for Conformally Flat Hypersurfaces

The Symmetry Group of Lam´ e’s System and the Associated Guichard Nets

for Conformally Flat Hypersurfaces

Jo˜ao Paulo dos SANTOS and Keti TENENBLAT

Departamento de Matem´atica, Universidade de Bras´ılia, 70910-900, Bras´ılia-DF, Brazil E-mail: j.p.santos@mat.unb.br, k.tenenblat@mat.unb.br

Received October 01, 2012, in final form April 12, 2013; Published online April 17, 2013 http://dx.doi.org/10.3842/SIGMA.2013.033

Abstract. We consider conformally flat hypersurfaces in four dimensional space forms with their associated Guichard nets and Lam´e’s system of equations. We show that the symmetry group of the Lam´e’s system, satisfying Guichard condition, is given by translations and dilations in the independent variables and dilations in the dependents variables. We obtain the solutions which are invariant under the action of the 2-dimensional subgroups of the symmetry group. For the solutions which are invariant under translations, we obtain the corresponding conformally flat hypersurfaces and we describe the corresponding Guichard nets. We show that the coordinate surfaces of the Guichard nets have constant Gaussian curvature, and the sum of the three curvatures is equal to zero. Moreover, the Guichard nets are foliated by flat surfaces with constant mean curvature. We prove that there are solutions of the Lam´e’s system, given in terms of Jacobi elliptic functions, which are invariant under translations, that correspond to a new class of conformally flat hypersurfaces.

Key words: conformally flat hypersurfaces; symmetry group; Lam´e’s system; Guichard nets 2010 Mathematics Subject Classification: 53A35; 53C42

1 Introduction

The investigation of conformally flat hypersurfaces has been of interest for quite some time. Any surface in R³ is conformally flat, since it can be parametrized by isothermal coordinates. For higher dimensional hypersurfaces, E. Cartan [2] gave a complete classification for the conformally flat hypersurfaces of an (n+ 1)-dimensional space form whenn+ 1≥ 5. He proved that such hypersurfaces are quasi-umbilic, i.e., one of the principal curvatures has multiplicity at leastn−1.

In the same paper, Cartan investigated the case n+ 1 = 4 . He showed that the quasi-umbilic surfaces are conformally flat, but the converse does not hold (for a proof see [13]). Moreover, he gave a characterization of the conformally flat 3-dimensional hypersurfaces, with three distinct principal curvatures, in terms of certain integrable distributions. Since then, there has been an effort to obtain a classification of hypersurfaces satisfying Cartan’s characterization.

Lafontaine [13] considered hypersurfaces of type M³ = M² ×I ⊂ R⁴. He obtained the following classes of conformally flat hypersurfaces: a)M³ is a cylinder over a surface,M² ⊂R³, with constant curvature; b) M³ is a cone over a surface in the sphere, M² ⊂S³, with constant curvature; c) M³ is obtained by rotating a constant curvature surface of the hyperbolic space, M² ⊂H³⊂R⁴, where H³ is the half space model.

Motivated by Cartan’s paper, Hertrich-Jeromin [8], established a correspondence between conformally flat three-dimensional hypersurfaces, with three distinct principal curvatures, and

?This paper is a contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants and Applications”. The full collection is available athttp://www.emis.de/journals/SIGMA/SDE2012.html

Guichard nets. These are systems of triply orthogonal surfaces originally considered by C. Gui- chard in [6], where he referred to those systems as the analogues of isothermal coordinates.

In view of Hertrich-Jeromin results, the problem of classifying conformally flat 3-dimensional hypersurfaces was transferred to the problem of classifying Guichard nets in R³. These are open sets of R³, with an orthogonal flat metricg=

3

P

i=1

l_i²dx²_i, where the functionsl_i satisfy the Guichard condition, namely,

l²₁−l₂²+l²₃ = 0,

and a system of second-order partial differential equations, which is called Lam´e’s system (see (2.2)).

Hertrich-Jeromin obtained an example of a Guichard net, starting from surfaces parallel to Dini’s helix and he proved that the corresponding conformally flat hypersurface was a new example, since it did not belong to the class described by Lafontaine.

In [20, 21, 22], Suyama extended the previous results by showing that the Guichard nets described by Hertrich-Jeromin are characterized in terms of a differentiable functionϕ(x₁, x₂, x₃) that determines, up to conformal equivalence, the first and second fundamental forms of the corresponding conformally flat hypersurfaces. Moreover, Suyama showed that if ϕ does not depend on one of the variables, then the hypersurface is conformal to one of the classes described by Lafontaine. He also showed that the function associated to the example given by Hertrich- Jeromin satisfied ϕ,x1x2 = ϕ,x2x3 = 0. Starting with this condition on ϕ, Suyama obtained a partial classification of such conformally flat hypersurfaces. The complete classification of conformally flat hypersurfaces, satisfying the above condition on the partial derivatives of ϕ, was obtained by Hertrich-Jeromin and Suyama in [10]. They showed that these hypersurfaces correspond to a special type of Guichard nets. The authors called them cyclic Guichard nets, due to the fact that one of the coordinates curves is contained in a circle.

In this paper, we obtain solutionsli satisfying Lam´e’s system and the Guichard condition, which are invariant under the action of the 2-dimensional subgroups of the symmetry group of the system. Moreover, we investigate the properties of the Guichard nets and of the conformally flat hypersurfaces associated to the solutionsli. We first determine the symmetry group of Lam´e’s system satisfying the Guichard condition. We prove that the group is given by translations and dilations of the independent variables x_i and dilations of the dependent variablesl_i.

We obtain the solutionsli,i= 1,2,3, which are invariant under the action of the 2-dimensional translation subgroup, i.e., l_i(ξ), where ξ =

3

P

i=1

α_ix_i. These solutions are given explicitly in Theorem 3 by Jacobi elliptic functions, whenever all the functions l_i are not constant and in Theorem4when one of the functionsli is constant. Moreover, we consider the solutionsliwhich are invariant under the 2-dimensional subgroup involving translations and dilations, i.e., l_i(η), where η =

3

P

j=1

ajxj/

3

P

k=1

bkxk. In this case, if we require the functions li(η) to depend on all three variables, thenli are constant functions. Otherwise, the solutionsli(η) are given explicitly in Theorem5. The symmetry subgroup of dilations on the dependent variables is irrelevant for the study of conformally flat hypersurfaces.

Considering the functionsliwhich are invariant under the action of translations, we study the corresponding Guichard nets. We show that their coordinate surfaces have constant Gaussian curvature and the sum of the three curvatures is equal to zero. Moreover the Guichard nets are foliated by flat surfaces, with constant mean curvature.

Finally, we investigate the conformally flat hypersurfaces associated to the functionsl_i which are invariant under the action of translations. We show that, whenever the basic invariant ξ depends on two variables, the hypersurface is conformal to one of the products considered by

Lafontaine. In this case, the three-dimensional conformally flat hypersurfaces are constructed from flat surfaces contained in the hyperbolic 3-space H³ or in the sphere S³. Whenever the basic invariantξ depends on all three independent variables, then the functions l_i(ξ), which are given in terms of Jacobi elliptic functions, produce a new class of conformally flat hypersurfaces.

In Section 2, we review the correspondence between conformally flat 3-dimensional hyper- surfaces with Lam´e’s system, and Guichard nets.

In Section3, we obtain the symmetry group of Lam´e’s system satisfying Guichard condition and the solutions which are invariant under 2-dimensional subgroups of the symmetry group.

The motivation and the technique used in this section were inspired by the fact that our system of differential equations is quite similar to the intrinsic generalized wave and sine-Gordon equations and the generalized Laplace and sinh-Gordon equations. The symmetry groups of these systems and the solutions invariant under subgroups were obtained by Tenenblat and Winternitz in [24]

and Ferreira [4]. The geometric properties of the submanifolds corresponding to the solutions invariant under the subgroups of symmetries can be found in [1] and [19].

In Sections 4 and 5, we describe the geometric properties of the Guichard nets and of the conformally flat hypersurfaces that are associated to the solutions of Lam´e’s system which are invariant under the action of the translation group.

The solutionsl_i of Lam´e’s system, satisfying Guichard condition, which are invariant under the subgroup of dilations of the independent variables and the corresponding geometric theory, will be considered in another paper. Such solutions are obtained by solving a (reduced) system of partial differential equations, in contrast to what occurs in this paper, where the Lam´e’s system is reduced to a system of ordinary differential equations.

2 Lam´ e’s system and conformally f lat hypersurfaces

Consider the Minkowski space R⁶₁ with coordinates (x₀, . . . , x₅) and the scalar product h , i given by

h, i: R⁶×R⁶ −→R, (v, w)7→ −v₀w₀+

5

X

i=1

v_iw_i.

LetL⁵=

y∈R⁶₁| hy, yi= 0 , be the light cone inR⁶₁and considerm_K ∈R⁶₁, withhm_K, m_Ki=K.

Then, it is not difficult to see that, the sets M_K⁴ =

y∈L⁵| hy, m_Ki=−1 ,

with the metric induced from R⁶1, are complete Riemannian manifolds with constant sectional curvature K. IfK <0, thenM_K⁴ consists of two connected components which can be isometri- cally identified (see [7, Lemma 1.4.1] for details).

With this approach, consider a Riemannian immersion f : M³ → M_K⁴ ⊂ L⁵, with unit normal n. Then hdf, ni ≡ 0, and n also satisfies hn, m_Ki = hn, fi = 0. Let ˜f : M³ → L⁵ be an immersion given by ˜f = e^uf, where u is a differentiable function on M. Observe that the metric induced on ˜f is conformal to the metric induced on the immersion f, i.e.,

hdf , d˜ fi˜ =e^2uhdf, dfi.

Definition 1. Let f : M³ → L⁵ be an immersion such that the induced metric, hdf, dfi, is positive definite. Let nbe a unit normal withhf, ni= 0 and consider differentiable functionsu and aon M³. Then the pair (f, n) is called a strip and the pair ( ˜f ,˜n) given by

f˜=e^uf, n˜ =n+af

is called a conformal deformation of the strip (f, n).

Therefore, we can deform a conformally flat immersion in a space formf :M³ →M_K⁴ ⊂L⁵ to a flat immersion in the light cone ˜f : M³ → L⁵, by considering a conformal deformation, and vice-versa. Hence the problem of investigating conformally flat hypersurfaces in space forms reduces to a problem of studying flat immersions in the light cone f : M³ → L⁵. We say that a conformally flat hypersurface in a space form M_k⁴ is generic if it has three distinct principal curvatures. Hertrich-Jeromin in [8] established a relation between generic conformally flat hypersurfaces inM_k⁴ and Guichard nets [6]. Namely, lete1,e2,e3 be an orthonormal frame tangent to M³⊂M_k⁴, such thate_i are principal directions. Letω₁,ω₂,ω₃ be the co-frame and let k₁,k₂,k₃ be the principal curvatures. Assume that locallyk₃> k₂ > k₁, then theconformal fundamental forms

α1=p

(k3−k1)(k2−k1)ω1, α2=p

(k2−k1)(k3−k2)ω2, α3=p

(k3−k2)(k3−k1)ω3

are closed, if and only if, the hypersurfaceM³ is conformally flat. Therefore, whenαi are closed forms, locally there existx₁,x₂,x₃ such thatα₁ =dx₁,α₂=dx₂ andα₃=dx₃. By integration, we obtain a special principal coordinate system x₁, x₂, x₃ for a conformally flat hypersurface inM_K⁴.

Definition 2. A triply orthogonal coordinate system in a Riemannian 3-manifold (M, g) x= (x1, x2, x3) : (M, g)→R³,

where the functionsl_i =p

g(∂_xi, ∂_xi) satisfy the Guichard condition

l²₁−l₂²+l²₃ = 0, (2.1)

is called a Guichard net.

Since we can deform a conformally flat immersion in a space form into a flat immersion in the light cone, we can consider Guichard nets for flat immersionsf :M³ →L⁵. For such a flat immersion, we express the induced metricg=hdf, dfi, in terms of the Guichard net, as

g=l₁²dx²₁+l²₂dx²₂+l²₃dx²₃.

Since the metric is flat, the functions li must satisfy the Lam´e’s system[14, pp. 73–78]:

∂²li

∂x_j∂x_k − 1 l_j

∂li

∂x_j

∂lj

∂x_k − 1 l_k

∂li

∂x_k

∂l_k

∂x_j = 0,

∂

∂xj

1 lj

∂li

∂xj

+ ∂

∂xi

1 li

∂l_j

∂xi

+ 1

l_k²

∂li

∂x_k

∂l_j

∂x_k = 0. (2.2)

for i, j, k distinct. Moreover, if f : M³ → L⁵ is flat, we can consider M³ as a subset of the Euclidean spaceR³ andf as an isometric immersion. Then we have a Guichard net on an open subset of R³, by considering as in Definition 2,x:U ⊂R³→R³, where the functionsli satisfy the Guichard condition (2.1) and the Lam´e’s system (2.2). At this point, one can ask if such a Guichard net determines a conformally flat hypersurface in a space form, or equivalently, a flat immersion in L⁵. The answer to this question was given by the following fundamental result due to Hertrich-Jeromin [8]:

Theorem 1. For any generic conformally flat hypersurface of a space form M_K⁴, there exists a Guichard net x : U ⊂ R³ → R³ on an open set U of the Euclidean space R³ (uniquely determined up to a M¨obius transformation of R³).

Conversely, given a Guichard net x = (x1, x2, x3) : U ⊂ R³ → R³ for the Euclidean space, with li =

q

g(∂xi, ∂xj), where g is the canonical flat metric, there exists a generic conformally flat hypersurface in a space form M_K⁴ (in this case, M¨obius equivalent Guichard nets are related to conformally equivalent immersions), whose induced metric is given by

g=e^2P(x)

l₁²dx²₁+l²₂dx²₂+l²₃dx²₃ , (2.3) where P(x) is a function depending on M_K⁴.

The converse is based on the fact that the functions l_i determine the connection forms of a flat immersion f : M³ → L⁵. In fact, these connection forms satisfy the Maurer–Cartan equations if, and only if, the functions li satisfy the Guichard condition and the Lam´e’s system.

Therefore, one way of obtaining generic conformally flat hypersurfaces in space formsM_K⁴ is finding solutions of Lam´e’s system, satisfying the Guichard condition. Then the hypersurfaces are constructed by using Theorem 1. Our objective is to obtain a class of such solutions and to investigate the associated Guichard nets as well as the conformally flat hypersurfaces. We will use the theory of Lie point symmetry groups of differential equations, to obtain the symmetry group of Lam´e’s system and their solutions invariant under the action of subgroups of the symmetry group. This is the content of the following sections.

3 The symmetry group of Lam´ e’s system

In this section, we obtain the Lie point symmetry group of Lam´e’s system. We start with a brief introduction of symmetry groups of differential equations. The reader who is familiar to the theory may skip this introduction.

The theory of Lie point symmetry group is an important tool for the analysis of differential equations developed by Lie at the end of the nineteen century [15]. Roughly speaking, Lie point symmetries of a system of differential equations consist of a Lie group of transformations acting on the dependent and independent variables, that transform solutions of the system into solutions.

A standard reference for the theory of symmetry groups of differential equations is Olver’s book [17], where a clear approach to the subject is given, with theoretical foundations and a large number of examples and techniques. We will describe here some basic concepts that will be used in this section.

A systemS of n-th order differential equations inp independent and q dependent variables is given as a system of equations

∆_r x, u⁽ⁿ⁾

= 0, v= 1, . . . , l, (3.1)

involvingx= (x₁, . . . , x_p),u= (u₁, . . . , u_q) and the derivativesu⁽ⁿ⁾ ofuwith respect tox up to order n.

A symmetry group of the system S is a local Lie group of transformations G acting on an open subset M ⊂ X×U of the space of independent and dependent variables for the system, with the property that whenever u = f(x) is a solution of S, and whenever gf is defined for g∈G, then u=gf(x) is also a solution of the system. A vector field v in the Lie algebra gof the group Gis called an infinitesimal generator.

Considervas a vector field on M ⊂X×U, with corresponding (local) one-parameter group exp(εv), i.e.,

exp(εv)≡Ψ(ε, x),

where Ψ is the flow generated by v. In this case, v will be the infinitesimal generator of the action.

The symmetry group of a given system of differential equation, is obtained by using the prolongation formulaand theinfinitesimal criterionthat are described as follows. Given a vector field on M ⊂X×U,

v=

p

X

i=1

ξⁱ(x, u) ∂

∂x_i +

q

X

α=1

φ_α(x, u) ∂

∂u^α, then-th prolongation of vis the vector field

pr⁽ⁿ⁾v=v+

q

X

α=1

X

J

φ^J_α x, u⁽ⁿ⁾ ∂

∂u^α_J.

It is defined on the correspondingjet space M⁽ⁿ⁾ ⊂X×U⁽ⁿ⁾, whose coordinates represent the independent variables, the dependent variables and the derivatives of the dependent variables up to order n. The second summation is taken over all (unordered) multi-indices J = (j1, . . . , j_k), with 1 ≤j_k ≤p, 1≤ k≤n. The coefficient functions φ^J_α of pr⁽ⁿ⁾v are given by the following formula:

φ^J_α x, u⁽ⁿ⁾

=D_J φα−

p

X

i=1

ξⁱu^α_J,i,

! ,

where u^α_i = ^∂u_∂x^α

i,u^α_J,i= ^∂u

α J

∂xi and DJ is given by the total derivatives D_J =D_j1D_j2· · ·D_jk,

with

Dif x, u⁽ⁿ⁾

= ∂f

∂xi

+

p

X

α1

X

J

u^α_J,i ∂f

∂u^α_J.

We say that the system (3.1) is a system of maximal rank overM ⊂X×U, if the Jacobian matrix

J∆ x, u⁽ⁿ⁾

= ∂∆r

∂xi

,∂∆r

∂u^α_,J

!

has rank l, whenever ∆r x, u⁽ⁿ⁾

= 0, where J= (j1, . . . , j_k) is a multi-index that denotes the partial derivatives of u^α.

Suppose that (3.1) is a system of maximal rank. Then the set of all vectors fieldsv on M such that

pr⁽ⁿ⁾v

∆r x, u⁽ⁿ⁾

= 0, r = 1, . . . , l, whenever ∆r x, u⁽ⁿ⁾

= 0, (3.2)

is a Lie algebra of infinitesimal generators of a symmetry group for the system. It is shown in [17] that the infinitesimal criterion (3.2) is in fact both a necessary and sufficient condition for a group G to be a symmetry group. Hence, all the connected symmetry groups can be determined by considering this criterion.

Since the prolongation formula is given in terms of ξⁱ and φα and the partial derivatives with respect to bothxand u, the infinitesimal criterion provides a system of partial differential equations for the coefficientsξⁱ and φ_α of v, called thedetermining equations. By solving these equations, we obtain the vector fieldvthat determines a Lie algebrag. The symmetry groupG is obtained by exponentiating the Lie algebra.

3.1 Obtaining the symmetry group of Lam´e’s system

From now on, we consider the following notation for derivatives of a function f =f(x1, . . . , xn) f,xi := ∂f

∂x_i and f,xixj := ∂²f

∂x_i∂x_j. With this notation, Lam´e’s system (2.2) is given by

li,xjx_k−l_i,xjl_j,xk

l_j −l_i,xkl_k,xj

l_k = 0, (3.3)

l_i,xj lj

,xj

+ l_j,xi

li

,xi

+l_i,xkl_j,xk

l²_k = 0, (3.4)

where i, j and k are distinct indices in the set {1,2,3}. We will also consider the following notation,

εs=

1 if s= 1 or s= 3,

−1 if s= 2. (3.5)

We can now rewrite Guichard condition as εil²_i +εjl_j²+ε_kl²_k= 0.

Next, we introduce auxiliary functions in order to reduce the system of second-order diffe- rential equations (3.3) and (3.4), into a first order one. Consider the functionsh_ij, with i6=j, given by

li,xj−hijlj = 0.

With these functions, we rewrite (3.3) and (3.4) as h_ij,xk−h_ikh_kj = 0, h_ij,xj+h_ji,xi+h_ikh_jk = 0.

for i, j, k distinct. Since the functionsl₁, l₂ and l₃ satisfy Guichard condition, there are other relations involving the derivatives of li and hij. Taking the derivative of Guichard condition with respect to x_i, we have

ε_il_i,xi+ε_jh_jil_j+ε_kh_kil_k= 0,

fori,j,kdistinct. The derivatives of the above equation with respect tox_j leads to εihij,xi +εjhji,xj +εkhkihkj = 0.

Therefore, we summarize the last six equations in the following system of first-order partial differential equations, equivalent to Lam´e’s system, that we call Lam´e’s system of first order

ε_il²_i +ε_jl_j²+ε_kl²_k= 0, (3.6)

li,xj−hijlj = 0, (3.7)

εili,xi+εjhjilj+εkhkilk= 0, (3.8)

h_ij,xk−h_ikh_kj = 0, (3.9)

h_ij,xj+h_ji,xi+h_ikh_jk = 0, (3.10)

ε_ih_ij,xi +ε_jh_ji,xj +ε_kh_kih_kj = 0. (3.11)

By consideringx= (x₁, x₂, x₃),l= (l₁, l₂, l₃) andhthe off-diagonal 3×3 matrix given byh_ij in our next two results, we obtain the Lie algebra of the infinitesimal generators and the symmetry group of Lam´e’s system of first order.

Theorem 2. Let V be the infinitesimal generator of the symmetry group of Lam´e’s system of first order (3.6)–(3.11), given by

V =

3

X

i=1

ξⁱ(x, l, h) ∂

∂xi

+

3

X

i=1

ηⁱ(x, l, h) ∂

∂li

+

3

X

i,j=1, i6=j

φ^ij(x, l, h) ∂

∂hij

. (3.12)

Then the functions ξⁱ, ηⁱ andφ^ij are given by ξⁱ =axi+ai, ηⁱ =cli, φ^ij =−ah_ij, where a, c, ai ∈R.

The proof of Theorem2is very long and technical. It consists of obtaining the functionsξⁱ,ηⁱ and φ^ij by solving the determining equations which are obtained as follows. We apply the first prolongation of V to each equation (3.6)–(3.11) and we eliminate the functional dependence of the derivatives of h andl caused by the system. Then we equate to zero the coefficients of the remaining unconstrained partial derivatives. The complete proof with, all the details, is given in AppendixA.

As a consequence of Theorem 2, by exponentiating V, we obtain the symmetry group of Lam´e’s system. Observe that the functionsφ^ij do not depend onx andl(see [18] for symmetry group of equivalent systems):

Corollary 1. The symmetry group of Lam´e’s system (3.6)–(3.11) is given by the following transformations:

1) translations in the independent variables: x˜_i =x_i+v_i; 2) dilations in the independent variables: x˜i=λxi; 3) dilations in the dependent variables: l˜i =ρli; where vi ∈R and λ, ρ∈R\ {0}.

3.2 Group invariant solutions

The knowledge of all the infinitesimal generators v of the symmetry group of a system of differential equations, allows one to reduce the system to another one with a reduced number of variables. Specifically, if the system haspindependent variables and ans-dimensional symmetry subgroup is considered, then the reduced system for the solutions invariant under this subgroup will depend on p−s variables (see Olver [17] for details). Finding all the s-dimensional sym- metry subgroups is equivalent to finding all thes-dimensional subalgebras of the Lie algebra of infinitesimal symmetriesv. For the remainder of this paper, we will consider the 2-dimensional subgroups of the symmetry group of Lam´e’s system. The first one will be the translation sub- group and the second one will be the subgroup involving translations and the dilations. The 1-dimensional subgroup given just by dilations and the solutions invariant under this subgroup are being investigated. We will report on our investigation in another paper. We observe that the symmetry subgroup of dilations in the dependent variables (Corollary 1(3)) is irrelevant for the geometric study of conformally flat hypersurfaces due to (2.3).

We start with the 2-dimensional subgroup of translations. The basic invariant of this group is given by

ξ =α1x1+α2x2+α3x3, (3.13)

where (α1, α2, α3) is a non zero vector. We will consider solutionsli such that li(x1, x2, x3) =li(ξ), 1≤i≤3,

whereξ is given by (3.13). For such solutions, Lam´e’s system reduces to a system of ODEs. We start with two lemmas:

Lemma 1. Let l_s(ξ), s = 1,2,3, where ξ = P3 s=1

α_sx_s, be a solution of Lam´e’s system (3.6)–

(3.11). Let i, k∈ {1,2,3} be two fixed and distinct indices such that αi=α_k= 0. Then li or l_k is constant.

Proof . Since α_i =α_k= 0, it follows from (3.7) that equation (3.10) reduces to α²_j

l_i,ξ l_j

,ξ

= 0,

which implies l_i,ξ =c_il_j, where c_i ∈R. Similarly, interchanging iwith k, we obtain l_k,ξ =c_kl_j. Finally, interchanging kwith j, we get

α²_jl_i,ξl_k,ξ

l²_j =α²_jc_ic_k= 0.

Therefore, we conclude that l_i orl_k is constant.

Lemma 2. Letl_s(ξ),s= 1,2,3,whereξ=

3

P

s=1

α_sx_s, be a solution of Lam´e’s system (3.6)–(3.11).

If there exists a unique j∈ {1,2,3} such that l_j is a non zero constant, thenα_j = 0.

Proof . Interchanging the indices in (3.9), we obtain the following two equations αjα_k

l_i,ξξ− li,ξlk,ξ

l_k

= 0, (3.14)

α_jα_i

l_k,ξξ− l_k,ξl_i,ξ l_i

= 0, (3.15)

and an identity.

Similarly, it follows from (3.10) that

α²_jl_i,ξξ = 0, (3.16)

α²_jl_k,ξξ = 0, (3.17)

α²_k li,ξ

l_k

,ξ

+α²_i lk,ξ

l_i

,ξ

+α²_jli,ξlk,ξ

l_j = 0. (3.18)

Suppose, by contradiction, that αj 6= 0. It follows from (3.16) and (3.17) that l_i,ξ =ci and l_k,ξ = c_k, where c_i 6= 0 and c_k 6= 0, since by hypothesis, l_i and l_k are non constants. Then, it follows from (3.14) and (3.15) that αi =αk = 0. From (3.18), we obtain α²_jcick = 0, which is

a contradiction.

The following theorem gives the solutions of Lam´e’s system, satisfying Guichard condition, which are invariant under the action of the translation group, whenever none of the functions l_i is constant.

Theorem 3. Let l_s(ξ), s = 1,2,3, where ξ =

3

P

s=1

α_sx_s, be a solution of Lam´e’s system (3.6)–

(3.11), such that l_s is not constant for all s. Then there exist c_s∈R\ {0}, such that,

l_i,ξ =cil_klj, i, j, k distinct, (3.19)

c1−c2+c3= 0, (3.20) α²₁c₂c₃+α²₂c₁c₃+α²₃c₁c₂ = 0. (3.21) Moreover, the functions l_i(ξ) are given by

l²_1,ξ =c₂(c₂−c₁)

l₁²− λ

c₂ l₁²− λ c₂−c₁

, (3.22)

l²₂ = c₂ c1

l²₁− λ

c2

, (3.23)

l²₃ = c2−c1

c1

l²₁− λ c2−c1

, (3.24)

where λ∈R.

Proof . By hypothesis, we are considering non constant solutions. Then, it follows from Lem- ma1, thatα_s6= 0 for at least two distinct indices. Suppose thatα_j andα_knon zero. From (3.7) and (3.9) we obtain

αjαk

( l_i,ξ

lj

,ξ

−l_i,ξ lj

l_k,ξ lk

)

= 0, which implies

l_i,ξ lj

,ξ

l_i,ξ lj

−1

= l_k,ξ lk

.

Integrating this equation, we obtain l_i,ξ =cil_klj, whereci 6= 0.

Ifαi6= 0, analogously considering the non zero pairs (α_i, αj) and (αi, αk), we conclude that l_k,ξ =c_kl_il_j and l_j,ξ =c_jl_il_k. Ifα_i= 0, then from equation (3.10) we have

l_i,xj l_j

,xj

+ li,xk

l_k lj,xk

l_k =α²_jcil_k,ξ+α²_kcilj

l_j,ξ l_k = 0.

Since ci 6= 0, we integrate the above expression to obtain α²_jl²_k+α²_kl_j²=λjk,

where λjk is a constant. This equation and Guichard condition (3.6) lead to l²_j = α²_j

α²_k λ_jk

α²_j −l_k²

!

, l_i²= ε_i α²_k

l²_k ε_jα²_j −ε_kα²_k

−ε_jλ_jk

. (3.25)

Taking the derivative of the last equation with respect to ξ, we have li(cilklj) = ε_i

α²_k

lklk,ξ εjα²_j −εkα²_k .

If ε_jα²_j −ε_kα²_k6= 0, we conclude that l_k,ξ = ciα²_k

εjα²_j−εkα²_kl_il_j =c_kl_il_j.

Applying this expression into the derivative of the first equation in (3.25) with respect toξ we obtain

l_jl_j,ξ=−α²_j

α²_kl_kl_k,ξ=−α²_j

α²_kl_k(c_kl_il_j), consequently,l_j,ξ =c_jl_il_k.

Next, we will show that εjα²_j −ε_kα_k² 6= 0 to conclude the proof of (3.19). Suppose by contradiction thatεjα²_j−ε_kα²_k= 0, then the first equation of (3.25) can be written asεjl²_j+ε_kl²_k=

εkλjk

α²_j . Then Guichard condition now implies that li is constant, which is a contradiction. The relations between the constants (3.20) and (3.21) follow from a straightforward computation using equations (3.8) and (3.10), respectively.

In order to complete the proof of the theorem, we start with

l_1,ξ =c₁l₂l₃, (3.26)

l_2,ξ =c2l1l3, (3.27)

l_3,ξ =c3l1l2. (3.28)

Multiplying (3.27) byl₂ and integrating we have l²₂ = c₂

c₁

l²₁− λ c₂

, (3.29)

where λis a constant. Therefore, it follows from (3.29) and Guichard condition that l²₃ = c2−c1

c1

l²₁− λ c2−c1

. (3.30)

Using (3.26), (3.29) and (3.30), we conclude that l_1,ξ²=c²₁

c2

c₁

l²₁− λ c₂

c2−c1

c₁

l₁²− λ c₂−c₁

=c₂(c₂−c₁)

l₁²− λ c2

l₁²− λ c2−c1

.

In our next theorem, we consider the solutionsli(ξ) when one of the functions li is constant.

Theorem 4. Let ls(ξ), s = 1,2,3, where ξ =

3

P

s=1

αsxs, be a solution of Lam´e’s system (3.6)–

(3.11). Suppose that only one of the functions ls is constant. Then one of the following occur:

a) l₁ =λ₁, l₂ =λ₁cosh(bξ+ξ₀), l₃ =λ₁sinh(bξ+ξ₀), where ξ =α₂x₂+α₃x₃, α²₂+α²₃ 6= 0 and b, ξ0 ∈R;

b) l2 =λ2, l1 =λ2cosϕ(ξ), l3 =λ2sinϕ(ξ), where ξ = α1x1 +α3x3, α²₁+α₃² 6= 0 and ϕ is one of the following:

b.1) ϕ(ξ) =bξ+ξ0, ifα²₁6=α²₃, where ξ0, b∈R; b.2) ϕ is any function ofξ, if α²₁ =α²₃;

c) l₃ =λ₃, l₂ =λ₃cosh(bξ+ξ₀), l₁ =λ₃sinh(bξ+ξ₀), where ξ =α₁x₁+α₂x₂, α²₁+α²₂ 6= 0 and b, ξ0 ∈R.

Proof . We will consider each case separately:

a) If l₁ = λ₁, then it follows from Lemma 2 that we must have ξ = α₂x₂ +α₃x₃. Now Guichard condition implies that l₂ = λ₁coshϕ(ξ) and l₃ = λ₁sinhϕ(ξ). In order to determi- ne ϕ, we use (3.10) with the following indices

h_23,x3 +h_32,x2+h₂₁h₃₁= 0, to obtain

α²₃

λ1ϕ,ξsinhϕ λ₁sinhϕ

,ξ

+α²₂

λ1ϕ,ξcoshϕ λ₁coshϕ

,ξ

= 0.

Since l₂ and l₃ are not constant, we have α²₂+α²₃ 6= 0, which implies ϕ_,ξξ = 0.Consequently, ϕ(ξ) =bξ+ξ₀.

b) If l2 = λ2, it follows from Lemma 2 that ξ = α1x1+α3x3. Then Guichard condition implies thatl₁ =λ₂cosϕ(ξ) andl₃=λ₂sinϕ(ξ). As in the casea), from equation (3.10) we get

α²₁−α²₃

ϕ_,ξξ = 0. Since l₁ and l₃ are non constant, we haveα²₁+α²₃ 6= 0. Then we have two cases to consider:

b.1) Ifα²₁ 6=α²₃, thenϕ(ξ) =bξ+ξ₀;

b.2) Ifα²₁ =α²₃, thenϕcan be any function of ξ.

c) The proof is the same as ina).

Next, we consider the solutions invariant under the 2-dimensional subgroup involving trans- lations and dilations. In this case, the basic invariant is given by

η= a1x1+a2x2+a3x3

b₁x₁+b₂x₂+b₃x₃, (3.31)

where the vectors (a1, a2, a3) and (b1, b2, b3) are linearly independent. If f =f(η) is a function depending on η, then

f_,xi =f_,ηη_xi = a_i−b_iη

b₁x₁+b₂x₂+b₃x₃f_,η.

In order to simplify the computations, we will use the following notation:

N_i :=a_i−b_iη and β =b₁x₁+b₂x₂+b₃x₃. (3.32) Then we have η_,xi = ^N_βⁱ.

In order to obtain the solutions of Lam´e’s systeml_i(η), which depend onη, we will need some lemmas.

Lemma 3. Let l₁(η), l₂(η), l₃(η), where η is given by (3.31), be a solution of Lam´e’s system (3.6)–(3.11). Suppose that for a fixed pair j, k∈ {1,2,3}, j 6=k, (aj, bj)6= (0,0) and (a_k, b_k)6=

(0,0). Then there existsc_i ∈Rsuch that l_i,η =c_i l_kl_j

NkNj

, i6=j, k, (3.33)

where N_k is given by (3.32).

Proof . From (3.7), we have that h_ij = ^li,η_l ^Nj

jβ . Then, equation (3.9) can be written as li,ηNkNj

l_j

η

−li,ηNkNj

l_j

l_k,η l_k = 0,

which implies li,ηNkNj

l_kl_j

,η

= 0.

Since (a_j, b_j)6= (0,0) and (a_k, b_k)6= (0,0), we have thatN_j 6= 0,N_k 6= 0 and the equation (3.33)

holds.

Lemma 4. Let l₁(η), l₂(η), l₃(η), where η is given by (3.31), be a solution of Lam´e’s system (3.6)–(3.11). If (a_i, b_i) = (0,0), for some i∈ {1,2,3}, then l_i is constant.

Proof . Since the vectors (a1, a2, a3) and (b1, b2, b3) are linearly independent, if (ai, bi) = (0,0) we must have (aj, bj) 6= (0,0) and (a_k, b_k)6= (0,0) for i, j, k distinct and we can use Lemma 3.

By considering equation (3.10), we have c_il_k

βN_k

,η

+ c_il_k

βN_k

l_j,ηNk βl_k = 0, which implies

c_i

l_k,ηN_j Nk

− l_k

N_k²(N_kβ)_,x

j+l_jl_j,ηN_k lkNj

= 0. (3.34)

By interchanging j withk, we have analogously ci

"

lj,ηNk

N_j − lj

N_j²(Njβ)_,x

k+l_kl_k,ηN_j l_jN_k

#

= 0. (3.35)

Suppose by contradiction thatci 6= 0. Then, it follows from (3.34) and (3.35) that l²_k

N_k² (N_kβ)_x

j = l_j²

N_j² (Njβ)_x

k. If a_i =b_i = 0, we must have

(a_kb_j−b_ka_j) l²_k N_k² + l²_j

N_j²

!

= 0,

which is a contradiction since (a_kb_j−b_ka_j)6= 0. Thereforec_i= 0 and l_i is constant.

Lemma 5. Let l₁(η),l₂(η),l₃(η), withη given by (3.31), be a solution of Lam´e’s system (3.6)–

(3.11). If there exists a unique function li which is a non zero constant, then(ai, bi) = (0,0).

Proof . Suppose by contradiction that (ai, bi)6= (0,0). Sincelj andl_kare not constant, fori,j,k distinct, it follows from Lemma4, that we must have (a_j, b_j)6= (0,0) and (a_k, b_k)6= (0,0). Then, Lemma 3implies that there are constants ci,cj and ck such that

l_i,η =c_i ljl_k

N_jN_k, l_j,η =c_j l_kli

N_kN_i and l_k,η =c_k l_kli

N_kN_i. Using equation (3.10) and interchanging the indices we have

c_klilj

N_j −(a_kb_i−b_ka_i)l_k

N_k = 0, (3.36)

cjc_k ljlk

N_jN_k − li

N_i² [cj(aib_k−a_kbi) +c_k(aibj−biaj)] = 0, (3.37) cj

lilk

N_k −(ajbi−bjai)lj

N_j = 0. (3.38)

Multiplying equation (3.36) bycjN_k

lk, (3.37) by ^N_lⁱ²

i and (3.38) by ckNj

lj , the sum will reduce to c_jc_k

(l_il_jN_k)²+ (l_il_kN_j)²+ (l_jl_kN_i)²

= 0,

which is a contradiction. Then, we must have (ai, bi) = (0,0) and the lemma is proved.

Remark 1. We observe that when all pairs (a_s, b_s) are different from zero, then the proof of Lemma 5shows that the solution li(η) of Lam´e’s system is constant.

We will now obtain the solutionsls(η), when one pair (as, bs) = (0,0).

Theorem 5. Let li(η), with η given by (3.31), be a solution of Lam´e’s system invariant under the 2-dimensional subgroup involving translation and dilations. Suppose that one of the pairs (a_s, b_s) = (0,0). Then one of the following occur:

a) If (a1, b1) = (0,0)then l1 =λ1, l2 =λ1coshϕ(η), l3 =λ1sinhϕ(η), where η= ^a_b^2x2+a3x3

2x2+b3x3

and ϕ is given by ϕ(η) = C0

a2b3−a3b2

arctan

b²₂+b²₃ a3b2−a2b3

η− a2b2+a3b3

b²₂+b²₃

+C1, (3.39)

where C₀, C₁ ∈R.

b) If (a2, b2) = (0,0) then l2 = λ2, l1 = λ2cosϕ(η), l3 = λ2sinϕ(η), where η = ^a_b^1x1+a3x3

1x1+b3x3

and ϕ is given as follows:

b.1) if b₁ =b₃ =b, then ϕ(η) = D₀

2b(a3−a1)log (2bη−a1−a3) +D1, (3.40) where D₀, D₁∈R;

b.2) if b₁ 6=b₃, then ϕ(η) = D2

2(a₁b₃−a₃b₁)log

(b3+b1)η−(a3+a1) (b₃−b₁)η−(a₃−a₁)

+D3, (3.41)

where D2, D3∈R.

c) If (a3, b3) = (0,0), then l3 = λ3, l2 = λ3coshϕ(η), l1 = λ3sinhϕ(η), with η = ^a_b^1x1+a2x2

1x1+b2x2

and ϕ is given by ϕ(η) = E0

a₂b₁−a₁b₂ arctan

b²₂+b²₁ a₂b₁−a₁b₂

η− a2b2+a1b1

b²₂+b²₁

+E1, where E₀, E₁ ∈R.

Proof . a) If (a₁, b₁) = (0,0) then Lemma4implies thatl₁ =λ₁and Guichard condition implies thatl2 =λ1coshϕ(η) andl3=λ1sinhϕ(η). In order to findϕ, we use equation (3.10) with the following indices

h32,x2 +h23,x3+h31h21= 0.