Bour's theorem and helicoidal surfaces with constant mean curvature in the Bianchi-Cartan-Vranceanu spaces

In this paper we generalize a classical result of Bour concerning helicoidal surfaces in the three-dimensional Euclidean space R^3 to the case of helicoidal surfaces in the Bianchi-Cartan-Vranceanu (BCV) spaces, i.e. in the Riemannian 3-manifolds whose metrics have groups of isometries of dimension $4$ or $6$, except the hyperbolic one. In particular, we prove that in a BCV-space there exists a two-parameter family of helicoidal surfaces isometric to a given helicoidal surface; then, by making use of this two-parameter representation, we characterize helicoidal surfaces which have constant mean curvature, including the minimal ones.


Introduction and preliminaries
Helicoidal surfaces in the Euclidean three dimensional space R 3 are invariant under the action of the 1-parameter group of helicoidal motions and are a generalization of rotation surfaces. Since the beginning of differential geometry of surfaces much attention has been given to the surfaces of revolution with constant Gauss curvature or constant mean curvature (CMC-surfaces). The surfaces of revolution with constant Gauss curvature seem to have been known to Minding (1839, [16]), while those with constant mean curvature have been classified by Delaunay (1841, [10]). Helicoidal minimal surfaces were studied by Scherk in 1835 (see [26] and, also, [29]), but it is rather recent the classification of the helicoidal surfaces in R 3 with nonzero constant mean curvature, given by Do Carmo and Dajczer in [12].
The starting point of their work in [12] is a result of Bour about helicoidal surfaces in R 3 (see [5], p. 82, Theorem II), for which he received the mathematics prize awarded by the Académie des Sciences de Paris in 1861 1 . Bour proved that there exists a 2-parameter family of helicoidal surfaces isometric to a given helicoidal surface in R 3 . For this, firstly he obtained orthogonal parameters (u, t) on a helicoidal surface M for which the families of u-coordinate curves are geodesics on M parametrized by arc length, and the t-coordinate curves are the trajectories of the helicoidal motion. Such parameters are called natural parameters and the first fundamental form with respect to them takes the form ds 2 = du 2 + U 2 (u) dt 2 . Reciprocally, given the natural parameters (u, t) on M and a function U (u), Bour determined a 2-parameter family of isometric helicoidal surfaces that have induced metric given by ds 2 = du 2 + U 2 (u) dt 2 , that includes rotation surfaces. An exposition of Bour's results about the theory of deformation of surfaces can be found in the Chapter IX of [9].
By using the result of Bour, in [12] Do Carmo and Dajczer established a condition for a surface of the Bour's family to have constant mean curvature. Also they obtained an integral representation (depending on three parameters) of helicoidal surfaces with nonzero constant mean curvature, that is a natural generalization of the representation for Delaunay surfaces, i.e. CMC rotation surfaces, given by Kenmotsu (see [15]).
In [25] the authors obtain a generalized Bour's theorem for helicoidal surfaces in the products H 2 ×R and S 2 × R, and use it to determine all isometric immersions in these spaces that give the surfaces which are helicoidal and have the same constant mean curvature.
In regard to the study of CMC helicoidal surfaces in BCV-spaces, in [13] and in [17,20] the authors use the equivariant geometry to classify the profile curves of these surfaces in the Heisenberg group H 3 and in H 2 ×R, respectively. The case of rotational minimal and constant mean curvature surfaces in the Heisenberg group is treated in [6]. J. Ripoll in [23,24] classified the CMC invariant surfaces in the 3-dimensional sphere S 3 and also in the hyperbolic 3-space H 3 .
The aim of this paper is to generalize the results obtained in [12] and [25]. The paper is organized as follows. Section 2 is devoted to give a short description of the Bianchi-Cartan-Vranceanu spaces and the helicoidal surfaces in these spaces. In Section 3 we establish a Bour's type theorem for helicoidal surfaces in the BCV-spaces (see Theorem 3.1) and, as an immediate consequence of this result, we have that every helicoidal surface in a BCV-space can be isometrically deformed into a rotation surface through helicoidal surfaces. Moreover, Corollary 3.4 refers to the particular case of isometric rotation surfaces.
In Section 4 we use standard techniques of equivariant geometry, in particular the Reduction Theorem of Back, do Carmo and Hsiang (see [2]), to deduce a differential equation that the function U (u) must satisfy in order that a helicoidal surface of the Bour's family determined by U (u) has constant mean curvature. We solve this equation by making a transformation of coordinates, treating separately the case of the space forms R 3 and S 3 from the other BCV-spaces. In this way, we obtain Theorem 4.3 that provides a description, in terms of natural parameters, of all helicoidal surfaces of constant mean curvature in a BCV-space, including the minimal ones. We conclude by showing that in R 3 these results give a natural parametrization of all the helicoidal minimal surfaces obtained by Scherk in [26].

Helicoidal surfaces in Bianchi-Cartan-Vranceanu spaces
A Riemannian manifold (M, g) is said to be homogeneous if for every two points p and q in M, there exists an isometry of M, mapping p into q. The classification of 3-dimensional simply connected homogeneous spaces is well-known and can be summarized as follows. First of all, the dimension of the isometry group must be equal to 6, 4 or 3 (see [3] or [14]). Then, if the isometry group is of dimension 6, M is a complete real space form, i.e. the Euclidean space E 3 , a sphere S 3 (k), or a hyperbolic space H 3 (k). If the dimension of the isometry group is 4, M is isometric to SU(2), the special unitary group, to SL(2, R), the universal covering of the real special linear group, to Nil 3 , the Heisenberg group, all with a certain left-invariant metric, or to a Riemannian product S 2 (k) × R or H 2 (k) × R. Finally, if the dimension of the isometry group is 3, M is also isometric to a simply connected Lie group with a left-invariant metric, for example that called SOL, one of the Thurston's eight models of geometry [27].
An explicit classification of 3-dimensional homogeneous Riemannian metrics based on the dimension of their isometry group was first given by Luigi Bianchi in 1897 (see [3] or [4]). Later Élie Cartan in [7] and Gheorghe Vranceanu in [28] proved that all the metrics whose group of isometries has dimension 4 or 6, except the hyperbolic one, can be represented in a concise form by the following two-parameter family of metrics for κ, τ ∈ R, and B = 1 + κ 4 (x 2 + y 2 ), (x, y, z) ∈ R 3 , positive. Thus, the family of metrics g κ,τ , that can rightfully be named the Bianchi-Cartan-Vranceanu metrics (BCV metrics) consists of all three-dimensional homogeneous metrics whose group of isometries has dimension 4 or 6, except for those of constant negative sectional curvature. In the following we shall denote by N κ,τ the open subset of R 3 where the metrics g κ,τ are defined.
With respect to (2.1) we have the following globally defined orthonormal frame 21,22]). The isometry group of g κ,τ admits the basis of Killing vector fields Therefore, the group of isometries of the BCV-spaces contains the helicoidal subgroup, whose infinitesimal generator is the Killing vector field given by We consider the surfaces in N κ,τ which are invariant under the action of the one-parameter group of isometries G X of g κ,τ generated by X. For convenience, we shall introduce cylindrical coordinates with r ≥ 0 and θ ∈ (0, 2π). In these coordinates the metric (2.1) becomes Moreover, the Killing vector field X takes the form X = ∂ ∂θ + a ∂ ∂z and a set of two invariant functions is Thus, the orbit space of the action of G X can be identified with and the orbital distance metric of B is given by Now, consider a helicoidal surface M (with pitch a) that, locally, with respect to the cylindrical coordinates, can be parametrized by and suppose that the profile curveγ(u) = (ξ 1 (u), ξ 2 (u)) is parametrized by arc-length in (B, g), so that it follows that the coefficients of the induced metric of the helicoidal surface are given by where ω(u) is the volume function of the principal orbit.

A Bour's type theorem
In this section, we show that every helicoidal surface in a BCV-space admits a reparametrization by natural parameters and, conversely, given a positive function U , it is possible to find a 2-parameter family of isometric helicoidal surfaces associate with it that are parameterized by natural parameters.
Theorem 3.1. In the BCV-space N κ,τ there exists a two parameter family of helicoidal surfaces that are isometric to a given helicoidal surface of the form (2.6) and that includes a rotation surface. More precisely, for a given positive function U (u) and arbitrary constants m = 0 and a, the helicoidal surfaces (2.6) whose profile curveγ(u) = (ξ 1 (u), ξ 2 (u)) is given by , are all to each other isometric and have first fundamental form given by du 2 + U (u) 2 dt 2 .
Proof. From (2.8) we have that the induced metric of a helicoidal surface (2.6), with pitch a 0 , is given by Now we introduce a new parameter t = t(u, θ) that satisfies: As the Jacobian |∂(u, t)/∂(u, θ)| is equal to 1, it follows that (u, t) are local coordinates on a helicoidal surface M and, also, that we can write (3.3) as We now observe that the u-coordinate curves are parametrized by arc length and also that are geodesics of M (see [19]) which are orthogonal to the t-coordinate curves, i.e. the helices. Consequently, the local parametrization ψ(u, θ(u, t)) is a natural parametrization of the helicoidal surface M .
Remark 3.3. The family of helicoidal surfaces Ψ(u, t) := ψ [U,m,a] (u, t) in the BCV-space N κ,τ obtained in the Theorem 3.1 depends on two parameters m = 0 and a, and for m = 1 and a = a 0 it contains the original helicoidal surface. Also, when m = 1 and a = 0, we obtain a rotational surface isometric to the given helicoidal surface. Therefore, by varying the constant a from a = 0 to a = a 0 , we get an isometric deformation from a rotational surface to a given helicoidal surface.
If we suppose m = 1, from the formulas (3.15) we obtain that By varying a from a = 0 to a = d we have the classical isometric deformation of the catenoid into the helicoid ψ [U,1,d] (u, t) = (u cos t, u sin t, t + b), b ∈ R, that are minimal surfaces. Also, the intermediate helicoidal surfaces are all minimal and their natural parametrization is given by Such surfaces are also called second Scherk's surfaces (see [11] and, also, the Example 3).
From the formula for the Gaussian curvature of an invariant surface obtained in [18], it follows that the helicoidal surfaces of the Bour's family in the BCV-space N κ,τ have all the same Gaussian curvature given by With regard to the mean curvature H of these surfaces, in next section we shall see that different values of a and m can give rise to different values of H.

Helicoidal surfaces of constant mean curvature
In this section, we will describe the helicoidal surfaces in the BCV-spaces that have the same constant mean curvature. We start by computing the mean curvature of a helicoidal surface (2.6). It turns out that the mean curvature of an invariant immersion is tightly related to the geodesic curvature of the profile curve, as shown by the remarkable following theorem. But first we recall that if on a three-dimensional connected Riemannian manifold (N 3 , g) we consider the 1-parameter subgroup G X of isometries generated by X, an orbit G(p) of p ∈ N is called principal if there exists an open neighborhood U ⊂ N of p such that all orbits G(q), q ∈ U , are of the same type as G(p) (i.e. the isotropy subgroups G q and G p are coniugated). This implies that G(q) is diffeomorphic to G(p). We denote with N r the regular part of N , that is, the subset consisting of points belonging to principal orbits [1]. Then we have Theorem 4.1 (Reduction Theorem [2]). Let H be the mean curvature of a G X -invariant surface M r ⊂ N r and k g the geodesic curvature of the profile curve M r /G X ⊂ B r . Then where n is the unit normal of the profile curve and ω = g(X, X) is the volume function of the principal orbit.
Let nowγ(u) = (ξ 1 (u), ξ 2 (u)) be a curve in B r , parametrized by arc-length, that under the action of G X generates the helicoidal surface. From (2.5), it follows that (4.1) ξ 1 = B cos σ, ξ 2 = ξ 2 1 + (a B − τ ξ 2 1 ) 2 sin σ ξ 1 and the geodesic curvature ofγ takes the expression (4.2) where σ is the angle thatγ makes with the ∂ ∂ξ 1 direction. Also, as the normal derivative is given by ] 2 and thus we obtain that the mean curvature is given by Proof. If we consider a helicoidal surface Ψ(u, t) of the Bour's family, from (4.1) we can write equation (4.3) as and therefore, using (3.12) we get Then, Now, deriving (3.11) we get and hence, taking into account (3.10), (3.11) and (3.12), we can write equation (4.7) as 4.1. The solution of the mean curvature equation. Next, we will give a description of the helicoidal surfaces in N κ,τ with constant mean curvature H. For this purpose, we assume that the helicoidal surfaces are parametrized by natural coordinates (u, t) and we determine explicitly the expression of the function U (u), that gives the metric, by integrating (4.4).
Proof. Using the transformation of coordinates given by where c is an arbitrary constant. Consequently, we have the following cases: Case 4τ 2 − κ = 0. From (4.8) and (4.10), if we suppose that 1 − 2a τ > 0, then we have Putting z = x 2 the above expression is transformed into the following We observe that when c = 0, from (3.15) it follows that the helicoidal surface ψ [U,m,a] is a helicoid. ii) If H 2 + 4τ 2 = 0, by integrating (4.11) we have that , up to a constant. Thus, as z = m 2 U 2 , it follows that Case 4τ 2 − κ = 0. In this case, as x(u) x (u) ∆(u) = ( ∆(u)) 4τ 2 − κ ,