kappa$ is a smooth function. Now, we consider the Riemannian metric g defined by $g(\xi _i,\xi _j)=\delta _{ij}$. Then $\eta _1=g(\xi _1,\cdot )$ and $\eta _2=g(\xi _2,\cdot )$. Moreover, if we define the (1, 1)-tensors $\varphi _1$ and $\varphi _2$ by

$$\begin{aligned} \varphi _1 \xi _1=0, \quad \varphi _1 \xi _2=\xi _3 , \quad \varphi _1 \xi _3=-\xi _2, \quad \varphi _2 \xi _2= 0, \quad \varphi _2 \xi _1=\varepsilon \xi _3, \quad \varphi _2 \xi _3=-\varepsilon \xi _1, \end{aligned}$$

then $(\eta _1,\xi _1, \varphi _1)$ and $(\eta _2,\xi _2, \varphi _2)$ are almost contact structures. Moreover, by using (7.2), we get $d\eta _1 = g(\cdot ,\varphi _1)$ and $d\eta _2 = g(\cdot ,\varphi _2)$, that is , $(\eta _1,\eta _2, g)$ is a bi-contact metric structure on M. $\square$

From Theorem 7.1 follows that for a bi-contact metric structure $(\eta _1,\eta _2, g)$ are uniquely determined the 1-form $\eta _3= d\eta _1(\cdot , \xi _2)$ and the smooth function $\kappa =(d\eta _3)(\xi _2,\xi _1)$. In particular, we deduce

Corollary 7.2

If a 3D Lie group G admits a left invariant bi-contact metric structure, then G is unimodular.

The next example shows that there exist taut contact hyperbolas/circles which do not define $(-\varepsilon )$-Cartan structures.

Example 7.3

Consider on ${\mathbb {R}}^3$ the 1-forms

$$\begin{aligned} \eta _1= (ay +bz) f(x)\,dx +dy \quad \text { and } \quad \eta _2= (\varepsilon by +az) f(x)\,dx +dz \end{aligned}$$

where f(x) is a positive smooth function and $a,b\in {\mathbb {R}}$, $a,b\ne 0$. Then

$$\begin{aligned} d\eta _1= a f(x)\,dy\wedge dx + bf(x)\,dz\wedge dx \quad \text { and } \quad d\eta _2= \varepsilon b f(x)\,dy\wedge dx + af(x)\,dz\wedge dx. \end{aligned}$$

Consequently,

$$\begin{aligned} \eta _1\wedge d\eta _1= bf(x)\,dx\wedge dy\wedge dz, \quad \eta _2\wedge d\eta _2=- \varepsilon \eta _1\wedge d\eta _1 \ne 0 \end{aligned}$$

and

$$\begin{aligned} \eta _1\wedge d\eta _2= -\eta _2\wedge d\eta _1= a f(x)\,dx\wedge dy\wedge dz \ne 0. \end{aligned}$$

So for $\varepsilon =1$, $(\eta _1, \eta _2)$ defines a taut contact hyperbola, and $\varepsilon =-1$, $(\eta _1, \eta _2)$ defines a taut contact circle. In both the cases $(\eta _1, \eta _2)$ does not define a $(-\varepsilon )$-Cartan structure. Besides, by Remark 4.5, we get that on the four manifold ${\mathbb {R}}^3\times {\mathbb {R}}_+$ the corresponding symplectic 2-forms define a symplectic pair for $\varepsilon =1$ and a conformal symplectic couple for $\varepsilon =-1$.

Remark 7.4

An interpretation of the function $\kappa$ in terms of the Webster scalar curvature. Consider the Webster scalar curvature ${\mathcal {W}}$ as defined by Chern and Hamilton [9] in their study on contact Riemannian three-manifolds. If $(M,\eta ,g)$ is a contact Riemannian three-manifold, the Webster scalar curvature is given by ( [9], p.284)

$$\begin{aligned} {\mathcal {W}} = ({1}/{8}) (w - Ric(\xi ,\xi ) + 4), \end{aligned}$$

where w is the usual scalar curvature and $Ric(\xi ,\xi )$ is the Ricci curvature in the direction of the Reeb vector field $\xi$. We note that the generalized Tanaka-Webster scalar curvature ${\hat{w}}$ (cf. [26]) is eight times the Webster scalar curvature ${\mathcal {W}}$ as defined by Chern and Hamilton. Moreover, a compact simply connected regular Sasakian $(2n + 1)$-manifold is a principal ${\mathbb {S}}^1$-bundle over a compact Kaehler manifold B of complex dimension n, and the generalized Tanaka-Webster scalar curvature ${\hat{w}}$ is the scalar curvature of the Kaehler manifold B ( [25] p.26). Of course, in dimension three, B is a Riemann surface and hence the Webster scalar curvature ${\mathcal {W}}$ determines the Gaussian curvature ($4{\mathcal {W}}$) and the Eulero-Poincaré characteristic of B.

Now, let $(\eta _1,\eta _2,g)$ be a bi-contact metric structure. Then, we have the following (cf. [24], p.224):

• if $(\eta _1,\eta _2)$ is a taut contact circle, i.e., $\varepsilon =-1$, the Webster scalar curvatures of $(\eta _1,g)$ and $(\eta _2,g)$ are given by the same function

  $$\begin{aligned} {\mathcal {W}} = (\kappa +1)/2 ; \end{aligned}$$

  (7.3)

• if $(\eta _1,\eta _2)$ is a taut contact hyperbola, i.e., $\varepsilon =+1$, the Webster scalar curvatures of $(\eta _1,g)$ and $(\eta _2,g)$ are given, respectively, by the functions

  $$\begin{aligned} {\mathcal {W}}_1 = (\kappa -1)/2 \quad \text { and } \quad {\mathcal {W}}_2 = -(\kappa +1)/2 . \end{aligned}$$

So, in both the cases the function $\kappa$ determines the Webster curvature and it does not depend on the associated metric g. Therefore, we call $\kappa$ the Webster function of the $(-\varepsilon )$-Cartan structure $(\eta _1,\eta _2)$. If $(\eta _1,\eta _2)$ is a taut contact circle, i.e., a Cartan structure, the Webster function $\kappa$ is invariant for an Euclidean rotation of constant angle (cf. [24]).

Now, we suppose that $(\eta _1,\eta _2)$ is a taut contact hyperbola and consider a hyperbolic rotation of constant angle of $(\eta _1,\eta _2)$, i.e.,

$$\begin{aligned} (\eta _1'=a_1\eta _1+a_2\eta _2, \eta _{2}'=r(a_2\eta _1+a_1\eta _2), r=\pm 1, \text { with } a=(a_1,a_2)\in {\mathbb {H}}_1^1. \end{aligned}$$

Since

$$\begin{aligned} \eta _1'\wedge d\eta _1'= & {} (a_1^2-a_2^2)\eta _1\wedge d\eta _1=\eta _1\wedge d\eta _1 , \quad \eta _2'\wedge d\eta _2' = (a_2^2-a_1^2)\eta _1\wedge d\eta _1= -\eta _1\wedge d\eta _1, \\ \eta _1'\wedge d\eta _2'= & {} r(a_1^2-a_2^2)\eta _1\wedge d\eta _2=0, \quad \eta _2'\wedge d\eta _1' = r(a_2^2-a_1^2)\eta _1\wedge d\eta _2=0, \end{aligned}$$

then $(\eta _1', \eta _{2}')$ is again a $(-1)$-Cartan structure. Moreover, the corresponding Reeb vector fields of the contact forms $(\eta _1', \eta _{2}')$ are $\xi _1'= a_1 \xi _1 - a_2 \xi _2$ and $\xi _2'= -r(a_2 \xi _1 - a_1 \xi _2)$. Consequently,

$$\begin{aligned} \eta _3'=-d\eta _1'(\xi _2',\cdot )= r(a_1d\eta _1+a_2d\eta _2)(a_2 \xi _1 - a_1 \xi _2, \cdot )= r\eta _3=...=-d\eta _2'(\xi _1',\cdot ), \end{aligned}$$

and thus $d\eta _3'= r d\eta _3= 2 r \kappa \eta _2\wedge \eta _1 = ...= 2 \kappa \eta _2'\wedge \eta _1'$. Therefore, the Webster function of the $(-1)$-Cartan structure $(\eta _1',\eta _2')$ is $\kappa '=\kappa$, i.e., for a $(-1)$-Cartan structure, the Webster function $\kappa$ is invariant for a hyperbolic rotation of constant angle.

About the 1-form $\eta _3$ and the Webster function $\kappa$, we have the following

Theorem 7.5

Let $(\eta _1,\eta _2, g)$ be a bi-contact metric structure on the three-manifold M. Then, for the 1-form $\eta _3$ hold the following properties.

(1):

$\eta _3$ is a Killing 1-form (with respect to g) if and only if $(\eta _1,\eta _2)$ is a taut contact circle.

(2):

$\eta _3$ is a contact form if and only if the Webster function $\kappa \ne 0$ everywhere.

(3):

If $\eta _3$ is a contact form, then

$$\begin{aligned} (\eta =-{\bar{\varepsilon }}\eta _3, g_\kappa = \varepsilon {\bar{\varepsilon }} \kappa g+(1-\varepsilon {\bar{\varepsilon }}\kappa )\eta \otimes \eta ), \text { where } {\bar{\varepsilon }}= \varepsilon (sign \kappa ), \end{aligned}$$

is a contact metric structure of the three-manifold M, and it is a Sasakian structure if and only if $(\eta _1,\eta _2)$ is a taut contact circle.

Proof

Let $(\eta _1,\eta _2, g)$ be a bi-contact metric structure on the three-manifold M. Denote by $(\eta _i,\varphi _i,\xi _i, g)$, $i=1,2$, the corresponding contact metric structures with $g(\xi _1,\xi _2)=0$. The 1-form $\eta _3$ defined by (III) of Theorem 7.1 is given by $\eta _3= - d\eta _1(\xi _2,\cdot ) \ne 0$ everywhere, and thus

$$\begin{aligned} \eta _3= d\eta _1(\cdot , \xi _2) = g(\cdot , \varphi _1\xi _2)= g(\xi _3,\cdot ),\quad \hbox {where} \xi _3:= \varphi _1\xi _2= \varepsilon \varphi _2\xi _1. \end{aligned}$$

Moreover, $g(\xi _3, \xi _3)=\eta _3(\xi _3)=g(\varphi _1\xi _2,\varphi _1\xi _2)=1$, $g(\xi _1, \xi _3)=-g(\varphi _1\xi _1,\xi _2)=0$ and $g(\xi _2, \xi _3)=- \varepsilon g(\varphi _2\xi _2,\xi _3)=0$. So $(\xi _1,\xi _2, \xi _3)$ is an orthonormal basis, dual to the basis of 1-forms $(\eta _1,\eta _2, \eta _3)$. Moreover, by proof of Theorem 7.1, the basis $(\xi _1,\xi _2, \xi _3)$ satisfies (7.2). Consequently, the fundamental tensors $h_1=(1/2){\mathcal {L}}_{\xi _1}\varphi _1$ and $h_2=(1/2){\mathcal {L}}_{\xi _2}\varphi _2$ of the contact metric structures $(\eta _1, g)$, $(\eta _2, g)$, respectively, satisfy

$$\begin{aligned} \left\{ \begin{array}{cc} h_1\xi _1 = 0,\quad h_1\xi _2 = (\kappa +\varepsilon )\xi _2, \quad h_1\xi _3 = -(\kappa +\varepsilon )\xi _3 , \\ h_2\xi _1 = \varepsilon (1-\kappa )\xi _1 , \quad h_2\xi _2= 0, \quad h_2\xi _3 = -\varepsilon (1-\kappa )\xi _3 . \end{array} \right. \end{aligned}$$

(7.4)

Now, recall that a 1-form $\eta$ on a Riemannian manifold (M, g) is a Killing form if and only if

$$\begin{aligned} i(X) d\eta = \nabla _X \eta \, \hbox {for any vector field}\, X\, \hbox {on}\, M, \end{aligned}$$

where $\nabla$ is the Levi-Civita connection of the metric g. Then, in our case, by using the notations introduced before, $\eta _3$ is a Killing form if and only if

$$\begin{aligned} d\eta _3(\xi _i,\xi _j) = (\nabla _{\xi _i} \eta _3)\xi _j\, \hbox {for}\, i,j=1,2,3. \end{aligned}$$

By (III) of Theorem 7.1, $d\eta _3=2\kappa \,\eta _2\wedge \eta _1$, thus

$$\begin{aligned} d\eta _3(\xi _i,\xi _j) =0 \hbox { when } i=3 or j=3, \hbox { and } d\eta _3(\xi _1,\xi _2) =- \kappa . \end{aligned}$$

Moreover, we have

$$\begin{aligned} (\nabla _{\xi _i} \eta _3)\xi _j = - g\big (\xi _3,\nabla _{\xi _i}\xi _j\big ). \end{aligned}$$

Since, by using (2.1), (7.4), and (7.2), we get

$$\begin{aligned} \left\{ \begin{array}{cc} \nabla _{\xi _1}\xi _1=0, \quad \nabla _{\xi _2}\xi _1= -(1+\kappa +\varepsilon )\xi _3, \quad \nabla _{\xi _3}\xi _1= (1-\kappa -\varepsilon )\xi _2, \\ \\ \nabla _{\xi _2}\xi _2=0, \quad \nabla _{\xi _1}\xi _2= -(1-\kappa +\varepsilon )\xi _3, \quad \nabla _{\xi _3}\xi _2= (\varepsilon -1+\kappa )\xi _1,\\ \\ \nabla _{\xi _1}\xi _3= (1-\kappa +\varepsilon )\xi _2, \quad \nabla _{\xi _2}\xi _3= (1+\kappa +\varepsilon )\xi _1, \quad \nabla _{\xi _3}\xi _3= 0. \end{array} \right. \end{aligned}$$

(7.5)

Then, we obtain

$$\begin{aligned} (\nabla _{\xi _1} \eta _3)\xi _2 = (1-\kappa +\varepsilon ), \ (\nabla _{\xi _2} \eta _3)\xi _1 = (1+\kappa +\varepsilon ) \ \hbox { and } (\nabla _{\xi _i} \eta _3)\xi _j =0 \hbox { in the other cases.} \end{aligned}$$

Therefore,

$$\begin{aligned} d\eta _3(\xi _i,\xi _j) = (\nabla _{\xi _i} \eta _3)\xi _j \hbox { for } i,j=1,2,3 \, \Longleftrightarrow \, \varepsilon +1=0, \end{aligned}$$

that is, the property (1). Since $\eta _3\wedge d\eta _3=2\kappa \,\eta _3\wedge \eta _2\wedge \eta _1$, we get the property (2).

Now, suppose that $\eta _3$ is a contact form, that is, the Webster function $\kappa \ne 0$ everywhere. Then, $\xi _3=\varphi _1 \xi _2$ is the Reeb vector field of the contact form $\eta _3$. If we define the tensor $\varphi _3$ by

$$\begin{aligned} \varphi _3 \xi _3=0, \varphi _3 \xi _1=-\varepsilon \xi _2, \varphi _3 \xi _2= \varepsilon \xi _1, \end{aligned}$$

since $\eta _3(\xi _i)=\delta _{3i}$, we have $\varphi _3^2=-I+\eta _3\otimes \xi _3$. Consider the tensors

$$\begin{aligned} \eta= & {} -{\bar{\varepsilon }}\eta _3, \quad \varphi =\varphi _3, \quad \xi =-{\bar{\varepsilon }}\xi _3,\\ g_\kappa= & {} \varepsilon {\bar{\varepsilon }} \kappa g+(1-\varepsilon {\bar{\varepsilon }}\kappa )\eta _3\otimes \eta _3 , \, \hbox { where we put } {\bar{\varepsilon }}= sign (\varepsilon \kappa ). \end{aligned}$$

It is not difficult to see that these tensors satisfy $\varphi ^2= -I +\eta \otimes \xi$, $\eta = g_\kappa (\xi , \cdot )$ and $d\eta = g_\kappa (\cdot , \varphi )$. Therefore, $(\eta ,\varphi ,\xi ,g_\kappa )$ is a contact metric structure on the three manifold M. Moreover, this structure is Sasakian if and only if the almost contact metric structure $(\eta ,\varphi ,\xi )$ is normal.

Recall that in dimension three, any almost CR structure is integrable, then by using Theorem 11 of [25] we get that the almost contact structure $(\eta ,\varphi ,\xi )$ is normal (equivalently, the induced almost CR structure is normal) if and only if $\xi$ is a CR Reeb vector field, that is, the tensor ${\mathcal {L}}_{\xi } J$ vanishes, where $J= \varphi _{|\ker \eta }$, $\ker \eta =$span$(\xi _1,\xi _2)$. By using (7.2),

$$\begin{aligned} -{\bar{\varepsilon }}\big ({\mathcal {L}}_{\xi } J\big )\xi _1= & {} \big ({\mathcal {L}}_{\xi _3}\varphi _3\big )\xi _1=[\xi _3,\varphi _3 \xi _1]- \varphi _3[\xi _3, \xi _1]= 2(\varepsilon +1)\xi _1, -{\bar{\varepsilon }}\big ({\mathcal {L}}_{\xi } J\big )\xi _2=\big ({\mathcal {L}}_{\xi _3}\varphi _3\big )\\\xi _2= & {} [\xi _3,\varphi _3 \xi _2]- \varphi _3[\xi _3, \xi _2]= -2(\varepsilon +1)\xi _2 . \end{aligned}$$

Then, $(\xi ,\eta ,\varphi )$ is normal if and only if $\varepsilon +1=0$, that is, $(\eta _1,\eta _2)$ is a taut contact circle. $\square$

Next, we examine the property (3) of Theorem 7.5. Consider the contact metric structure $(\eta =- {\bar{\varepsilon }}\eta _3, g_\kappa = \varepsilon {\bar{\varepsilon }} \kappa g+(1-\varepsilon {\bar{\varepsilon }}\kappa )\eta _3\otimes \eta _3)$, ${\bar{\varepsilon }}= sign (\varepsilon \kappa )$, which is Sasakian if and only if $\varepsilon = -1$. We note that the metric $g_\kappa =g$ if and only if the Webster function $\kappa =\pm 1$.

Now, for $\kappa =\pm 1$, the vector fields $(\xi _1,\xi _2, \xi _3)$ are g-orthonormal and satisfy (7.2), for which ${{\widetilde{M}}}$ has a Lie group structure isomorphic to SU(2) or ${\widetilde{SL}}(2,{\mathbb {R}})$; moreover,

$$\begin{aligned} (\eta _1,\eta _2,g), \quad (\eta _1,-{\bar{\varepsilon }}\eta _3,g), \quad (\eta _2,- {\bar{\varepsilon }}\eta _3,g) \end{aligned}$$

(7.6)

are three left invariant bi-contact metric structures with

$$\begin{aligned} \eta _3\wedge d\eta _3= -\varepsilon \kappa \ \eta _2\wedge d\eta _2= \kappa \ \eta _1\wedge d\eta _1, \quad \eta _i\wedge d\eta _j=0, i\ne j . \end{aligned}$$

So, we distinguish the following cases.

a):

  $(\kappa ,\varepsilon )=(1,-1)$. In this case $(\eta ,\xi ,\varphi ,g_\kappa )=(\eta _3,\xi _3,\varphi _3,g)$ and ${{\widetilde{M}}}$ is the Lie group SU(2). Moreover, the tensors $(\xi _i,\eta _i,\varphi _i)$, $i=1,2,3$, related to the bi-contact metric structures (7.6), satisfy the condition (2.3). Thus, $(\eta _1,\eta _2,\eta _3,g)$ is a contact metric 3-structure. In particular, by Remark 5.2, the triple $(\eta _1,\eta _2,\eta _3)$ defines a taut contact 2-sphere. On the other hand, a contact metric 3-structure is a Sasakian 3-structure (see, for example, [3] p.293) and g is of constant sectional curvature $+1$.

b):

  $(\kappa ,\varepsilon )=(-1,1), (1,1), (-1,-1)$. In this case ${{\widetilde{M}}}$ is the Lie group ${\widetilde{SL}}(2,{\mathbb {R}})$.

$b_1$):

  If $(\kappa ,\varepsilon )=(-1,1)$, then $(\eta ,\xi ,\varphi ,g_\kappa )=(\eta _3,\xi _3,\varphi _3,g)$ is not Sasakian and thus $(\eta _1,\eta _2,\eta _3,g)$ is not a contact metric 3-structure. Moreover, since $\kappa +\varepsilon =0$, from (7.4) we have that $(\eta _1, g)$ is Sasakian. Besides, by using (7.5) a direct computation gives that the Ricci tensor of g is $Ric=-6g+8\eta _1\otimes \eta _1$. Then, if we consider the corresponding Lorentzian-Sasakian structure $(\eta _1, g_L=g-2\eta _1\otimes \eta _1)$, from formula (22) of [25], the corresponding Ricci tensor is given by $Ric_L=Ric+4 g - 4\eta _1\otimes \eta _1=-2 g_L$, and thus $g_L$ is a Lorentzian metric of constant sectional curvature $-1$.

$b_2$):

If $(\kappa ,\varepsilon )=(1,1)$, also in this case $(\eta ,\xi ,\varphi ,g_\kappa )=(-\eta _3,-\xi _3,\varphi _3,g)$ is not Sasakian and thus $(\eta _1,\eta _2,-\eta _3,g)$ is not a contact metric 3-structure. Moreover, since $\kappa =\varepsilon$, from (7.4) we have that $(\eta _2, g)$ is Sasakian. Then, as in the case $b_1)$, we get that $(\eta _2, g_L=g-2\eta _2\otimes \eta _2)$ is a Lorentzian-Sasakian structure with $g_L$ Lorentzian metric of constant sectional curvature $-1$.

$b_3$):

If $(\kappa ,\varepsilon )=(-1,-1)$, the structures $(\eta _1, g)$,$(\eta _2, g)$ are not Sasakian, thus $(\eta _1,\eta _2,-\eta _3,g)$ is not a contact metric 3-structure, but the structure $(\eta ,\xi ,\varphi ,g_\kappa )$ =$(-\eta _3,-\xi _3,\varphi _3,g)$ is Sasakian. Then, as in the case $b_1)$, we get that $(-\eta _3, g_L=g-2\eta _3\otimes \eta _3)$ is a Lorentzian-Sasakian structure with $g_L$ Lorentzian metric of constant sectional curvature $-1$.

Finally, in all the cases $b_i)$, $i=1,2,3$, from Proposition 5.1, follows that the 1-forms $\eta _1,\eta _2,\eta _3$ define a taut contact 2-hyperboloid.

Summing up, we get

Corollary 7.6

Let $(\eta _1,\eta _2,g)$ be a bi-contact metric structure on the three-manifold M with the Webster function $\kappa \ne 0$ everywhere. Then, the metric $g_\kappa =g$ if and only if $\kappa =$const.$=\pm 1$.

In this case, i.e., for $\kappa =$const.$=\pm 1$,

$$\begin{aligned} (\eta _1,\eta _2,g), (\eta _1,-\kappa \varepsilon \eta _3,g) \text { and } (\eta _2, -\kappa \varepsilon \eta _3,g) \end{aligned}$$

are three left invariant bi-contact metric structures and ${{\widetilde{M}}}$ is either SU(2) or ${\widetilde{SL}}(2,{\mathbb {R}})$.

More precisely, we have the following.

• If $(\kappa ,\varepsilon )=(1,-1)$, ${{\widetilde{M}}}$ is SU(2) and $(\eta _1,\eta _2,\eta _3,g)$ is a 3-Sasakian structure on it, where $(\eta _1,\eta _2,\eta _3)$ is a taut contact 2-sphere and g is of constant sectional curvature $+1$.

• If $(\kappa ,\varepsilon )=(-1,1), (1,1),(-1,-1)$, ${{\widetilde{M}}}$ is ${\widetilde{SL}}(2,{\mathbb {R}})$ and the 1-forms $\eta _1,\eta _2,\eta _3$ define a taut contact 2-hyperboloid. Besides, for $(\kappa ,\varepsilon )=(-1,-1)$ (resp. $(\kappa ,\varepsilon )=(-1,1), (1,1)$) the structure $(-\eta _3,g_L=g- 2\eta _3\otimes \eta _3)$ (resp. $(\eta _1,g_L=g- 2\eta _1\otimes \eta _1)$, $(\eta _2,g_L=g- 2\eta _2\otimes \eta _2)$) is a Lorentzian-Sasakian structure with $g_L$ Lorentzian metric of constant sectional curvature $-1$.

Remark 7.7

From Corollary 7.6 we get that the Lie group ${\widetilde{SL}}(2,{\mathbb {R}})$ admits three left invariant bi-contact metric structures, with the same associated metric, which do not define a contact metric 3-structure.

8 Generalized Finsler structures and bi-contact metric structures

The main purpose of this section is to see how the taut contact hyperbolas are related to generalized Finsler structures, and construct examples of bi-contact metric structures $(\eta _1,\eta _2,g)$ with $(\eta _1,\eta _2)$ taut contact hyperbola. On the other hand, in [24], we posed the question to find examples (if there exist) of bi-contact metric structures $(\eta _1,\eta _2,g)$ on 3-manifolds which are not homogeneous, and thus with the Webster function $\kappa$ non-constant, where the 1-forms $(\eta _1,\eta _2)$ satisfy the conditions that define a taut contact hyperbola. So, by this study we give, in particular, a positive answer to this question (see Example 8.2).

Let M be a three-manifold. Following R. Bryant [5, 6], a coframe $(\omega _1,\omega _2,\omega _3)$ on M is said to be a (I, J, K)-generalized Finsler structure if it satisfies the following structure equations

$$\begin{aligned} \left\{ \begin{array}{cc} d\omega _1 = -\omega _2\wedge \omega _3 , \\ d\omega _2 = \omega _1\wedge \omega _3 + I \omega _3\wedge \omega _2 , \\ d\omega _3 = - K \omega _1\wedge \omega _2 - J \omega _2\wedge \omega _3 , \\ \end{array} \right. \end{aligned}$$

(8.1)

where (I, J, K) are smooth functions on M, known as the main scalar, the Landsberg curvature and the flag curvature, respectively.

We note that if $(\omega _1,\omega _2,\omega _3)$ is a (I, J, K)-generalized Finsler structure, then $(\omega _1,-\omega _2,-\omega _3)$ is a $(-I,-J,K)$-generalized Finsler structure. As remarked in [5] and [6], the difference between the notions of Finsler structure and generalized Finsler structure is global in nature, that is, any generalized Finsler structure is locally diffeomorphic to a Finsler structure, hence M can be realized locally as the unit sphere bundle of a Finsler surface (N, F) in such a way that the given coframing is the canonical coframing induced on M by the (local) Finsler structure F.

In the sequel, for a smooth function f on M equipped with a generalized Finsler structure, we put $df= \sum ^{3}_{i=1}f_i\,\omega _i$. Computing the exterior derivative of the structure equations (8.1), one gets the so called Bianchi identities (cf. [5], Section 1; [6], Section 2.2)

$$\begin{aligned} I_1=J, \qquad J_1=-K_3-K I . \end{aligned}$$

(8.2)

In particular, $I=$ const. implies $J=0$;   $J=$const. and $K=$const.$\ne 0$ imply $I=J=0$. When $I=J=0$, the generalized Finsler structure is locally a Riemannian structure.

Denote by $\Omega$ the volume form $\omega _1\wedge \omega _2\wedge \omega _3$. From (8.1), a simply computation gives

$$\begin{aligned}&\omega _2\wedge d\omega _2= \omega _1\wedge d\omega _1= - \Omega ,\quad \omega _3\wedge d\omega _3= K \omega _1\wedge d\omega _1= -K\Omega ,\quad \omega _1\wedge d\omega _2=- I\,\Omega ,\\&\omega _2\wedge d\omega _1=0,\quad \omega _1\wedge d\omega _3=- J\,\Omega , \quad \omega _3\wedge d\omega _1=0, \quad \omega _2\wedge d\omega _3= \omega _3\wedge d\omega _2=0. \end{aligned}$$

Then, from Theorem 7.1, we get

Proposition 8.1

Let $(\omega _1,\omega _2,\omega _3)$ be a (I, J, K)-generalized Finsler structure on a three-manifold M. Then

a):

  $(\omega _1,\omega _2)$ defines a bi-contact metric structure   if and only if   $\varepsilon =-1$, $I=J=0$.

b):

  $(\omega _1,\omega _3)$ defines a bi-contact metric structure   if and only if   $K=-\varepsilon$, $J=I=0$.

c):

  $(\omega _2,\omega _3)$ defines a bi-contact metric structure   if and only if   $K=-\varepsilon$.

Bryant et al. [6] studied Finsler surfaces of constant flag curvature $K= 0,\pm 1$, with a Killing field.

Next, we discuss separately the cases a), b), c).

• The case a) : $\varepsilon =-1,\, I=J=0$. In this case the triple of 1-forms $(\eta _1,\eta _2,\eta _3)= (1/2)(\omega _1,\omega _2,\omega _3)$ satisfies (III) of Theorem 7.1, where $(\eta _1,\eta _2)$ is a taut contact circle with the Webster function $\kappa =K$. So, if the flag curvature K is $\ne 0$ everywhere, from Theorem 7.5follows that $\omega _3$ is a Killing contact form. Moreover, in this case, we have a generalized Riemann structure in the sense of [7]. Of course, a bi-contact metric structure with $\varepsilon =-1$ defines a $(0,0,\kappa )$ generalized Finsler structure.

  A model for this type of structure is implicitly given in [24]. More precisely, consider the space ${\mathbb {R}}^3(x_1,x_2,t)$, a smooth function $\sigma =\sigma (x_1,x_2)$ and put $\sigma _1= \partial \sigma /\partial x_1$, $\sigma _2= \partial \sigma /\partial x_2$, $\sigma _{11}= \partial ^2\sigma /\partial x_1^2$ and $\sigma _{22}= \partial ^2\sigma /\partial x_2^2$. Then, the 1-forms

  $$\begin{aligned} \omega _1&= {e^{\sigma }}\big ((\cos t)dx_1 + ({\sin t})dx_2\big ),\\ \omega _2&= {e^{\sigma }}\big (-({\sin t})dx_1 + ({\cos t})dx_2\big ),\\ \omega _3&= -\sigma _2 d x_1 +{\sigma _1} d x_2 + d t, \end{aligned}$$

define a coframe on ${\mathbb {R}}^3(x_1,x_2,t)$. Moreover, they satisfy the structure equations (8.1) of a generalized Finsler structure with $I=J=0$ and $K =- e^{-2\sigma }(\sigma _{11}+ \sigma _{22})$. We note that if (N, G) is a Riemannian surface, using isothermal local coordinates $(x_1,x_2)$ on N, the Riemannian metric G is given by $G = e^{2\sigma }(dx_1^2+dx_2^2)$ and, in terms of these coordinates, the function $K =- e^{-2\sigma }(\sigma _{11}+ \sigma _{22})$ is its Gaussian curvature.

• In the case b): $K=-\varepsilon , I=J=0$, the triple of 1-forms $(\eta _1,\eta _2,\eta _3)= (1/2)(\omega _1,\omega _3, -\omega _2)$ satisfies (III) of Theorem 7.1 with $\kappa =1$, where $(\eta _1,\eta _2)$ is a taut contact hyperbola (resp. circle) if $\varepsilon =1$ (resp. $\varepsilon =-1$).

• In the case c): $K=-\varepsilon$, the most interesting case for our study, the structure equations become

  $$\begin{aligned} \left\{ \begin{array}{cc} d\omega _1 = -\omega _2\wedge \omega _3 , \\ d\omega _2 = \omega _1\wedge \omega _3 + I \omega _3\wedge \omega _2 , \\ d\omega _3 = \varepsilon \omega _1\wedge \omega _2 - J \omega _2\wedge \omega _3 . \\ \end{array} \right. \end{aligned}$$

Then, the 1-forms

$$\begin{aligned} \eta _1 = (1/2) \omega _3, \quad \eta _2 = (1/2) \omega _2 \, \ \hbox {and} \, \ \eta _3=(1/2)(\varepsilon \omega _1 + J\omega _3- \varepsilon I\omega _2) \end{aligned}$$

satisfy:

$$\begin{aligned} d\eta _1= & {} (1/2)d\omega _3=(1/2) (\varepsilon \omega _1\wedge \omega _2 - J \omega _2\wedge \omega _3) = 2\eta _3\wedge \eta _2,\\ -2\varepsilon \eta _1\wedge \eta _3= & {} - (1/2) \omega _3\wedge (\omega _1 + J\varepsilon \omega _3 -I\omega _2)= (1/2)d\omega _2= d\eta _2. \end{aligned}$$

Besides,

$$\begin{aligned} 2 d\eta _3&= (\varepsilon d\omega _1 + dJ\wedge \omega _3 + J d\omega _3 - \varepsilon dI\wedge \omega _2 - \varepsilon Id\omega _2) \\&= (J_2 -J^2 + \varepsilon I_3 + \varepsilon I^2 - \varepsilon )\omega _2\wedge \omega _3 +( J_1 -\varepsilon I) \omega _1\wedge \omega _3 +(\varepsilon J -\varepsilon I_1)\omega _1\wedge \omega _2 , \end{aligned}$$

and thus, by using (8.2), we have

$$\begin{aligned} d\eta _3 = 2\kappa \, \eta _2\wedge \eta _1, \hbox { where } \kappa = (J_2 -J^2 + \varepsilon I_3 + \varepsilon I^2 - \varepsilon ) . \end{aligned}$$

Therefore, for $K=-1$ (resp. $K=1$), we get taut contact hyperbolas (resp. circles) with the Webster function $\kappa$, in general, non-constant. If $K=1$ and $\kappa \ne 0$ everywhere, from Theorem 7.5 follows that $\eta _3$ is a Killing contact form.

Next, we give an explicit example of bi-contact metric structure $(\eta _1,\eta _2,g)$ where $(\eta _1,\eta _2)$ is a taut contact hyperbola with the Webster function $\kappa$ non-constant.

Example 8.2

Let M be a connected open subset of ${\mathbb {R}}^3$. On M we consider the following 1-forms

$$\begin{aligned} \omega _1= dx +x dy +dz, \quad \omega _2= -\dfrac{\cosh z}{f(x)} dx +f(x) (\sinh z) dy,\quad \omega _3= -\dfrac{\sinh z}{f(x)} dx +f(x) (\cosh z) dy \end{aligned}$$

where f(x) is a positive smooth function defined on M. These forms satisfy:

$$\begin{aligned} d\omega _1= & {} dx\wedge dy= \omega _3\wedge \omega _2,\\ d\omega _2= & {} -\dfrac{\sinh z}{f(x)} dz\wedge dx +f'(x) (\sinh z) dx\wedge dy +f(x) (\cosh z) dz\wedge dy ,\\ d\omega _3= & {} -\dfrac{\cosh z}{f(x)} dz\wedge dx +f'(x) (\cosh z) dx\wedge dy +f(x) (\sinh z) dz\wedge dy,\\ \omega _1\wedge \omega _3= & {} -\dfrac{\sinh z}{f(x)} dz\wedge dx +f(x) \cosh z\,(dz\wedge dy) +\big (f(x) (\cosh z)+x\dfrac{\sinh z}{f(x)} \big )dx\wedge dy,\\ \omega _1\wedge \omega _2= & {} -\dfrac{\cosh z}{f(x)} dz\wedge dx + f(x) (\sinh z) dz\wedge dy +\big (f(x) (\sinh z)+x\dfrac{\cosh z}{f(x)} \big )dx\wedge dy.\\ \end{aligned}$$

So, one gets

$$\begin{aligned} d\omega _1= \omega _3\wedge \omega _2, \quad d\omega _2=\omega _1\wedge \omega _3 + I \omega _3\wedge \omega _2, \quad d\omega _3= \omega _1\wedge \omega _2 - J\omega _2\wedge \omega _3, \end{aligned}$$

where

$$\begin{aligned} I= (f'- (x/f))\sinh z - f \cosh z \,\, \hbox {and} \,\, J=(f'- (x/f))\cosh z - f \sinh z. \end{aligned}$$

Therefore, $(\omega _1,\omega _2,\omega _3)$ is a $(I,J,-1)$ generalized Finsler structure on M. Then, by the discussion of the case c) of Proposition 8.1, $(\eta _1,\eta _2)= (1/2)(\omega _3, \omega _2)$ is a taut contact hyperbola. In this case, the 1-form $\eta _3$ is given by

$$\begin{aligned} \eta _3= (1/2)\big ( \omega _1 + J \omega _3 -I\omega _2\big )= ... =(1/2)\big ( dz + f'(x) f(x) dy\big ). \end{aligned}$$

Since $2d\eta _3 = (f' f)'(x)\,dx\wedge dy= (f' f)'(x) \omega _3\wedge \omega _2=-4 (f' f)'(x) \eta _2\wedge \eta _1$, we get

$$\begin{aligned} d\eta _3 = 2\kappa \, \eta _2\wedge \eta _1 \end{aligned}$$

where the Webster $\kappa$ function is given by

$$\begin{aligned} \kappa (x) =-(f' f)' (x)= - (1/2)(f^2)''(x). \end{aligned}$$

The frame $(\xi _1,\xi _2,\xi _3)$, dual of the coframe $(\eta _1,\eta _2,\eta _3)$, is given by

$$\begin{aligned} \xi _1= & {} 2\Big (f(x) (\sinh z) \partial _x +\dfrac{\cosh z}{f(x)} \partial _y - f'(x) (\cosh z)\partial _z\Big ),\\ \xi _2= & {} -2\Big (f(x) (\cosh z) \partial _x +\dfrac{\sinh z}{f(x)} \partial _y - f'(x) (\sinh z)\partial _z\Big ), \quad \xi _3=2\partial _z, \end{aligned}$$

where $\xi _1,\xi _2$ are the Reeb vector fields of $\eta _1,\eta _2$.

Then, if g is the Riemannian metric defined by $g(\xi _i,\xi _j)=\delta _{ij}$, $(\eta _1,\eta _2,g)$ is a bi-contact metric structure with the Webster scalar curvatures

$$\begin{aligned} {\mathcal {W}}_1= -\big ((f^2)''(x) +2\big )/4 \quad \, \mathrm{and} \quad \, {\mathcal {W}}_2= \big ((f^2)''(x) -2\big )/4 . \end{aligned}$$

(8.3)

In this example, we have a family of $(I_f,J_f,-1)$ generalized Finsler structures, depending on a function f(x), that define taut contact hyperbolas. In particular, this family contains generalized Finsler structures that define left invariant taut contact hyperbolas on the Lie groups $Sol^3$ and ${\widetilde{SL}}(2,R)$. More precisely,

• Consider $M={\mathbb {R}}^3$ and the $(I_f,J_f,-1)$ generalized Finsler structure corresponding to the function $f(x)=1$. Then, the 1-forms $(\eta _1,\eta _2)= (1/2)(\omega _3, \omega _2)$ and $\eta _3$ are given by

  $$\begin{aligned} \eta _1= (-(\sinh z)dx + (\cosh z) dy)/2 , \quad \eta _2= (-(\cosh z) dx + (\sinh z) dy)/2, \quad \eta _3= (1/2) dz, \end{aligned}$$

  and the dual vector fields

  $$\begin{aligned} \xi _1= & {} 2( (\sinh z) \partial _x + (\cosh z) \partial _y),\quad \xi _2= - 2((\cosh z) \partial _x +(\sinh z) \partial _y),\quad \\ \xi _3= & {} 2\partial _z \end{aligned}$$

  satisfy (3.7), and hence define on ${\mathbb {R}}^3$ a Lie group structure isomorphic to $Sol^3$.

• Consider $M={\mathbb {R}}^3_{+}= \{(x,y,z)\in {\mathbb {R}}^3: x>0\}$ and the $(-1,I_f,J_f)$ generalized Finsler structure corresponding to the function $f(x)=x$. Then, the 1-forms $(\eta _1,\eta _2)= (1/2)(\omega _3, \omega _2)$ and $\eta _3$ are given by

  $$\begin{aligned} \eta _1= & {} \big (-\dfrac{\sinh z}{x} dx + x (\cosh z) dy\big )/2 , \quad \eta _2= \big (-\dfrac{\cosh z}{x} dx + x (\sinh z) dy\big )/2, \quad \\ \eta _3= & {} (1/2)(x dy + dz), \end{aligned}$$

  and the dual vector fields

  $$\begin{aligned} \xi _1= & {} 2\big (x(\sinh z) \partial _x + \dfrac{\cosh z}{x} \partial _y - (\cosh z)\partial _z\big ), \quad \xi _2= - 2\big (x(\cosh z) \partial _x +\dfrac{\sinh z}{x} \partial _y -(\sinh z)\partial _z\big ), \, \\ \xi _3= & {} 2\partial _z, \end{aligned}$$

  satisfy (3.8), and hence define on ${\mathbb {R}}^3_+$ a Lie group structure isomorphic to ${\widetilde{SL}}(2,{\mathbb {R}})$.

Remark 8.3

About the Webster curvature, we recall that every contact structure on a compact orientable three-manifold has a contact form and an associated Riemannian metric whose Webster scalar curvature is either a constant $\le 0$ or is everywhere strictly positive (see the main result of [9]). On the other hand, every compact orientable three-manifold M has a contact structure [19]. Therefore, every compact orientable three-manifold M has a contact Riemannian structure whose Webster scalar curvature is either a constant $\le 0$ or is everywhere strictly positive.

Now, from (8.3), it is not difficult to find a positive function f(x) for which ${\mathcal {W}}_1$ be a strictly negative function and ${\mathcal {W}}_2$ be a strictly positive function. On the other hand, on a three-manifold, a contact Riemannian structure determines a non-degenerate CR structure with the same Webster scalar curvature (cf., for example, [25] p.30). So, we get the following

Proposition 8.4

Any connected open subset of ${\mathbb {R}}^3$ admits a non-degenerate CR structure whose Webster scalar curvature is a strictly negative function and a non-degenerate CR structure whose Webster scalar curvature is a strictly positive function.

Final remark

Of course, it is an open question to give a classification of three-manifolds which admit a taut contact hyperbola. Recall that the homothety class of a taut contact circle is defined by multiplication by the same positive function and by a rotation of constant angle [13]. Similarly, we can define the homothety class of a taut contact hyperbola. If f is a positive smooth function and $(\eta _1,\eta _2)$ a pair of contact forms on a three-manifold M, the contact forms $({\tilde{\eta }}_1= f\eta _1,{\tilde{\eta }}_2=f\eta _2)$ satisfy

$$\begin{aligned} {\tilde{\eta }}_i\wedge d{\tilde{\eta }}_i= f^2 \eta _i\wedge d\eta _i,\, (i=1,2), \qquad {\tilde{\eta }}_1\wedge d{\tilde{\eta }}_2 + {\tilde{\eta }}_2\wedge d{\tilde{\eta }}_1 =f^2 (\eta _1\wedge d\eta _2+ \eta _2\wedge d\eta _1). \end{aligned}$$

Then,

•   $(\eta _1,\eta _2)$ is a taut contact hyperbola if and only if $({\tilde{\eta }}_1,{\tilde{\eta }}_2)$ is a taut contact hyperbola.

  Moreover, if $(\eta _1,\eta _2)$ is a taut contact hyperbola and $(\eta _1',\eta _2')$ is obtained from $(\eta _1,\eta _2)$ by a hyperbolic rotations of constant angle, it is not difficult to see that

•   $(\eta _1,\eta _2)$ is a taut contact hyperbola if and only if $(\eta _1',\eta _2')$ is a taut contact hyperbola.

This suggests to define the homothety class of a taut contact hyperbola by multiplication by the same positive function and by a hyperbolic rotation of constant angle. Hence, to classify taut contact hyperbolas is equivalent to classify their homothety classes.