A variational characterization of profile curves of invariant linear Weingarten surfaces

Differential Geometry and its Applications 68 (2020) 101564

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A variational characterization of profile curves of invariant linear Weingarten surfaces A. Pámpano Deparment of Mathematics, Faculty of Science and Technology, University of the Basque Country, Bilbao, Spain

a r t i c l e

i n f o

Article history: Received 31 March 2019 Received in revised form 24 August 2019 Accepted 17 September 2019 Available online xxxx Communicated by E. Garcia-Rio MSC: primary 53C42 secondary 53C50, 53C17 Keywords: Constant Gaussian curvature surfaces Extremal curves Linear Weingarten surfaces Sub-Riemannian geodesics Total curvature type energy Unit tangent bundles

a b s t r a c t We show that the profile curve of any invariant linear Weingarten (LW) surface of a semi-Riemannian 3-space form, Mr3 (ρ), is an extremal curve for a curvature energy. Moreover, we also give a construction procedure of LW surfaces from extremal curves of this energy, which allows us to understand invariant LW surfaces as binormal evolution surfaces with prescribed velocity. In particular, if the profile curve is planar, the variational problem depends on a total curvature type energy. We use this characterization to give a new approach to locally get all invariant constant Gaussian curvature surfaces whose profile curve is planar. Indeed, we also prove that the orbits of these surfaces are critical curves of the same energy acting on a different space of curves. Finally, restricting ourselves to the Riemannian case, M 3 (ρ), we study the existence of closed rotational surfaces with constant Gaussian curvature and we see that the planar profile curves of these rotational surfaces of M 3 (ρ) are just the horizontal projections of sub-Riemannian geodesics of semiRiemannian unit tangent bundles. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The concept of Gaussian curvature of a surface was named after Carl Friedrich Gauss, who published his Theorema Egregium in 1827. This theorem states that the Gaussian curvature is an intrinsic measure of curvature, that is, it can be computed only with the aid of distances that can be measured on the surface. Indeed, it does not depend on the way the surface is isometrically immersed in an ambient space. On the other hand, for an immersed surface, the most important extrinsic invariant is, probably, the mean curvature. This curvature was first coined by Marie-Sophie Germain in 1831, who applied the total mean curvature to describe elastic shells, although it was previously used by Jean Baptiste Meusnier to characterize surfaces that locally minimize area. E-mail address: [email protected]. https://doi.org/10.1016/j.difgeo.2019.101564 0926-2245/© 2019 Elsevier B.V. All rights reserved.

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A. Pámpano / Differential Geometry and its Applications 68 (2020) 101564

The family of surfaces with constant mean curvature (CMC) have played a major role in Mathematics, with a special impact on Analysis and Differential Geometry. In many cases, the mean curvature gives us enough information to understand the extrinsic geometry of an immersed surface into the ambient space. This has stimulated the interest of many authors in studying CMC surfaces along the centuries. In particular, the rich symmetry of CMC invariant surfaces makes them ideal for modeling physical systems. Thus, there have been many works in the literature studying different properties of them. A nice result about CMC invariant surfaces of R3 is that they belong to a suitable Bour’s family, that is, they are isometric to a given surface of revolution with the same CMC (see, for instance, [8]). In order to solve the problem of finding all surfaces that are isometric to a given surface of revolution, Julius Weingarten introduced a class of surfaces satisfying a relation Υ(H, K) = 0 between the mean curvature, H, and the Gaussian curvature, K, [28]. In the literature, these surfaces are usually referred as Weingarten surfaces and their study occupies an important role in classical Differential Geometry. In this setting, one of the first results is due to Chern [13], who proved that the sphere is the only closed smooth surface of constant Gaussian curvature. In any case, the classification of Weingarten surfaces in the general case is almost completely open today, even though the works of Chern, Hartman, Hopf, Winter and others in the fifties (see, for instance, [12], [19] and [20]). One of the simplest relation Υ(H, K) = 0 is a linear relation of the type, a H + b K = c, where a, b and c are real constants such that a2 + b2 = 0. These surfaces are called linear Weingarten surfaces and some of the most studied examples are given by 1. Constant Mean Curvature Surfaces. They appear when b = 0 and for any c ∈ R. 2. Constant Gaussian Curvature Surfaces. This is the case where a = 0 and c is any real constant. For the linear case, there have been many achievements in last decades. Indeed, linear Weingarten surfaces have also been studied in different ambient spaces (see, for instance, [25] and references therein). Moreover, after assuming some kind of symmetry we refer to [6], [23] and [24] among many others. As a special case, in [3], it was proved that CMC invariant surfaces of both Riemannian and Lorentzian 3-space forms can be constructed, locally, as the binormal evolution of extremal curves of curvature energy functionals which generalize a variational problem studied by Blaschke in R3 , with an appropriate velocity. Motivated by this, throughout this paper we give a variational characterization of profile curves of any invariant linear Weingarten surface of semi-Riemannian 3-space forms, Mr3 (ρ). We begin by introducing the basic facts about the theory of curves and surfaces in 3-space forms in Section §2. Moreover, in this section, we also fix the notation that is going to be used thorough the whole paper. In Section §3, we show that the profile curve of any invariant linear Weingarten surface has either constant curvature or it is an extremal of a curvature energy. Then, using this fact, we give a local classification of these surfaces, proving that invariant linear Weingarten surfaces can be locally constructed by evolving these extremal curves under their binormal flow with a prescribed velocity. Moreover, it turns out that if the profile curve is planar (that is, if its torsion vanishes) invariant linear Weingarten surfaces are locally binormal evolution surfaces with initial curve being extremal, as a curve in the ambient space, for a total curvature type energy Θ∗ (γ) =

  ε ((κ − α)2 + β) ds γ

where the constants α and β are closely related with the constants a, b and c in the linear relation. Here, ε denotes the sign of (κ − α)2 + β.

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As an application of our findings we study invariant constant Gaussian curvature surfaces (α = 0 in above energy) with planar profile curves in Section §4. In this case, we also show that the fibers of these surfaces are also extremal curves of the same energy, when acting on a suitable space of curves immersed in the surface. In particular, restricting ourselves to Riemannian ambient spaces, we have that planarity of filaments implies that surfaces are invariant under the action of a one-parameter group of rotations, that is, these surfaces are rotational constant Gaussian curvature surfaces. In this setting, we analyze flat surfaces using our previous characterization of profile curves. Finally, as an illustration we obtain that the profile curve of a rotational surface with constant Gaussian curvature of the Euclidean 3-space, R3, is critical for Θ∗ with α = 0, which is quite illuminating for all previously proved results. Up to this point, we have obtained some results which are local in nature. However, our variational characterization of profile curves can also be used to get global properties of these rotational surfaces. Indeed, in Section §5 we prove the existence of non-isoparametric closed rotational surfaces with constant Gaussian curvature with genus zero in each Riemannian 3-space form. What is more, using the topological properties of profile curves, we show that in the round 3-sphere, S3 (ρ), there exist non-isoparametric closed rotational surfaces with constant Gaussian curvature having the shape of sufficiently many times pinched rotational torus. We point out that without assuming rotational invariance some nice global results about constant Gaussian curvature surfaces are already known in the literature (see, for instance, [16] and references therein). We end up the paper proving the existence of a one-to-one correspondence between the profile curves of constant Gaussian curvature surfaces and geodesics of some unit tangent bundles in Section §6. This correspondence is proved by comparing the length functional of some particular sub-Riemannian geodesics and the total curvature type energy that describes the profile curves. That is, it comes as an intrinsic consequence of the variational characterization explained in previous paragraphs. A nice application of this correspondence can be found in [4] (see also [7]) where these sub-Riemannian geodesics of the unit tangent bundle of the plane are used for image reconstruction purposes. 2. Preliminaries m Consider the Euclidean semi-space Em endowed with the canonical metric of index ν, ν . That is, R ¯ Let M 3 (ρ), r = 0, 1 be a denoted by ·, ·, and the corresponding Levi-Civita connection, denoted by ∇. r complete, connected, simply connected, semi-Riemannian 3-manifold of index r, with constant sectional curvature ρ (a semi-Riemannian 3-space form). If r = 0, Mo3 (ρ) is a Riemannian 3-space form (usually, simply denoted by M 3 (ρ)) and if r = 1, M13 (ρ) is a Lorentzian 3-space form. Then, Mr3 (ρ), r = 0, 1, can be isometrically immersed in E4ν , the 4-dimensional Euclidean semi-space, in a standard way. In fact, the flat case, Mr3 (ρ) = E3r , ρ = 0, r = 0, 1, corresponds to either the Euclidean R3 or to the Minkowski space R31 ≡ L3 . They can be isometrically immersed in L4 = R41 endorsed with the metric

g = dx21 + dx22 + dx23 − dx24 , in an obvious manner; R3 = {(x1 , x2 , x3 , x4 ) ∈ L4 ; x4 = 0} ,

L3 = {(x1 , x2 , x3 , x4 ) ∈ L4 ; x1 = 0} .

When ρ > 0, Mr3 (ρ) corresponds to the round 3-sphere S3 (ρ), if r = 0, and to the de Sitter 3-space S13 (ρ), if r = 1, which are defined by Sr3 (ρ) = {x ∈ E4r ; x, x =

1 }, ρ

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where x = (x1 , x2 , x3 , x4 ). Finally, for ρ < 0 we obtain the hyperbolic 3-space, H3 (ρ) if r = 0, and the anti de Sitter 3-space, H13 (ρ), if r = 1 Hr3 (ρ) = {x ∈ E4r+1 ; x, x =

1 }. ρ

The standard isometric immersions of Mr3 (ρ) into E4ν ([11], p. 20) will be all denoted by i and the induced ¯ and ∇,  respecmetrics also by ·, ·, while the Levi-Civita connections on E4ν and Mr3 (ρ) are denoted by ∇ 3 tively. As usual, the cross product of two vector fields X, Y in Mr (ρ), denoted by X × Y , is defined so that X × Y, Z = det(X, Y, Z) for any other vector field Z of Mr3 (ρ), where det(X, Y, Z) stands for the determinant. If γ : I → Mr3 (ρ) is a smooth immersed curve in Mr3 (ρ), γ  (t) will represent its velocity vector dγ(t) dt 1  and the covariant derivative of a vector field X(t) along γ will be denoted by DX dt (t) := ∇T X(t). A C immersed curve in a semi-Riemannian manifold is spacelike (respectively, timelike; respectively, lightlike) if γ  (t), γ  (t) > 0, ∀t ∈ I (respectively, γ  (t), γ  (t) < 0, ∀t ∈ I; respectively, γ  (t), γ  (t) = 0, ∀t ∈ I). Of course, there exist curves whose causal character changes as t moves along the parameter interval, but this kind of curves will not be considered here. Lightlike vectors are also known as null vectors. A non-null curve can be parametrized by the arc-length, denoted here by s, and this natural parameter is called proper time. We will also denote by T (s) to the unit velocity vector of a curve. Now, for a given isometric immersion of a surface, x : Nν2 → Mr3 (ρ), ν ∈ {0, 1}, we denote by ∇ the LeviCivita connection of the immersion (Nν2 , x). As it is also customary, for a surface Nν2 in any 3-dimensional space form Mr3 (ρ), we require the first fundamental form to be non-degenerate. Take X, Y, Z, W tangent vector fields to Nν2 and choose η a normal vector field to Nν2 in Mr3 (ρ). Then the formulas of Gauss and Weingarten are, respectively  X Y − ρ X, Y  x = ∇X Y + h(X, Y ) − ρ X, Y  x , ¯ XY = ∇ ∇

(1)

 X η = −Aη X + ∇

(2)

⊥ DX η,

where x = i ◦x is the position vector (we will often resort to the standard abuse of notation and identification tricks in the theory of submanifolds), h denotes the second fundamental form of Nν2 in Mr3 (ρ), and D⊥ denotes the connection on the normal bundle of Nν2 . We use the notation Aη for the shape operator. A surface is said to be isoparametric if the shape operator Aη has the same characteristic polynomial at all points of the surface. In the Riemannian case Aη is diagonalizable and it has constant eigenvalues, usually called principal curvatures, with constant algebraic multiplicities. In [10], Cartan classified isoparametric surfaces in Riemannian 3-space forms and showed that they are either totally umbilical or spherical cylinders. In Lorentzian backgrounds, Magid [26] studied isoparametric hypersurfaces in Lorentz-Minkowski space Ln ; Xiao [29] studied isoparametric hypersurfaces in the anti-de Sitter space H1n (ρ), ρ < 0; and Li and Wang [22] studied isoparametric surfaces in the de Sitter space S13 (ρ), ρ > 0.  the Riemann curvature tensors associated to ∇ and ∇,  By using (1) and (2), and denoting by R and R respectively, the following relation holds  R(X, Y )Z = ρ(< Y, Z > X− < X, Z > Y ) ,

(3)

while the equations of Gauss and Codazzi are given respectively by  R(X, Y )Z, W  = R(X, Y )Z, W  − h(X, W ), h(Y, Z) + h(X, Z), h(Y, W ) , (∇h)(X, Y, Z) = (∇h)(Y, X, Z) , where ∇h is defined by

(4) (5)

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⊥ (∇h)(X, Y, Z) = DX h(Y, Z) − h(∇X Y, Z) − h(Y, ∇X Z) .

Now choosing an adapted local orthonormal frame {e1 , e2 , e3 } in Mr3 (ρ) such that the vectors e1 , e2 are tangent to Nν2 and e3 is normal to Nν2 in Mr3 (ρ), the intrinsic Gaussian curvature of Nν2 is given by K = ε1 ε2 R(e1 , e2 )e1 , e2  ,

(6)

where εj =< ej , ej > is the causal character of ej . Moreover, denoting by {ω 1 , ω 2 , ω 3 } the dual frame of {e1 , e2 , e3 }, the Cartan connection forms are defined by  X ei = ∇



εj ωij (X)ej ,

j

for i, j ∈ {1, 2, 3}. Then, ωij = −ωji and h(ei , ej ) = ε3 hij e3 ,

 e e3 , ej >= ω 3 (ei ) , hij = − < ∇ j i

i, j ∈ {1, 2}. The mean curvature vector H of a surface Nν2 isometrically immersed in Mr3 (ρ) is defined by H = while the mean curvature function, H, is defined so that the following equation is verified H = ε3 He3 .

(7)

1 2

trace h

(8)

A surface Nν2 immersed in Mr3 (ρ) is said to be a linear Weingarten surface if the Gaussian curvature K, (6), and the mean curvature H of Nν2 , (8), verify the linear relation aH + bK = c,

(9)

where a, b and c are real constants such that a2 + b2 = 0. On the other hand, an immersed surface Nν2 is said to be an invariant surface if it stays invariant under the action of a one-parameter group of isometries of Mr3(ρ). The one-parameter group of isometries of Mr3 (ρ) is determined by the flow of a Killing vector field of Mr3 (ρ). If the Killing field ξ is null, then Nν2 is isometric to the Minkowski 2-space, L2 . Isometric immersions of L2 into M13 (ρ) have been studied in [14], [15] and [18]. Thus, let’s consider that ξ is a non-null Killing vector field on Mr3 (ρ). Since Mr3 (ρ) is complete, we can consider that the flow of ξ is Gξ = {φt ; t ∈ R}. Therefore, Nν2 is a ξ-invariant surface of Mr3 (ρ) if we have φt (Nν2 ) = Nν2 . However, if ξ changes its causal character, there may exist points such that for all their neighborhoods there are orbits of different causal characters. Thus, throughout this paper we are also going to assume that we are working in any local neighborhood of a point p, such that ξ(p), ξ(p) is non-zero. More precisely, if we call S = Nν2 − {p ∈ Nν2 ; ξ(p), ξ(p) = 0}, then S is a ξ-invariant surface, which can be locally parametrized as x(s, t) := φt (γ(s)) ,

(10)

where γ(s) is a curve in S everywhere orthogonal to the Killing vector field ξ, called profile curve. That is, locally any ξ-invariant surface of Mr3 (ρ) can be described as a surface Sγ with profile curve γ and parametrized by (10). From now on, we will denote by Sγ to any local description of a ξ-invariant surface of Mr3 (ρ) whose profile curve is γ. Notice that the length of the Killing field ξ in these cases is εξ, ξ = εxt , xt  = G2 (s), where ε is the constant causal character of ξ. If the profile curve, γ(s), of a ξ-invariant surface, Sγ , is a non-null curve whose acceleration vanishes (in this case we will say that the curvature vanishes, i.e. κ(s) = 0), then γ is a geodesic in Mr3 (ρ), and

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Sγ is a ruled surface. By ruled surface we refer to a surface of Mr3 (ρ) foliated by geodesics of Mr3 (ρ). In this case, we call δs (t) the integral curves of the field ξ through γ(s) and consider the orthonormal frame {Tγ , Tδ , n} where Tγ and Tδ , are unit tangent fields to γ and δs (t), respectively, and n is determined by the cross product of both. We are going to use 1 for the causal character of γ, 3 for that of δ, and finally, n, n = 2 . Then, by standard computations we obtain the following Frenet type equations,  T Tγ = 0 , ∇ γ

 T Tδ = −2 f (s)n , ∇ γ

 T n = 3 f (s)Tδ , ∇ γ

and also  T Tδ = −1 3 Gs Tγ + 2 h22 n , ∇ δ G

 T n = 2 ∇  T (Tγ × Tδ ) = 1 f Tγ − 2 3 h22 Tδ . ∇ δ δ

Thus, Sγ is completely determined by its first fundamental form, g = 1 ds2 + 3 G2 (s)dt2 , and the second following fundamental form h = −2f (s)G(s)dsdt +h22 G2 (s)dt2 . The functions involved in both fundamental forms are connected by the Gauss-Codazzi equations (4) and (5), which, in this case, can be written as f 2 (s) = (−1)r ρ + 2 3

Gss , G

(11)

Gs f (s) = 0 , G Gs h22 (s) . h22 (s) = G

f  (s) + 2

(12) (13)

On the other hand, if the profile curve, γ(s), is a unit speed non-geodesic smooth curve immersed in DT Mr3 (ρ) with non-null velocity Dγ ds (s) = T (s) and non-null acceleration, ds (s), then γ(s) is a Frenet curve of rank 2 or 3 and the standard Frenet frame along γ(s) is given by {T, N, B}(s), where N and B are the unit normal and unit binormal to the curve, respectively, and B is chosen so that det(T, N, B) = 1. Then the Frenet equations DT (s) = ε2 κ(s)N (s) , ds DN (s) = −ε1 κ(s)T (s) + ε3 τ (s)B(s) , ds DB (s) = −ε2 τ (s)N (s) , ds

(14) (15) (16)

define the curvature, κ(s) (κ(s) > 0 if n = 3), and torsion, τ (s), along γ(s), where εi , 1 ≤ i ≤ 3 are the causal characters of T , N and B, respectively. Notice that {εi , i = 1, 2, 3} are three numbers satisfying i) at most one of them is negative ,

ii) εi = ±1 ,

iii) ε1 ε2 ε3 = (−1)r .

(17)

Now, the following relations hold T = ε1 N × B ,

N = ε2 B × T ,

B = ε3 T × N .

In a semi-Riemannian space form any local geometrical scalar defined along Frenet curves can always be expressed as a function of their curvatures and derivatives. Notice that, even if the rank of γ is 2 (i.e., τ = 0), the binormal B = ε3 T × N is still well defined and above formulas (14)-(16) still make sense when τ = 0. A curve with vanishing torsion, τ = 0, is going to be referred as a planar curve. Moreover, in a

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3-space form, a planar curve can be assumed to lie in a totally geodesic surface Mr2(ρ). On the other hand, if a curve has constant curvature and torsion it is called a Frenet helix. Then, the ξ-invariant surface Sγ admits a warped product metric. Indeed, with respect to the natural parametrization of Sγ , x(s, t) := φt (γ(s)), the induced metric can be written as g = ε1 ds2 + ε3 G2 (s)dt2 ,

(18)

where the function G(s) is the warping function. The Christoffel symbols of the Levi-Civita connection of (18) can be expressed in terms of the metric coefficients gij . Then, since γt (s) are Frenet curves with non-null velocity and acceleration their Frenet frame {T (s, t), N (s, t), B(s, t)} satisfies (14)-(16) and we can choose the following adapted frame on Sγ ; e1 = xs = T ,

e2 =

xt =B, G

e3 = η = T × B = −ε2 N,

(19)

e1 , e2 being tangent to Sγ whilst e3 = η is a unit normal on it. In this case, Sγ is a binormal evolution surface with velocity G(s) (for more details see, [2] and [17]). Observe that, since the evolution is by isometries, then the curvature and torsion of γt (s), and the velocity G, are independent of time t; κ(s), τ (s) and G(s). Then, after long straightforward computations using the Gauss and Weingarten formulae, (1) and (2), and the simplicity of the curvature tensor in Mr3 (ρ), (3) (for details see [11]), the second fundamental form (7) of the natural parametrization (10) admits the following expression h = −ε2 κ(s)ds2 + 2ε2 τ (s)G(s)dsdt + h22 (s)G2 (s)dt2 ,

(20)

1

where G(s) = (ε3 xt , xt ) 2 and h22 (s) is given by (7) (see [17]) h22

1 = κ(s)





Gss (s) ε3 − ε2 τ 2 (s) + ε1 ε3 ρ G(s)

.

(21)

Direct computations using (20) and (21) give that, with respect to the above coordinate system, the mean curvature function, (8), can be written as H=

 1  Gss − G ε1 ε2 κ2 + ε2 ε3 τ 2 − ε1 ρ . 2κG

(22)

Moreover, one can check that from the definition of the Riemannian curvature tensor (see, for instance, [11]), the Gaussian curvature, K, (6), of any ξ-invariant surface Sγ in terms of the natural parametrization, is given by (see also [17]) K(s, t) = −ε1

Gss (s) . G(s)

(23)

3. Characterization as binormal evolution surfaces In this paper, we are going to study ξ-invariant linear Weingarten surfaces of Mr3 (ρ). Typical examples of linear Weingarten surfaces are surfaces with constant mean curvature (b = 0). The profile curve of ξ-invariant constant mean curvature surfaces of Mr3 (ρ) has been characterized in [3] as an extremal curve of a curvature dependent variational problem. Thus, from now on we will assume that b = 0. Then, we can divide (9) by b, to obtain  aH + K =  c.

(24)

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Therefore, from now on we are going to study ξ-invariant surfaces of Mr3 (ρ) verifying the relation (24). Moreover, in order to simplify the notation, we will introduce the parameters α = ε1  a/2 and β = −α2 + c). Notice that β has a clear geometric meaning. In fact, restricting ourselves to R3 , the discriminant ε2 (ρ −  Δ which classifies linear Weingarten surfaces in elliptic, parabolic and hyperbolic (see, for instance, [23]) can be rewritten as Δ = −4 β b2 . By the analogy with the Euclidean case, we are going to call elliptic (respectively, parabolic or hyperbolic) linear Weingarten surfaces to those for which β < 0 (respectively, β = 0 or β > 0). 3.1. Variational characterization of profile curves Linear Weingarten ξ-invariant surfaces of semi-Riemannian 3-space forms, Mr3 (ρ), which satisfy the relation (24) have a nice geometric property. In fact, we have Theorem 3.1. Let ξ be a non-null Killing vector field of Mr3 (ρ) and let’s assume that Sγ is a local description of a ξ-invariant surface of Mr3 (ρ) verifying that  aH + K =  c. Then, locally, Sγ is either a ruled surface or a warped product surface whose profile curve is either a Frenet helix or it verifies the Euler-Lagrange equations

for the functional Θ(γ) = γ P (κ) ds, where P (κ) is a solution of the following ordinary differential equation  2 P˙ (κ) (κ − α) + β +

 = (κ − α) P (κ) , P˙ (κ)3

(25)

for some  ∈ R. Proof. Consider Nν2 ⊂ Mr3 (ρ) an invariant isometrically immersed surface in any semi-Riemannian 3-space form Mr3 (ρ) with local orientation determined by the normal vector η, and verifying the relation (24) between the mean curvature, H, and the Gaussian curvature, K. Then, locally on Sγ , the local description of Nν2 , ∂ we can choose Fermi geodesic coordinates (U, x), x : U → Nν2 , x(s, t), so that ξ = ∂t and s measures the arc-length along geodesics orthogonal to ξ. Thus, calling γ(s) := x(s, 0), we have that x(U ) := Sγ ⊂ Nν2 is parametrized by x(s, t) = φt (γ(s)),

(26)

where φt ∈ Gξ . Observe that γ(s) and all its copies by the action of Gξ , γt (s) := φt (γ(s)), t ∈ R, are arc-length parametrized geodesics of Sγ which are orthogonal to ξ, so that Sγ is foliated by geodesics having κ(s, t) and τ (s, t) as curvature and torsion in Mr3 (ρ). If γt (s) were also geodesics in Mr3 (ρ), ∀t, then Sγ would be foliated by geodesics of the ambient space what would make it a ruled surface. Hence, we assume that the orthogonal curves to the Killing field ξ, γt (s), are not geodesics of the ambient space. Then, since ξ is non-null, γt (s) are Frenet curves with non-null velocity and acceleration and defined over them we have a Frenet frame {T (s, t), N (s, t), B(s, t)} satisfying (14)-(16). At this point, after long straightforward computations, one can see that the Gauss and Weingarten formulae, (1) and (2), and the simplicity of the curvature tensor in Mr3 (ρ), expressed in (3), lead to a PDE system to be satisfied by the position vector, (see, for instance, [2]). The compatibility conditions for this system are given by the Gauss-Codazzi equations, (4) and (5), which in our case, since φt are isometries, can be shown to boil down to 0 = −2Gs τ − τs G ,

(27)

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 0=

1 ε2 ε3 Gss + ε1 ε3 G(κ2 − ε1 ε3 τ 2 + ε2 ρ) κ

9

 − ε1 ε3 κs G ,

(28)

s

where the involved functions depend only on s. Now, the mean curvature, H, and the Gaussian curvature, K, can be computed as above, (22) and (23), so that the relation (24) becomes  2 ( a − 2ε1 κ) Gss − ε1 ε2 G  aκ + ε1 ε3  aτ 2 − ε2  aρ + 2ε1 ε2  cκ = 0 ,

(29)

for constants  a,  c ∈ R. Now, assume first that γ has constant curvature κ(s) = α with α = 0 in Mr3 (ρ). We combine (28) and (29) to obtain  2

τ (s) = ε1 ε3



 a2 − + ε2 (ρ −  c) 4

= ε1 ε3 β .

That is, the torsion of γ must be constant too. Thus, γ is a Frenet helix. Moreover, if τ = 0 (i.e. β = 0), equation (28) reduces to Gss (s) + ε1 (2ρ −  c) G(s) = λ ,

(30)

for some real constant λ. By a simple manipulation we can check that, if λ = 0, then the Gaussian curvature of Sγ is constant, K = 2ρ −  c, and therefore from (22), H = −ε1 ε2 α and Sγ is isoparametric. In case that λ = 0, we can solve (30), obtaining three different cases, c = 2ρ, then 1. If  λ 2 s + μs + η , 2

(31)

λ + μ cos As + η sin As , A2

(32)

λ  + η sinh As  , + μ cosh As 2 A

(33)

G(s) = where μ, η ∈ R. 2. If ε1 (2ρ −  c) = A2 > 0, then G(s) = where μ, η ∈ R. 2 < 0, then 3. Finally, if ε1 (2ρ −  c) = −A G(s) = − where, again, μ, η ∈ R.

On the other hand, if τ = 0, from equation (27), G must be constant and Sγ is flat. Furthermore, H, (22) is also constant, so we are dealing with a flat isoparametric surface. Consider now that the curvature of γ is constant and that κ = 0, α. If G(s) is constant, then from (27) we obtain that τ (s) is also constant and γ is a Frenet helix. Furthermore, equation (23) tells us that Sγ is flat and using now (22) to compute the mean curvature H of Sγ we get that H is also constant. That is, Sγ is a flat isoparametric surface. Now, if G(s) is not constant, we can integrate (27) to obtain G2 (s)τ (s) = e for some e ∈ R. Notice that if e = 0, then τ = 0 and from (29) we have that K is constant. Moreover, to compute H, above argument can be applied. Thus, we get that Sγ is isoparametric. Therefore, let’s assume

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that e = 0. In this case, combining equations (28) and (29) together with the expression of the torsion we get that G must be constant, what contradicts our assumption. Finally, suppose that κ is not constant. Locally, by the Inverse Function Theorem we can suppose that s is a function of κ, and, calling G(κ) = P˙ (κ), where the upper dot denotes derivative with respect to κ, we have that (27), (28) and (29) can be expressed in the following way d ˙2 (P τ ) = 0 , ds P˙ss + ε1 ε2 P˙ κ2 − ε2 ε3 P˙ τ 2 + ε1 ρP˙ − ε1 ε2 κP − ε1 ε2 λκ = 0 ,  2 ( a − 2ε1 κ) P˙ss − ε1 ε2 P˙  aτ 2 − ε2  aρ + 2ε1 ε2  cκ = 0 , aκ + ε1 ε3 

(34) (35) (36)

for some λ ∈ R. Now, equations (34) and (35) are the Euler-Lagrange equations for γ (P (κ) + λ)ds in Mr3 (ρ) (see equations (50)-(51) in Section §3). Moreover, substituting them in equation (36) and using the parameters α and β we obtain (25), as desired. 2 Previous theorem states the variational characterization of profile curves of ξ-invariant linear Weingarten surfaces of Mr3 (ρ). Moreover, its proof can also be used to obtain a classification of non-ruled ξ-invariant linear Weingarten surfaces. However, for the sake of completeness, we also need to study ruled linear Weingarten surfaces. For this particular case we have Proposition 3.2. Let Sγ be a local description of a ξ-invariant ruled linear Weingarten surface of Mr3 (ρ), where ξ is a non-null Killing vector field orthogonal to the rulings. Then, Sγ must have constant Gaussian curvature. Proof. As in previous theorem, let’s consider that γ(s) (and therefore, all its copies γt (s) for all t ∈ R) is arc-length parametrized geodesic of Sγ . From the statement of the proposition, γ(s) is also a geodesic of Mr3 (ρ), and then, κ(s) = 0. That is, we can consider the associated Gauss-Codazzi equations (11)-(13). Let’s assume first that the function h22 (s) is identically zero, then H = 0. Moreover, from (24), Sγ is a minimal isoparametric surface. What is more, using equation (11), we see that f = 0 and, therefore, Sγ is a totally geodesic surface. Now, we consider a constant non-zero h22 . It is easy to check that (13) implies that G(s) must be constant, and therefore the surface is flat. Furthermore, as h22 is constant, Sγ has also constant mean curvature. Finally, we consider the case where h22 (s) is a non-constant function. In this case, Gauss-Codazzi equations can be integrated once h22 (s) = λ G(s) μ f (s) = 2 , G (s) Gss (s) μ2 = 2 3 4 − 1 ρ , G(s) G (s)

(37) (38) (39)

where λ = 0 and μ ∈ R. Moreover, we can rewrite equation (24) as  a Gss (s) h22 (s) − 1 = c. 2 G(s)

(40)

Thus, combining (39) with (40), we get that  a = 0, μ = 0 and  c = ρ, since G(s) cannot be constant. Therefore, we have that K = ρ, which finishes the proof. However, what is more, G(s) can be solved obtaining,

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11

1. If ρ = 0, then G(s) = ν s +  ,

(41)

G(s) = ν cos As +  sin As ,

(42)

 +  sinh As  , G(s) = ν cosh As

(43)

where ν = 0 and  ∈ R. 2. If 1 ρ = A2 > 0, then

where ν,  ∈ R. 2 < 0, then 3. Finally, if 1 ρ = −A

where, again, ν,  ∈ R. And, thus, these ruled surfaces are completely determined by the following fundamental forms; g = 1 ds2 + 3 G2 (s)dt2 and h = λG3 (s)dt2 for some λ > 0. 2 More precisely, if the curvature κ(s), of the profile curve, γ(s), is constant, we have from Theorem 3.1 and Proposition 3.2 that γ is either a geodesic or a Frenet helix. In particular, we obtain the following classification of the surfaces Sγ ; 1. If κ = 0, Sγ is a ruled invariant linear Weingarten surface. In this case we have, • Totally geodesic surfaces, if h22 = 0. • Flat isoparametric surfaces, if h22 is a non-zero constant. • Surfaces with K = ρ determined by the functions G(s) given in (41)-(43), if h22 is non-constant. 2. If κ = α with α = 0, then we have two different options depending on the torsion, τ , of γ, • Flat isoparametric surfaces when τ = 0 (this case appears, if and only if,  = 0). • Whenever τ = 0 (that is, if and only if, β =  = 0) we obtain either isoparametric surfaces or warped product surfaces with non-constant Gaussian curvature. For the warping function G(s) of these surfaces, there are three different possibilities, (31), (32) and (33). Notice that in R3 these surfaces are just the usual tori of revolution. 3. For other constant options of κ, G must also be constant and then, from (23), we obtain flat isoparametric surfaces. On the other hand, if the curvature of the profile curve, κ(s), is not constant, then from Theorem 3.1 we have that the profile curve, γ, verifies the Euler-Lagrange equations for a suitable curvature energy. It turns out that there exist a natural way of constructing invariant surfaces of semi-Riemannian 3-space forms by evolving extremal curves under the binormal flow, [2] and [17]. Indeed, using this binormal evolution procedure, in the following subsection we are going to show that these binormal evolution surfaces are invariant linear Weingarten. This fact allows us to understand all invariant linear Weingarten surfaces, Sγ , whose profile curve has non-constant curvature as binormal evolution surfaces. 3.2. Binormal evolution of extremal curves As we will see throughout this section, the characterization of the profile curves of ξ-invariant linear Weingarten surfaces of semi-Riemannian 3-space forms introduced in Theorem 3.1 can also be used to construct invariant surfaces that verify the relation (24). In fact, let’s assume from now on that all our

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12

curves satisfy that γ  (t) and curvature energy functional

Dγ  dt (t)

are not lightlike vectors along the curve, and let’s consider the following  Θ(γ) :=

L P (κ) =

γ

P (κ(s)) ds ,

(44)

0

where P (κ) is a smooth function in an adequate domain and where, as usual, the arc-length or natural parameter is represented by s ∈ [0, L], L being the length of γ. Then, we consider Θ acting on the following spaces of curves, satisfying given boundary conditions in (Mr3 (ρ), ·, ·). We shall denote by Ωrρ po p1 the space of smooth immersed curves of Mr3 (ρ), joining two given points of it, that is: 3 Ωrρ po p1 = {ι : [0, 1] → Mr (ρ) ; ι(i) = pi , i ∈ {0, 1},

dι (t) = 0, ∀t ∈ [0, 1]}, dt

where pi ∈ Mr3 (ρ), i ∈ {0, 1}, are arbitrary given points of Mr3 (ρ). 3 We take Θ acting on Ωrρ po p1 . For a Frenet curve γ : [0, 1] → Mr (ρ) we take a variation of γ, Γ = 3 Γ(t, ζ) : [0, 1] × (−, ) → Mr (ρ) with Γ(t, 0) = γ(t). Associated to this variation we have the vector ∂Γ field W = W (t) = ∂Γ ∂ζ (t, 0) along the curve γ(t). We also write V = V (t, ζ) = ∂t (t, ζ), W = W (t, ζ), v = v(t, ζ) = |V (t, ζ)|, T = T (t, ζ),... with the obvious meanings and put V (s, ζ), W (s, ζ),... for the corresponding reparametrizations by arc-length. Then, the following general formulas for the variations of v, κ and τ in γ in the direction of the variation vector field, W , can be obtained using standard computations that involve the Frenet equations (14) (see, [3], [21] and references therein)  T W, T , W (v) = ε1 v∇

(45)

 T W, T  + ε1 ρW, N ,  2T W, N  − 2ε1 κ∇ W (κ) = ∇   1 2  T W, T  + ε1 κ∇  T W, B , ∇ W + ε1 ρ W, B − ε1 τ ∇ W (τ ) = ε2 κ T s

(46) (47)

where the subscript s denotes differentiation with respect to the arc-length. Then, after a standard computation involving integration by parts and formulae (45)-(47), the First Variation Formula is obtained d Θ(ς)|ς=o = dς

L L

E(γ), W ds + B [W, γ]0 , 0

L

with E(γ), B [W, γ]0 denoting the Euler-Lagrange operator and boundary term, respectively. These are given by DJ DJ + R(K, T )T = + ε1 ρ K , ds ds L

DW L  − J , W  , B [W, γ]0 = K, ds 0 E(γ) =

where K(γ) = P˙ (κ) N , J (γ) =

 DK + ε1 2κP˙ (κ) − P (κ) T . ds

(48) (49)

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We will call critical curve or extremal curve to any curve γ ⊂ Ωrρ po p1 such that E(γ) = 0. Notice that this is an abuse of notation, since proper criticality depends on the boundary conditions, as it can be checked from the First Variation Formula. However, under suitable boundary conditions, curves verifying E(γ) = 0 are going to be proper critical curves. Therefore, since for the purposes of this paper we just need to consider curves that verify E(γ) = 0, for the sake of simplicity, from now on, we are going to use the name critical curve to denote any curve γ ⊂ Ωrρ po p1 verifying E(γ) = 0. Now, using the Gauss formula (1) and the Frenet-Serret equations (14)-(16), we can see that E(γ) has no component in T while its normal and binormal components can be expressed in terms of the curvature and torsion. Thus, after long straightforward computations, E(γ) = 0 boils down to  P˙ss + ε1 ε2 P˙ κ2 − ε1 ε3 τ 2 + ε2 ρ − ε1 ε2 κP = 0 ,

(50)

2τ P˙ s + τs P˙ = 0 ,

(51)

which are the Euler-Lagrange equations of the curvature energy functional Θ, (44), acting on Ωrρ po p1 . A vector field W along γ, which infinitesimally preserves unit speed parametrization is said to be a Killing vector field along γ (in the sense of [21]) if γ evolves in the direction of W without changing shape, only position. In other words, the following equations must hold W (v)(s, 0) = W (κ)(s, 0) = W (τ )(s, 0) = 0 ,

(52)

(v = |γ  | = | dγ ds | being the speed of γ) for any variation of γ having W as variation field. Let us define the following vector field, I, along γ I = ε3 T × K , where × denotes the cross product and K is defined in (48). Combining the Frenet equations (14)-(16) and (48), we see that I is given by I = ε3 T × K = P˙ (κ) B .

(53)

It turns out that extremals of Θ, (44), have a naturally associated Killing vector field defined along them, as we summarize in the following proposition (for a proof, see [3] and/or [21]) Proposition 3.3. Assume that γ is an immersed curve in Mr3 (ρ) with non-null velocity and acceleration which is an extremal of Θ, (44). Consider the vector field (53) I = P˙ (κ) B , defined on γ, B being its Frenet binormal vector field. Then I is a Killing vector field along γ. At this point, using an argument similar to that of [21] we can extend I, (53), to a Killing vector field on the whole Mr3 (ρ). Let’s denote it by I again. Since Mr3 (ρ) is complete, we can consider the one-parameter group of isometries determined by the flow of I, {φt ; t ∈ R}, and define the surface Sγ := {φt (γ(s))} obtained as the evolution of γ under the I-flow. Observe that Sγ is an I-invariant surface, which is foliated by congruent copies of γ, γt (s) := φt (γ(s)). Moreover, since φt are isometries of Mr3 (ρ), we have xt (s, t) = P˙ (κ) B(s, t) ,

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κ being the curvature of γ(s), and B(s, t) the unit Frenet binormals of γt (s). Thus, Sγ obtained as the 1 flow evolution of γ, x(s, t) = φt (γ(s)), is a binormal evolution surface with velocity V (s) := (ε3 xt , xt ) 2 = 1 (ε3 I, I) 2 (for more details see, [2] and [17]). Indeed, using equations (22), (23) and (24), we can prove the converse of Theorem 3.1 for these I-invariant surfaces,

Theorem 3.4. Let γ be an extremal curve of the energy Θ(γ) = γ P (κ) ds, (44), where P (κ) is a solution of the ordinary differential equation (25) and let Sγ denote the I-invariant surface in Mr3 (ρ) obtained by evolving γ under the flow of the Killing field I which extends (53) to Mr3 (ρ). Then the Gaussian and mean curvature of Sγ verify (24), that is, Sγ is a linear Weingarten surface. We remind that Theorem 3.4 gives a way of constructing invariant linear Weingarten surfaces of Mr3(ρ) as the binormal evolution surfaces swept out by an extremal of Θ, (44), for any P (κ) solution of (25). 3.3. Particular case of planar profile curves In all the cases described above,  = −ε1 ε3 e2 , where e ∈ R is the constant of integration of the equation (34). Therefore, fixing e = 0, we have that τ = 0. Furthermore, since  also vanishes, the ordinary differential equation (25) can be solved, obtaining P (κ) =



ε ((κ − α)2 + β) ,

(54)

 where ε represents the sign of (κ − α)2 + β . Therefore, from (44), for the torsion-free case we obtain a total curvature type energy functional Θ∗ (γ) =

 

ε ((κ − α)2 + β) ds ,

(55)

γ

acting on the space of curves immersed in Mr2 (ρ)  rρ = {ι : [0, 1] → M 2 (ρ) ; ι(i) = pi , i ∈ {0, 1}, dι (t) = 0, ∀t ∈ [0, 1]}, Ω po p1 r dt

(56)

2

for any arbitrary points pi in Mr2 (ρ). Notice that the case (κ − α) + β = 0 represents a global minimum provided that Θ∗ is acting on the space of curves L1 ([0, L]). Moreover, the fact that τ = 0 implies that the second Euler-Lagrange equation vanishes, and thus, (50) and (51) simplify to 

d2 ε ((κ − α)2 + β) 2 ds

 



κ−α ε ((κ − α)2 + β)

+ ε1 (κ − α) (ε2 ακ + ρ) = ε1 ε2 βκ .

(57)

Now, if we consider that κ(s) is constant, the first term in above equation, (57), disappears. Thus, we have (κ − α)(ακ + ε2 ρ) = βκ. Let’s consider now α = 0, then either the curvature is zero, κ(s) = 0; or  c=0 (that is, β = ε2 ρ). Moreover, if α = 0, the curvature is given by κ(s) = whenever  c2 + 4ε2 α2 ρ ≥ 0.

−ε2  c±

  c2 + 4ε2 α2 ρ , 2α

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On the other hand, when κ(s) is not constant we can obtain a first integral of the Euler-Lagrange equation (57). For this purpose, let’s introduce the following parameters in terms of the sectional curvature ρ, the energy parameters α and β, and the constant of integration d ∈ R ¯b = 2ε1 ε2 αβ ,

a ¯ = ε2 εdβ − ε1 ε2 β 2 ,

c¯ = ε2 εd − ε1 ε2 α2 − ε1 ρ .

(58)

Now, we have

  rρ , (56). Proposition 3.5. Let γ be an extremal for Θ∗ (γ) = γ ε ((κ − α)2 + β) ds, (55), acting on Ω po p1 Assume that γ has non-constant curvature κ(s). Then, with the notation introduced in (58), the function x(s) = κ(s) − α must be a solution of the following differential equation x (s) =

x2 (s) + β εβ

 c¯x2 (s) + ¯bx(s) + a ¯.

(59)

Proof. A direct long computation using the Frenet equations (14)-(16), formulas (48) and (49) for P (κ) given by (54), and the Euler-Lagrange equation (57), shows that the derivative of the function J , J  + ε1 ε2 ε3 ρ I, I along the critical curve is zero. Thus, the equation J , J  + ε1 ε2 ε3 ρ I, I = d,

(60)

with d being a real constant represents a first integral of the Euler-Lagrange equation (57). Now, notice that if β = 0 then, from (57), the curvature κ(s) must be constant. So, since κ is not constant, we can assume β = 0. Then, by substituting (48), (49) and (54) in (60) and simplifying one gets 

dκ ds



2 =

2

(κ − α) + β β2

2



 2 2 2 ε2 εd (κ − α) + β − ε1 ε2 (α (κ − α) − β) − ε1 ρ (κ − α) .

(61)

In order to solve these equations, we make the change of variable, x = κ − α. Then, we see that (61) reduces  2 to β 2 x2s = x2 + β Q(x), where Q(x) = c¯x2 + ¯bx + a ¯ where c¯, ¯b and a ¯ are defined in (58). That is, we obtain the differential equation (59). 2 However, notice that equation (61) (and therefore (59)) only makes sense if Q(x) ≥ 0 what imposes some conditions on the parameters (58). Corollary 3.6. The non-constant curvature κ(s) of an extremal curve of (55) is implicitly given by εs = β

 (x2

dx  + β) c¯x2 + ¯bx + a ¯

(62)

where x(s) = κ(s) − α and the parameters a ¯, ¯b and c¯ have been defined in (58). 4. Application to constant Gaussian curvature surfaces As an application of the characterization introduced in previous paragraphs, throughout this section we are going to study invariant surfaces with constant Gaussian curvature whose profile curve is planar. That is, consider γ a planar (τ = 0) curve of Mr3 (ρ). Then, since its torsion vanishes, it can be assumed to lie fully in a totally geodesic surface, Mr2 (ρ). Moreover, Sγ will represent the binormal evolution surface generated by γ that verifies (24) for  a = 0 (therefore, α is also zero). Then, we obtain

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Proposition 4.1. Let γ be the planar profile curve of an invariant surface with constant Gaussian curvature, then, either the curvature of γ is constant κ(s) = κo ; or it is given by one of the following options; ¯ = 0, then 1. If a √

β e2 c¯ ε s √ κ (s) = . ε − e2 c¯ ε s 2

(63)

2. If a ¯ = β c¯ = 0, then κ2 (s) =

β c¯ s2 . β 2 − c¯ s2

(64)

3. Finally, if a ¯ = 0 and a ¯ = β c¯, the curvature of γ is,   c¯ |s a ¯ β f ε | a¯−β β   , κ2 (s) = c¯ (¯ a − β c¯) − a ¯f ε | a¯−β | s β where f (z) = sin2 z, if

β a ¯−β c¯

> 0; and f (z) is either − sinh2 z, − csch2 z, or, −ez , if

(65)

β a ¯−β c¯

< 0.

Proof. Let Sγ be a local description of a ξ-invariant constant Gaussian curvature surface of Mr3 (ρ), for a non-null Killing vector field ξ. That is, Sγ can be seen as an invariant linear Weingarten surface since it verifies (24) for  a = 0. Then, from Theorem 3.1 we know that the profile curve γ must be critical for (44). Moreover, as γ is planar, the energy (44) becomes (55). Now, applying Corollary 3.6, we have that if the curvature of γ, κ(s), is not constant, then it is implicitly determined by equation (62). However, for constant Gaussian curvature, the parameters of this integral (58), simplify to a ¯ = ε2 εd β − ε1 ε2 β 2 ,

¯b = 0 ,

c¯ = ε2 εd − ε1 ρ .

Now, as noticed before, the quadratic polynomial Q(x) = c¯x2 + a ¯ must be greater or equal zero. That is, there are no solutions when c¯ ≤ 0 and a ¯ ≤ 0. For the other cases, the integral on the right hand side of (62) can be solved, obtaining ¯ = 0 (then, necessarily c¯ > 0), 1. If a εx2 1 √ log 2 . x +β 2β c¯ 2. If a ¯ = β c¯ = 0, x √ . c¯x2 + a ¯ 3. If a ¯ = 0 and a ¯ = β c¯, 1   h |β| |¯ a − β c¯| where h(z) = arcsin z, if

β a ¯−β c¯

    a x ¯ − β c ¯   ,  a ¯  ε(x2 + β)

> 0; or, h(z) is either − arcsinh z, − arccosh z or − log z, if

β a ¯−β c¯

< 0.

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In any case, x(s) = κ(s) can be obtained by inverting the resulting equality of (62), proving the statement. 2 Notice that Proposition 4.1 geometrically describes the profile curve γ of all invariant constant Gaussian curvature surfaces that admit planar generators. Then, since the evolution is made by isometries, we obtain that all leaves of these invariant surfaces are critical for the total curvature type energy Θ∗ , (55), with α = 0, as curves in the ambient space. Furthermore, the orbits of these invariant constant Gaussian curvature surfaces of Mr3 (ρ), Sγ , can also be characterized in terms of the same energy functional Θ∗ , (55) with α = 0. In fact, if all the orbits are critical curves of (55) with α = 0 for a space of curves immersed in the surface Sγ joining two fixed points po and p1 of the surface, Ω∗po p1 , then KSγ = εδ2 β, where the index δ denotes the corresponding geometric objects on the orbits δs . That is, let’s define Ω∗po p1 = {ι : [0, 1] → Sγ ; ι(i) = pi , i ∈ {0, 1},

dι = 0, ∀t ∈ [0, 1]} , dt

where pi are points in Sγ . Then, we can sum up this result in the following proposition Proposition 4.2. Let Sγ be an invariant surface of Mr3 (ρ) all whose orbits are extremal curves of Θ∗ , (55) with α = 0. Then, either every orbit is a geodesic of Sγ (and then Sγ is flat) or it has constant Gaussian curvature KSγ = εδ2 β. Proof. Let’s call δs (v) to the integral curves of the Killing vector field I. Then, κδ is constant on any orbit (see, for instance, [2]). Now, assume that for all s, the orbit δs (v) is an extremal of Θ∗ , (55), with α = 0, acting on Ω∗po p1 . Then, E(δs ) = 0 for all s, where now E represents the Euler-Lagrange operator of Θ∗ with α = 0 acting on Ω∗po p1 . That is, for instance, the curvature R(K, T )T is not necessarily constant. In fact, we have the following equation  0=

d2 ε (κ2δ + β) 2 dv



εκδ  ε (κ2δ + β)



 + εδ1 κδ KSγ − εδ2 β ,

and since the geodesic curvature, κδ , is constant on the orbits we get  κδ (v) KSγ − εδ2 β (s, v) = 0 , on an open set U . Then, either the Gaussian curvature is constant on U , KSγ = εδ2 β; or every orbit δs (v) is a geodesic of Sγ . In the latter case, from (23) and [2], we conclude that Sγ is flat. 2 Moreover, these two properties of constant Gaussian curvature invariant surfaces generated by planar curves can be combined. Then, from Proposition 4.1 and Proposition 4.2, we have c, whose profile Corollary 4.3. Let Sγ ⊂ Mr3 (ρ) be an invariant constant Gaussian curvature surface, K =  curve, γ, is planar. Then, the congruent copies of γ, γt , are critical for Θ∗ , (55), with α = 0 acting on ∗ ∗ Ωrρ po p1 and all the orbits δs are critical for the same energy Θ , (55), with α = 0 acting on Ωpo p1 , if and δ only if, either Sγ is a flat surface ( c = 0), or ρ = (ε2 ε2 + 1) c. Proof. Let’s consider Sγ to be an invariant constant Gaussian curvature surface whose profile curve, γ, is planar. And, let’s assume that the constant Gaussian curvature is K =  c. Then, Sγ verifies the relation (24) for  a = 0, and, therefore, from Theorem 3.1 (see also Proposition 4.1), γ is an extremal of Θ∗ , (55), for α = 0, when acting on Ωrρ po p1 .

18

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Moreover, as every γt is a congruent copy of γ, since γ can be considered to be evolving under isometries, every leave γt is an extremal of Θ∗ , (55), for α = 0, as a curve in Mr3 (ρ). Now, if in addition, we also require that every orbit δs of Sγ is critical for the same energy, but this time acting on Ω∗po p1 , then, by Proposition 4.2, either every orbit is a geodesic of Sγ (and Sγ is flat), or the Gaussian curvature verifies K= c = εδ2 β .  Finally, from the definition of the parameter β, we conclude that ρ = εδ2 ε2 + 1  c.

2

To end up this subsection, observe that if we restrict ourselves to the Riemannian case, M 3 (ρ), invariant surfaces whose profile curve is planar are nothing but rotational surfaces, as it is proved in Proposition 5.3 of [3] (see also [9]). Therefore, from now on, we are going to work in Riemannian backgrounds, in order to study rotational constant Gaussian curvature surfaces. 4.1. Flat rotational surfaces When working in any Riemannian 3-space form, Proposition 4.1 completely characterized and classified rotational constant Gaussian curvature surfaces of M 3 (ρ). In particular, among them, there is a family who has special interest, flat rotational surfaces. Corollary 4.4. Let Sγ be a flat rotational surface of M 3 (ρ), then either Sγ is isoparametric or, 1. There are no more flat rotational surfaces in R3 . 2. In the round 3-sphere, S3 (ρ), there is a one-parameter family of non-isoparametric surfaces, {Sγ }d , where the profile curves γd are described by κ2d (s) =

ρ (d − ρ) s2 , ρ2 − (d − ρ) s2

 ρ ρ for d > ρ. Furthermore, in these cases, s ∈ − √d−ρ , √d−ρ , and, consequently the surfaces Sγ they generate are not complete. 3. In the hyperbolic 3-space, H3 (ρ), there are two one-parameter families of non-isoparametric surfaces, {Sγ }d,ε , where γd are determined by their curvatures κ2d (s) =

ρ (εd − ρ) s2 , − (εd − ρ) s2

ρ2

and, where ε denotes the sign of κ2d (s) + ρ. Moreover, if ε = 1,   then d > ρ and the arc-length parameter ρ ρ s is defined in two symmetric intervals, s ∈ −∞, √d−ρ ∪ − √d−ρ , ∞ , and, therefore the binormal evolution surfaces with these initial conditions are non-complete; and, if ε = −1, then, either d > −ρ ρ ρ , − √−d−ρ . for any s ∈ R, or d < −ρ and s ∈ √−d−ρ Proof. Notice that since the invariant surface Sγ is flat, then  c = 0 and, therefore β = ρ. Furthermore, the parameters a ¯ and c¯ above, simplify to a ¯ = (εd − ρ) ρ ,

c¯ = εd − ρ .

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Thus, it is clear that a ¯ = β c¯. Now, if M 3 (ρ) = R3 , then a ¯ = 0, but β is also zero, so Point 1 of Proposition 4.1 do not give any extra curvature. On the other hand, if ρ = 0, a ¯ = β c¯ = 0 and we are in the second point of Proposition 4.1. Thus, substituting we obtain the curvatures of this corollary. Finally, in order to determine the restriction on the constant of integration d, let’s remind first that κ2 +ρ is always positive in S3 (ρ) and a ¯ and c¯ have the same sign, so we need both of them to be positive, since 2 Q(x) = c¯x + a ¯ must be positive too. This happens when d > ρ. Now, in H3 (ρ), we have the two options for ε. Let’s assume first that ε = 1, that is κ2 + ρ =

ρ2

ρ3 > 0. + (εd − ρ) s2

Thus, either d > −ρ and then s ∈ R, or d < −ρ, and then s is restricted to lie on the interval  ρ √ ρ , − √−d−ρ . On the other hand, if ε = −1, arguing in the same way we conclude that d > ρ −d−ρ   ρ ρ , ∞ . This finishes the proof. 2 and s ∈ −∞, √d−ρ ∪ − √d−ρ

Fig. 1. Stereographic Projection of a Flat Rotational Surface of S3 (ρ) (Left) and its Profile Curve (Right).

Fig. 2. Stereographic Projection of a Flat Rotational Surface of H3 (ρ) (Left) and its Profile Curve (Right).

In Fig. 1 we illustrate the stereographic projection of a flat rotational surface of S3 (ρ) and its profile curve, which is completely determined by the curvature of Point 2 in previous corollary. At the same time, Fig. 2 shows the stereographic projection of a flat rotational surface of H3 (ρ) and its profile curve for ε = −1 and d < −ρ. We recall that helicoidal flat surfaces of the round 3-sphere, S3 (ρ), have recently been studied in [27].

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4.2. Euclidean ambient space Previous results are particularly illuminating in the Euclidean 3-space, R3 , as we explain in the following remark. Remark 4.5. Due to Corollary 4.4 we have that the only flat rotational surfaces in R3 are isoparametric, and therefore, either totally geodesic Euclidean planes or circular cylinders. Moreover, we recall that the case √ κ = −β for negative β determines a sphere, and it is a global minimum of Θ∗ when acting on L1 ([0, L]). On the other hand, by using Proposition 4.1 and the characterization introduced in previous sections, we recover classical results about the families of rotational constant Gaussian curvature surfaces of R3 . 5. Closed rotational surfaces with constant Gaussian curvature As suggested by classical examples, existence of closed surfaces needs a deeper analysis. Recall that rotational surfaces with constant Gaussian curvature have been classified in Sections §3 and §4, proving that they are either isoparametric or binormal evolution surfaces where the initial condition is critical for Θ∗ , (55), with α = 0. Then, for non-isoparametric closed rotational surfaces, first of all, we need that the orbits of the rotation must be closed, that is, Euclidean circles. This only happens when the constant of integration d is positive (see [5]). Then, we need either the profile curve to be closed (in order to obtain tori) or that it cuts the axis of rotation in at least two points (to get closed surfaces of genus zero). Now, since the rotational surfaces are generated by evolving the critical curves γ under the flow of I = P˙ B for P (κ) given in (54) and with α = 0, we can check when this evolution has fixed points along γ. Indeed, fixed points of this evolution represent the points where γ cuts the axis of rotation, and this happens exactly when κ(s) = 0 because in these cases P˙ (κ) also vanishes. Then, for each possible curvature of Proposition 4.1 in M 3 (ρ) (that is, for εi = 1, i ∈ {1, 2, 3}) we get that; 1. For the first case in Proposition 4.1, there are no values of s where the curvature vanishes. Consequently, these critical curves do not give rise to closed surfaces of genus zero. 2. For the second case, the curvature vanishes just at the point s = 0, therefore, it is not enough to obtain closed surfaces of genus zero. β β 3. Finally, for the last case we need to distinguish two options depending on the sign of a¯−β c¯ . If a ¯−β c¯ < 0,

β κ(s) is zero only at s = 0 which again is not enough. However, if a¯−β c¯ > 0, we have that there exist cases where there are two cuts with the axis of rotation at each period of the curvature and where the curvature is well-defined. In fact, we have,

Theorem 5.1. If a non-isoparametric rotational surface of constant Gaussian curvature K =  c in M 3 (ρ) is closed of genus zero, then both d and  c are positive. Moreover, they exist, if and only if, ε d < β or ε d > β and ρ > ε d, for positive constant of integration d. Proof. Consider a closed rotational surface in M 3 (ρ) with constant Gaussian curvature of genus zero, then β as explained above we need a¯−β c > 0. Of course, the orbits of the rotation must c¯ > 0 which implies that  be Euclidean circles too, and therefore d > 0, proving the first part of the result. Then, in order to obtain the second statement we need to have at least two cuts with the axis of rotation (we recall that they appear when κ(s) = 0) at some s belonging to the domain of definition of the arc-length parameter.   Notice that (restricting ourselves to the period of the curvature s ∈ 0, 2π  β exactly at s = 0 and s = π a¯−β c¯ .

β a ¯−β c¯

) the cuts appear

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21

Moreover, it is easy to check that s = 0 is always in the domain where the curvature (65) is well-defined, since  c is positive. On the other hand, the other cut belongs to the domain of definition of the arc-length parameter, if and only if, either ε d < β or ε d > β and ρ > ε d. Finally, under any of these conditions, we have exactly two points in each period of the curvature where the profile curve cuts the axis of rotation, proving the result. 2 This theorem proves the existence of closed rotational surfaces with constant Gaussian curvature in each Riemannian 3-space form, with genus zero. However, it may happen that there exist also torus with rotational symmetry and having constant Gaussian curvature. These cases appear for positive d (that is, where orbits are Euclidean circles) and whenever the profile curve (recall that it is critical for Θ∗ ) is closed. Therefore, in order to look for closed profile curves, we begin by checking if its curvature is periodic, since if it is not, the curve cannot be closed. β Thus, looking at the curvatures given in Proposition 4.1, there are only periodic curvatures when a¯−β c¯ > 0. In this case, κ(s), is given by   c¯ s a ¯ β sin2 ε a¯−β β ,   κ2 (s) = c¯ 2 (¯ a − β c¯) − a ¯ sin ε a¯−β s β

(66)

 β which is a periodic function with period  = 2επ a¯−β c¯ . Now, adapting the computations of [1] and [5], we have Proposition 5.2. Let γ ⊂ M 3 (ρ) be a planar critical curve of Θ∗ , (55), with α = 0 and with periodic curvature given by (66) for any d > 0. Then, the rotational surface Sγ swept out by γ is a closed non-isoparametric surface with closed profile curve, if and only if, the function  Λ(d) = 0

equals

2 n√π mβ ρd



ε (κ2 + β) ds (d − ερ) κ2 + d β

(67)

for any integers n and m, when ρ > 0; or, if and only if, Λ(d) vanishes when ρ ≤ 0.

The integers n and m have a clear geometric meaning. The number of rounds the critical curve gives around the pole of the parametrization in order to close up is encoded in n, while m denotes the number of lobes the curve has, that is, analytically, the number of periods of the curvature needed to close up. Notice that, if ρ ≤ 0, the integrand of (67) is always positive (respectively, negative) if ε = 1 (resp., ε = −1). Then, the following result is clear Proposition 5.3. There are no closed non-isoparametric surfaces with closed profile curves and constant Gaussian curvature if ρ ≤ 0. On the other hand, if M 3 (ρ) = S3 (ρ), things may be different as suggested by Fig. 3. For the round β 3-sphere, S3 (ρ), we have that from a¯−β c < 2ρ. Moreover, from (66) we obtain two different options, c¯ > 0,  since the right hand side of the equation must be positive. First, the case ε = 1 and 0 < β < d < ρ + 2β; and second, ε = −1 and d > −β > 0. Now, let us define  Λ(d) = εβ



ρ d Λ(d) ,

(68)

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Fig. 3. Stereographic Projection of a Closed Rotational Surface of S3 (ρ) with Constant Gaussian Curvature (Left) and its Closed Profile Curve (Right) for m = 2.

where Λ(d) is given in (67). Furthermore, taking limits in (68) we have √ 2π ρ  lim Λ(d) = √ , d→β ρ+β

lim

d→ρ+2β

√ √ ρ 4 β  Λ(d) = √ arctgh √ , ρ+β ρ + 2β

(69)

for the first case (ε = 1 and 0 < β < d < ρ + 2β); whereas, for the second case (ε = −1 and d > −β > 0), we have √ 2π ρ   =√ = 0. (70) lim Λ(d) , lim Λ(d) d→−β d→∞ ρ+β Thus, for each case there are always closed profile curves (and, therefore closed rotational surfaces with constant Gaussian curvature in S3 (ρ)), since it is always possible to find integers n and m such that  m Λ(d) = 2 n π, and therefore, the statement of Proposition 5.2 is verified. These surfaces are always  immersed in S3 (ρ) and they may have self-intersections. However, if m Λ(d) = 2 π, the rotational surface will 3 have the shape of a 2m-pinched rotational torus in S (ρ). For the first case (ε = 1 and 0 < β < d < ρ + 2β), we need that √ √ √ ρ 2π ρ 4 β 2π √ <√ arctgh √ < , (71) m ρ+β ρ + 2β ρ+β that is, if the rotational surface Sγ of constant Gaussian curvature K =  c verifies above inequalities for 3 any m ≥ 1, then Sγ is a 2m-pinched rotational torus in S (ρ). Similarly, for the second case (ε = −1 and d > −β > 0), 0<

√ 2π ρ 2π <√ , m ρ+β

for any m ≥ 1. However, in this second case, above inequalities do not give any extra information, since they are trivially true. Therefore, there exist 2m-pinched rotational torus Sγ for any constant Gaussian curvature K =  c ∈ (ρ, 2ρ), if β < 0. (See Fig. 4.) Finally, combining both cases with Proposition 5.3 we get the following Theorem 5.4. Let Sγ be a closed rotational surface with closed profile curve and with constant Gaussian curvature, K =  c, immersed in a Riemannian 3-space form, M 3 (ρ). Then, M 3 (ρ) = S3 (ρ), and,  c < 2ρ. Moreover, if ρ >  c, Sγ is a 2m-pinched rotational torus, if and only if, there exist an integer m verifying  (71), such that m Λ(d) = 2 π. On the other hand, if ρ <  c < 2ρ, Sγ is a 2m-pinched rotational torus, if and  only if, d is determined by m Λ(d) = 2 π, for any m ≥ 1.

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Fig. 4. Stereographic Projections of Closed Rotational Surfaces of S3 (ρ) with Constant Gaussian Curvatures and Their Closed Profile Curves for m = 3 (Left) and m = 4 (Right).

6. Correspondence with geodesics of unit tangent bundles Finally, in this last section, we are going to characterize the planar profile curves of rotational constant Gaussian curvature surfaces of any Riemannian 3-space form, M 3 (ρ), as the horizontal projections of subRiemannian geodesics of semi-Riemannian unit tangent bundles. In fact, this characterization is going to be made by relating the energy Θ∗ , (55), for α = 0 with the length functional in the unit tangent bundles with some sub-Riemannian structure. Therefore, this relation comes intrinsically from the variational characterization described in Theorem 3.1 and Theorem 3.4. Let’s begin by introducing some basic facts about sub-Riemannian geometry. Let M n be a smooth n-manifold. A sub-bundle of the tangent bundle T M n is called a distribution D on M n . Once we have chosen D, a D-curve on M n is a smooth immersed curve δ : [a, b] → M n which is always tangent to D; that is, δ  (t) ∈ Dδ(t) for all t ∈ [a, b]. A distribution D is said to be bracket-generating if for every p ∈ M n the sections of D near p together with all their commutators span the tangent space of M n at p, Tp M n . By a well-known theorem of Chow-Rashevskii, there is a D-curve joining any two points of M n if D is bracket-generating (check [7] for the smooth version of this theorem). Now, a sub-Riemannian metric is a smoothly varying non-degenerate bilinear form ·, · on D. Notice that with this definition we are also allowing metrics with indexes different from zero. As a particular case, if D were equal to the whole tangent bundle, ·, · would give a semi-Riemannian metric on M n . A sub-Riemannian manifold, (M n , D, ·, ·), is a smooth n-dimensional manifold M n equipped with a subRiemannian metric ·, · on a bracket-generating distribution D of rank m > 0. In this case, the length of a D-curve δ : [a, b] → M n is defined to be b L(δ) =

|δ  (t)| dt .

(72)

a

Since D is bracket-generating, it is possible to endow M n with a distance d. The distance d(p, q) between any two points p and q of M n is defined by d(p, q) = inf {L(δ) ; δ is a D-curve joining p to q} . δ

Now, let M 2 (ρ) be a totally geodesic surface of a Riemannian 3-space form. Then, M 2 (ρ) will denote the Euclidean plane, R2 , if ρ = 0; the 2-dimensional round sphere, S2 (ρ), if ρ > 0; or, the hyperbolic plane, H2 (ρ), if ρ < 0. In each case, we are going to use the coordinates (x, y) to parametrize them, as follows

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24

1. If ρ = 0, R2 = {(x, y, z) ∈ R3 ; z = 0} and x(x, y) = (x, y, 0) , where (x, y) ∈ Uo = R × R. 2. If ρ > 0, S2 (ρ) = {(x, y, z) ∈ R3 ; x2 + y 2 + z 2 = ρ1 } and 1 x(x, y) = √ (sin x cos y, sin x sin y, cos x) , ρ where (x, y) ∈ Uρ>0 = (0, π) × (0, 2π). 3. If ρ < 0, then H2 (ρ) = {(x, y, z) ∈ R3 ; x2 + y 2 − z 2 = ρ1 } and 1 x(x, y) = √ (sinh x cos y, sinh x sin y, cosh x) , ρ where (x, y) ∈ Uρ<0 = R × (0, 2π). √ √ In order to simplify the notation, let’s define c = 1, if ρ = 0; c = ρ, when ρ > 0; and, c = −ρ, if ρ < 0. Moreover, f (x) will denote, 1 if ρ = 0; sin x, if ρ > 0; and sinh x, if ρ < 0. And, finally,  will be 1 when ρ ≥ 0 and −1, if ρ < 0. With this notation, the induced metric of the three spaces above can be written as 1 f 2 (x) 2 2 dx + dy . c2 c2

g=

(73)

Now, let γ(s) = (x(s), y(s)) denote an arc-length parametrized curve of M 2 (ρ), then, its tangent vector ∂ ∂ is T (s) = γ(s) ˙ = x(s) ˙ ∂x + y(s) ˙ ∂y . Moreover, fixing θ(s) as the angle between γ(s) ˙ and the x-axis, we obtain x(s) ˙ = c cos θ(s) ,

y(s) ˙ =c

sin θ(s) . f (x(s))

(74)

Finally, after long computations, we can obtain the curvature of γ, κ(s) in above coordinates, df (x(s)) y(s) ˙ . dx

˙ + κ(s) = θ(s)

(75)

Let’s call Mρ3 to the product spaces Uρ × S1 , for each Uρ . Then, in order to construct a sub-Riemannian structure on Mρ3 = Uρ × S1 , we take the distribution  D = ker

sin θ dx − cos θ dy f (x)

 ,

(76)

where x and y are the coordinates on M 2 (ρ) and θ is the coordinate on S1 . This distribution is spanned by the vector fields X1 = c cos θ

sin θ ∂ ∂ +c , ∂x f (x) ∂y

X2 =

∂ . ∂θ

(77)

Consider on D, (76), the non-degenerate metric ·, · defined by sin2 θ X1 , X1  = cos θ +  2 f (x) 2





ρ − β df 2 1+ (x) β dx

,

X1 , X2  =

df (x) c sin θ dx , β f (x)

X2 , X2  =

1 , β

(78)

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where β is a non-zero constant. Every D-curve δ(t) = (x(t), y(t), θ(t)) with δ ∗ (cos θ dx + f (x) sin θ dy) = 0 is the lift of a regular curve γ(t) = (x(t), y(t)) in M 2 (ρ), whose tangent vector γ  (t) forms the angle θ(t) with the x-axis, i.e.,   sin θ(t) ∂ ∂ +c = v(t)X1 (t) , γ  (t) = v(t) c cos θ(t) ∂x f (x) ∂y where v(t) is the speed of γ(t). Conversely, every regular curve γ(t) in M 2 (ρ) may be lifted to a D-curve δ(t) = (x(t), y(t), θ(t)) by setting θ(t) equal to the angle between γ  (t) and the x-axis. Now, using the curvature of γ, (75), we have that the tangent vector δ  (t) of the D-curve δ(t) has squared length δ  (t), δ  (t) = ε

v 2 (t)   2 ε κ +β , β

(79)

where ε denotes the sign of κ2 + β. Then, Theorem 6.1. Let γ be the profile curve of a rotational constant Gaussian curvature surface of M 3 (ρ). That is, a critical point of Θ∗ , (55), with α = 0,   Θ (γ) = ε (κ2 + β) ds , ∗

γ

then 1. If β > 0 and ε = 1, the sub-Riemannian manifold Mρ3 has a Riemannian metric, and γ can be lifted to a sub-Riemannian geodesic of Mρ3 . 2. If β < 0 and ε = 1, Mρ3 has a Lorentzian metric, and γ is in one-to-one correspondence with timelike sub-Riemannian geodesics. 3. Finally, if β < 0 and ε = −1, then the metric of Mρ3 is Lorentzian and its spacelike geodesics project down to critical curves γ. Proof. Notice that if γ is a planar critical curve of Θ∗ , (55) with α = 0, by above argument, it can be lifted to a D-curve, δ(t), on Mρ3 . Moreover, the squared length of δ(t) is given in (79). Thus, the length functional (72) on Mρ3 is nothing but Θ∗ in M 2 (ρ). Therefore, there exist a one-to-one correspondence between critical curves in each space and for each functional, that is, between the geodesics on Mρ3 and extremal curves of Θ∗ , (55) with α = 0 acting on Ωrρ po p1 . Finally, if β > 0, then necessarily, ε = 1 and the metric (78) on Mρ3 is Riemannian. Furthermore, δ  (t), δ  (t) is positive, since the right hand side of (79) is positive in this case. If β < 0, then the metric (78) is not positive definite. In fact, in this case, Mρ3 has a Lorentzian metric since (78) has index one. Now, if ε = 1, then we get from (79) that δ is timelike. And, on the other hand, from the same equation (79), if ε = −1, δ is spacelike. 2 That is, the D-curves with δ ∗ (cos θ dx + f (x) sin θ dy) = 0 that realize the distance between two points (xo , yo , θo ) and (x1 , y1 , θ1 ) of Mρ3 are the lifts of curves γ in M 2 (ρ) joining (xo , yo ) to (x1 , y1 ) with initial angle θo and final angle θ1 that minimize the functional Θ∗ (γ) =

  ε (κ2 + β) ds , γ

(80)

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that is, (55) with α = 0. In other words, goedesics of Mρ3 are obtained by lifting to Mρ3 extremal curves of (80) in M 2 (ρ). Moreover, in all the cases, the sub-Riemannian spaces Mρ3 are in close relation with unit tangent bundles of M 2 (ρ). In fact, we can prove the following Proposition 6.2. Both Riemannian and Lorentzian sub-Riemannian spaces Mρ3 come from the unit tangent bundles of M 2 (ρ), M 2 (ρ) × Sr1 with r = 0, 1 (depending if β > 0 or β < 0). Indeed, up to a change of coordinates, they are just the restriction of the product metric of M 2 (ρ) × Sr1 to the distribution D, (76). Proof. Let’s consider the standard metric g, given by (73), in a totally geodesic surface of a Riemannian 3-space form, M 2 (ρ). Now, consider also that the coefficient of the metric on Sr1 is 1/β. Then, it is clear that r = 0 (respectively, r = 1), that is, Sr1 is Riemannian (resp. Lorentzian), if β > 0 (resp. β < 0). Finally, let’s assume that the metrics on M 2 (ρ) × Sr1 , the unit tangent bundles of M 2 (ρ), are just the product metrics. Now, after a suitable change of coordinates, we can pick up the following reference frame, df ∂ ∂ ∂ ∂ { ∂x , ∂y + dx (x) ∂θ , ∂θ }. And, with respect to this frame, the product metric on M 2 (ρ) × Sr1 is given by the matrix ⎛1 ⎞ 0 0 c2 ⎜ ⎟ ρ−β df 2 ε 1 df ⎜0 ⎟ c2 + ε c2 β dx (x) β dx (x) ⎠ . ⎝ 1 df 1 0 β dx (x) β The proof finishes after checking that above matrix restricted to the distribution D, (76), is precisely the metric defined on Mρ3 , (78). 2 Acknowledgements Research partially supported by MINECO-FEDER grant PGC2018-098409-B-100, Gobierno Vasco grant IT1094-16 and Programa Posdoctoral del Gobierno Vasco, 2018. The author also wants to thank Professor Óscar J. Garay for his kind advice and recommendations. References [1] J. Arroyo, Presión Calibrada Total: Estudio Variacional y Aplicaciones al Problema de Willmore-Chen, PhD Thesis, UPV-EHU, 2001. [2] J. Arroyo, O.J. Garay, A. Pámpano, Binormal motion of curves with constant torsion in 3-spaces, Adv. Math. Phys. 2017 (2017). [3] J. Arroyo, O.J. Garay, A. Pámpano, Constant mean curvature invariant surfaces and extremals of curvature energies, J. Math. Anal. Appl. 462 (2018) 1644–1668. [4] J. Arroyo, O.J. Garay, A. Pámpano, Curvature-dependent energies minimizers and visual curve completion, Nonlinear Dyn. 86 (2016) 1137–1156. [5] J. Arroyo, O.J. Garay, A. Pámpano, Delaunay surfaces in S 3 (ρ), Filomat 33 (4) (2019). [6] A. Barros, J. Silva, P. Sousa, Rotational linear Weingarten surfaces into the Euclidean sphere, Isr. J. Math. 192 (2012) 819–830. [7] G. Ben-Yosef, O. Ben-Shahar, Tangent bundle elastica and computer vision, IEEE Trans. Pattern Anal. Mach. Intell. 37 (1) (2015) 164–174. [8] M.P. do Carmo, M. Dajczer, Helicoidal surfaces with constant mean curvature, Tohoku Math. J. 34 (1982) 425–435. [9] M. do Carmo, M. Dajczer, Rotation hypersurfaces in spaces of constant curvature, Trans. Am. Math. Soc. 277 (1983) 685–709. [10] E. Cartan, Famillies de surfaces isoparametriques dans les espaces a courbure constante, Ann. Mat. 17 (1938) 177–191. [11] B-Y. Chen, Pseudo-Riemannian Geometry, δ-Invariants and Applications, World Scientific, Singapore, 2011. [12] S.S. Chern, On special W-surfaces, Proc. Am. Math. Soc. 6 (1955) 783–786. [13] S.S. Chern, Some new characterization of the Euclidean sphere, Duke Math. J. 12 (1945) 279–290. [14] M. Dajczer, K. Nomizu, On flat surfaces in S13 and H13 , in: Manifolds and Lie Groups, Notre Dame, 1980, pp. 71–108.

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