The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces

We consider conformally flat hypersurfaces in four dimensional space forms with their associated Guichard nets and Lamé's system of equations. We show that the symmetry group of the Lamé's system, satisfying Guichard condition, is given by translations and dilations in the independent vari...

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Datum:	2013
Hauptverfasser:	dos Santos, J.P., Tenenblat, K.
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spelling	nasplib_isofts_kiev_ua-123456789-1492032025-02-09T14:57:15Z The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces dos Santos, J.P. Tenenblat, K. We consider conformally flat hypersurfaces in four dimensional space forms with their associated Guichard nets and Lamé's system of equations. We show that the symmetry group of the Lamé'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 Lamé's system, given in terms of Jacobi elliptic functions, which are invariant under translations, that correspond to a new class of conformally flat hypersurfaces. This paper is a contribution to the Special Issue “Symmetries of Dif ferential Equations: Frames, Invariants and Applications”. The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html. The authors were partially supported by CAPES/PROCAD and CNPq. 2013 Article The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces / J.P. dos Santos, K. Tenenblat // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 24 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53A35; 53C42 DOI: http://dx.doi.org/10.3842/SIGMA.2013.033 https://nasplib.isofts.kiev.ua/handle/123456789/149203 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We consider conformally flat hypersurfaces in four dimensional space forms with their associated Guichard nets and Lamé's system of equations. We show that the symmetry group of the Lamé'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 Lamé's system, given in terms of Jacobi elliptic functions, which are invariant under translations, that correspond to a new class of conformally flat hypersurfaces.
format	Article
author	dos Santos, J.P. Tenenblat, K.
spellingShingle	dos Santos, J.P. Tenenblat, K. The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces Symmetry, Integrability and Geometry: Methods and Applications
author_facet	dos Santos, J.P. Tenenblat, K.
author_sort	dos Santos, J.P.
title	The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces
title_short	The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces
title_full	The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces
title_fullStr	The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces
title_full_unstemmed	The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces
title_sort	symmetry group of lamé's system and the associated guichard nets for conformally flat hypersurfaces
publisher	Інститут математики НАН України
publishDate	2013
url	https://nasplib.isofts.kiev.ua/handle/123456789/149203
citation_txt	The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces / J.P. dos Santos, K. Tenenblat // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 24 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 033, 27 pages The Symmetry Group of Lamé’s System and the Associated Guichard Nets for Conformally Flat Hypersurfaces? João Paulo dos SANTOS and Keti TENENBLAT Departamento de Matemática, Universidade de Braśılia, 70910-900, Braśılia-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 Lamé’s system of equations. We show that the symmetry group of the Lamé’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 Lamé’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; Lamé’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 when n + 1 ≥ 5. He proved that such hypersurfaces are quasi-umbilic, i.e., one of the principal curvatures has multiplicity at least n−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 at http://www.emis.de/journals/SIGMA/SDE2012.html mailto:j.p.santos@mat.unb.br mailto:k.tenenblat@mat.unb.br http://dx.doi.org/10.3842/SIGMA.2013.033 http://www.emis.de/journals/SIGMA/SDE2012.html 2 J.P. dos Santos and K. Tenenblat 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 metric g = 3∑ i=1 l2i dx 2 i , where the functions li satisfy the Guichard condition, namely, l21 − l22 + l23 = 0, and a system of second-order partial differential equations, which is called Lamé’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 function ϕ(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 solutions li satisfying Lamé’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 solutions li. We first determine the symmetry group of Lamé’s system satisfying the Guichard condition. We prove that the group is given by translations and dilations of the independent variables xi and dilations of the dependent variables li. We obtain the solutions li, i = 1, 2, 3, which are invariant under the action of the 2-dimensional translation subgroup, i.e., li(ξ), where ξ = 3∑ i=1 αixi. These solutions are given explicitly in Theorem 3 by Jacobi elliptic functions, whenever all the functions li are not constant and in Theorem 4 when one of the functions li is constant. Moreover, we consider the solutions li which are invariant under the 2-dimensional subgroup involving translations and dilations, i.e., li(η), where η = 3∑ j=1 ajxj/ 3∑ k=1 bkxk. In this case, if we require the functions li(η) to depend on all three variables, then li are constant functions. Otherwise, the solutions li(η) are given explicitly in Theorem 5. The symmetry subgroup of dilations on the dependent variables is irrelevant for the study of conformally flat hypersurfaces. Considering the functions li which 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 functions li 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 The Symmetry Group of Lamé’s System and Conformally Flat Hypersurfaces 3 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 invariant ξ depends on all three independent variables, then the functions li(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 Lamé’s system, and Guichard nets. In Section 3, we obtain the symmetry group of Lamé’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 Lamé’s system which are invariant under the action of the translation group. The solutions li of Lamé’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 Lamé’s system is reduced to a system of ordinary differential equations. 2 Lamé’s system and conformally flat hypersurfaces Consider the Minkowski space R6 1 with coordinates (x0, . . . , x5) and the scalar product 〈 , 〉 given by 〈 , 〉 : R6 × R6 −→ R, (v, w) 7→ −v0w0 + 5∑ i=1 viwi. Let L5 = { y ∈ R6 1 \| 〈y, y〉=0 } , be the light cone in R6 1 and consider mK ∈ R6 1, with 〈mK ,mK〉=K. Then, it is not difficult to see that, the sets M4 K = { y ∈ L5 \| 〈y,mK〉 = −1 } , with the metric induced from R6 1, are complete Riemannian manifolds with constant sectional curvature K. If K < 0, then M4 K 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 → M4 K ⊂ L5, with unit normal n. Then 〈df, n〉 ≡ 0, and n also satisfies 〈n,mK〉 = 〈n, f〉 = 0. Let f̃ : M3 → L5 be an immersion given by f̃ = euf , where u is a differentiable function on M . Observe that the metric induced on f̃ is conformal to the metric induced on the immersion f , i.e., 〈df̃ , df̃〉 = e2u〈df, df〉. Definition 1. Let f : M3 → L5 be an immersion such that the induced metric, 〈df, df〉, is positive definite. Let n be a unit normal with 〈f, n〉 = 0 and consider differentiable functions u and a on M3. Then the pair (f, n) is called a strip and the pair (f̃ , ñ) given by f̃ = euf, ñ = n+ af is called a conformal deformation of the strip (f, n). 4 J.P. dos Santos and K. Tenenblat Therefore, we can deform a conformally flat immersion in a space form f : M3 →M4 K ⊂ L5 to a flat immersion in the light cone f̃ : 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 M4 k is generic if it has three distinct principal curvatures. Hertrich-Jeromin in [8] established a relation between generic conformally flat hypersurfaces in M4 k and Guichard nets [6]. Namely, let e1, e2, e3 be an orthonormal frame tangent to M3 ⊂M4 k , such that ei are principal directions. Let ω1, ω2, ω3 be the co-frame and let k1, k2, k3 be the principal curvatures. Assume that locally k3 > k2 > k1, then the conformal fundamental forms α1 = √ (k3 − k1)(k2 − k1)ω1, α2 = √ (k2 − k1)(k3 − k2)ω2, α3 = √ (k3 − k2)(k3 − k1)ω3 are closed, if and only if, the hypersurface M3 is conformally flat. Therefore, when αi are closed forms, locally there exist x1, x2, x3 such that α1 = dx1, α2 = dx2 and α3 = dx3. By integration, we obtain a special principal coordinate system x1, x2, x3 for a conformally flat hypersurface in M4 K . Definition 2. A triply orthogonal coordinate system in a Riemannian 3-manifold (M, g) x = (x1, x2, x3) : (M, g)→ R3, where the functions li = √ 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 immersions f : M3 → L5. For such a flat immersion, we express the induced metric g = 〈df, df〉, in terms of the Guichard net, as g = l21dx 2 1 + l22dx 2 2 + l23dx 2 3. Since the metric is flat, the functions li must satisfy the Lamé’s system [14, pp. 73–78]: ∂2li ∂xj∂xk − 1 lj ∂li ∂xj ∂lj ∂xk − 1 lk ∂li ∂xk ∂lk ∂xj = 0, ∂ ∂xj ( 1 lj ∂li ∂xj ) + ∂ ∂xi ( 1 li ∂lj ∂xi ) + 1 l2k ∂li ∂xk ∂lj ∂xk = 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 space R3 and f 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 functions li satisfy the Guichard condition (2.1) and the Lamé’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 M4 K , there exists a Guichard net x : U ⊂ R3 → R3 on an open set U of the Euclidean space R3 (uniquely determined up to a Möbius transformation of R3). The Symmetry Group of Lamé’s System and Conformally Flat Hypersurfaces 5 Conversely, given a Guichard net x = (x1, x2, x3) : U ⊂ R3 → R3 for the Euclidean space, with li = √ g(∂xi , ∂xj ), where g is the canonical flat metric, there exists a generic conformally flat hypersurface in a space form M4 K (in this case, Möbius equivalent Guichard nets are related to conformally equivalent immersions), whose induced metric is given by g = e2P (x) { l21dx 2 1 + l22dx 2 2 + l23dx 2 3 } , (2.3) where P (x) is a function depending on M4 K . The converse is based on the fact that the functions li 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 Lamé’s system. Therefore, one way of obtaining generic conformally flat hypersurfaces in space forms M4 K is finding solutions of Lamé’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 Lamé’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 Lamé’s system In this section, we obtain the Lie point symmetry group of Lamé’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 system S of n-th order differential equations in p independent and q dependent variables is given as a system of equations ∆r ( x, u(n) ) = 0, v = 1, . . . , l, (3.1) involving x = (x1, . . . , xp), u = (u1, . . . , uq) and the derivatives u(n) of u with respect to x 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 g of the group G is called an infinitesimal generator. Consider v as a vector field on M ⊂ X ×U , with corresponding (local) one-parameter group exp(εv), i.e., exp(εv) ≡ Ψ(ε, x), 6 J.P. dos Santos and K. Tenenblat 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 formula and the infinitesimal criterion that are described as follows. Given a vector field on M ⊂ X × U , v = p∑ i=1 ξi(x, u) ∂ ∂xi + q∑ α=1 φα(x, u) ∂ ∂uα , the n-th prolongation of v is the vector field pr(n)v = v + q∑ α=1 ∑ J φJα ( x, u(n) ) ∂ ∂uαJ . It is defined on the corresponding jet 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, . . . , jk), with 1 ≤ jk ≤ p, 1 ≤ k ≤ n. The coefficient functions φJα of pr(n)v are given by the following formula: φJα ( x, u(n) ) = DJ ( φα − p∑ i=1 ξiuαJ,i, ) , where uαi = ∂uα ∂xi , uαJ,i = ∂uαJ ∂xi and DJ is given by the total derivatives DJ = Dj1Dj2 · · ·Djk , with Dif ( x, u(n) ) = ∂f ∂xi + p∑ α1 ∑ J uαJ,i ∂f ∂uαJ . We say that the system (3.1) is a system of maximal rank over M ⊂ X × U , if the Jacobian matrix J∆ ( x, u(n) ) = ( ∂∆r ∂xi , ∂∆r ∂uα,J ) has rank l, whenever ∆r ( x, u(n) ) = 0, where J = (j1, . . . , jk) is a multi-index that denotes the partial derivatives of uα. Suppose that (3.1) is a system of maximal rank. Then the set of all vectors fields v 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 both x and u, the infinitesimal criterion provides a system of partial differential equations for the coefficients ξi and φα of v, called the determining equations. By solving these equations, we obtain the vector field v that determines a Lie algebra g. The symmetry group G is obtained by exponentiating the Lie algebra. The Symmetry Group of Lamé’s System and Conformally Flat Hypersurfaces 7 3.1 Obtaining the symmetry group of Lamé’s system From now on, we consider the following notation for derivatives of a function f = f(x1, . . . , xn) f,xi := ∂f ∂xi and f,xixj := ∂2f ∂xi∂xj . With this notation, Lamé’s system (2.2) is given by li,xjxk − li,xj lj,xk lj − li,xk lk,xj lk = 0, (3.3)( li,xj lj ) ,xj + ( lj,xi li ) ,xi + li,xk lj,xk l2k = 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 εil 2 i + εjl 2 j + εkl 2 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 functions hij , with i 6= j, given by li,xj − hijlj = 0. With these functions, we rewrite (3.3) and (3.4) as hij,xk − hikhkj = 0, hij,xj + hji,xi + hikhjk = 0. for i, j, k distinct. Since the functions l1, 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 xi, we have εili,xi + εjhjilj + εkhkilk = 0, for i, j, k distinct. The derivatives of the above equation with respect to xj 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 Lamé’s system, that we call Lamé’s system of first order εil 2 i + εjl 2 j + εkl 2 k = 0, (3.6) li,xj − hijlj = 0, (3.7) εili,xi + εjhjilj + εkhkilk = 0, (3.8) hij,xk − hikhkj = 0, (3.9) hij,xj + hji,xi + hikhjk = 0, (3.10) εihij,xi + εjhji,xj + εkhkihkj = 0. (3.11) By considering x = (x1, x2, x3), l = (l1, l2, l3) and h the off-diagonal 3× 3 matrix given by hij in our next two results, we obtain the Lie algebra of the infinitesimal generators and the symmetry group of Lamé’s system of first order. 8 J.P. dos Santos and K. Tenenblat Theorem 2. Let V be the infinitesimal generator of the symmetry group of Lamé’s system of first order (3.6)–(3.11), given by V = 3∑ i=1 ξi(x, l, h) ∂ ∂xi + 3∑ i=1 ηi(x, l, h) ∂ ∂li + 3∑ i,j=1, i 6=j φij(x, l, h) ∂ ∂hij . (3.12) Then the functions ξi, ηi and φij are given by ξi = axi + ai, ηi = cli, φij = −ahij , where a, c, ai ∈ R. The proof of Theorem 2 is very long and technical. It consists of obtaining the functions ξi, ηi and φij 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 and l 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 Appendix A. As a consequence of Theorem 2, by exponentiating V , we obtain the symmetry group of Lamé’s system. Observe that the functions φij do not depend on x and l (see [18] for symmetry group of equivalent systems): Corollary 1. The symmetry group of Lamé’s system (3.6)–(3.11) is given by the following transformations: 1) translations in the independent variables: x̃i = xi + vi; 2) dilations in the independent variables: x̃i = λxi; 3) dilations in the dependent variables: l̃i = 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 has p independent variables and an s-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 the s-dimensional subalgebras of the Lie algebra of infinitesimal symmetries v. For the remainder of this paper, we will consider the 2-dimensional subgroups of the symmetry group of Lamé’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 solutions li such that li(x1, x2, x3) = li(ξ), 1 ≤ i ≤ 3, The Symmetry Group of Lamé’s System and Conformally Flat Hypersurfaces 9 where ξ is given by (3.13). For such solutions, Lamé’s system reduces to a system of ODEs. We start with two lemmas: Lemma 1. Let ls(ξ), s = 1, 2, 3, where ξ = 3∑ s=1 αsxs, be a solution of Lamé’s system (3.6)– (3.11). Let i, k ∈ {1, 2, 3} be two fixed and distinct indices such that αi = αk = 0. Then li or lk is constant. Proof. Since αi = αk = 0, it follows from (3.7) that equation (3.10) reduces to α2 j [ li,ξ lj ] ,ξ = 0, which implies li,ξ = cilj , where ci ∈ R. Similarly, interchanging i with k, we obtain lk,ξ = cklj . Finally, interchanging k with j, we get α2 j li,ξlk,ξ l2j = α2 jcick = 0. Therefore, we conclude that li or lk is constant. � Lemma 2. Let ls(ξ), s = 1, 2, 3, where ξ = 3∑ s=1 αsxs, be a solution of Lamé’s system (3.6)–(3.11). If there exists a unique j ∈ {1, 2, 3} such that lj is a non zero constant, then αj = 0. Proof. Interchanging the indices in (3.9), we obtain the following two equations αjαk ( li,ξξ − li,ξlk,ξ lk ) = 0, (3.14) αjαi ( lk,ξξ − lk,ξli,ξ li ) = 0, (3.15) and an identity. Similarly, it follows from (3.10) that α2 j li,ξξ = 0, (3.16) α2 j lk,ξξ = 0, (3.17) α2 k ( li,ξ lk ) ,ξ + α2 i ( lk,ξ li ) ,ξ + α2 j li,ξlk,ξ lj = 0. (3.18) Suppose, by contradiction, that αj 6= 0. It follows from (3.16) and (3.17) that li,ξ = ci and lk,ξ = ck, where ci 6= 0 and ck 6= 0, since by hypothesis, li and lk 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 Lamé’s system, satisfying Guichard condition, which are invariant under the action of the translation group, whenever none of the functions li is constant. Theorem 3. Let ls(ξ), s = 1, 2, 3, where ξ = 3∑ s=1 αsxs, be a solution of Lamé’s system (3.6)– (3.11), such that ls is not constant for all s. Then there exist cs ∈ R \ {0}, such that, li,ξ = cilklj , i, j, k distinct, (3.19) 10 J.P. dos Santos and K. Tenenblat c1 − c2 + c3 = 0, (3.20) α2 1c2c3 + α2 2c1c3 + α2 3c1c2 = 0. (3.21) Moreover, the functions li(ξ) are given by l21,ξ = c2(c2 − c1) ( l21 − λ c2 )( l21 − 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- ma 1, that αs 6= 0 for at least two distinct indices. Suppose that αj and αk non zero. From (3.7) and (3.9) we obtain αjαk {[ li,ξ lj ] ,ξ − li,ξ lj lk,ξ lk } = 0, which implies[ li,ξ lj ] ,ξ [ li,ξ lj ]−1 = lk,ξ lk . Integrating this equation, we obtain li,ξ = cilklj , where ci 6= 0. If αi 6= 0, analogously considering the non zero pairs (αi, αj) and (αi, αk), we conclude that lk,ξ = cklilj and lj,ξ = cjlilk. If αi = 0, then from equation (3.10) we have[ li,xj lj ] ,xj + li,xk lk lj,xk lk = α2 jcilk,ξ + α2 kcilj lj,ξ lk = 0. Since ci 6= 0, we integrate the above expression to obtain α2 j l 2 k + α2 kl 2 j = λjk, where λjk is a constant. This equation and Guichard condition (3.6) lead to l2j = α2 j α2 k ( λjk α2 j − l2k ) , l2i = εi α2 k [ l2k ( εjα 2 j − εkα2 k ) − εjλjk ] . (3.25) Taking the derivative of the last equation with respect to ξ, we have li (cilklj) = εi α2 k [ lklk,ξ ( εjα 2 j − εkα2 k )] . If εjα 2 j − εkα2 k 6= 0, we conclude that lk,ξ = ciα 2 k εjα2 j − εkα2 k lilj = cklilj . The Symmetry Group of Lamé’s System and Conformally Flat Hypersurfaces 11 Applying this expression into the derivative of the first equation in (3.25) with respect to ξ we obtain ljlj,ξ = − α2 j α2 k lklk,ξ = − α2 j α2 k lk (cklilj) , consequently, lj,ξ = cjlilk. Next, we will show that εjα 2 j − εkα 2 k 6= 0 to conclude the proof of (3.19). Suppose by contradiction that εjα 2 j−εkα2 k = 0, then the first equation of (3.25) can be written as εjl 2 j+εkl 2 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 l1,ξ = c1l2l3, (3.26) l2,ξ = c2l1l3, (3.27) l3,ξ = c3l1l2. (3.28) Multiplying (3.27) by l2 and integrating we have l22 = c2 c1 ( l21 − λ c2 ) , (3.29) where λ is 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 l1,ξ 2 = c2 1 [ c2 c1 ( l21 − λ c2 )][ c2 − c1 c1 ( l21 − λ c2 − c1 )] = c2 (c2 − c1) ( l21 − λ c2 )( l21 − λ c2 − c1 ) . � In our next theorem, we consider the solutions li(ξ) when one of the functions li is constant. Theorem 4. Let ls(ξ), s = 1, 2, 3, where ξ = 3∑ s=1 αsxs, be a solution of Lamé’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 = λ1 cosh(bξ + ξ0), l3 = λ1 sinh(bξ + ξ0), where ξ = α2x2 + α3x3, α2 2 + α2 3 6= 0 and b, ξ0 ∈ R ; b) l2 = λ2, l1 = λ2 cosϕ(ξ), l3 = λ2 sinϕ(ξ), where ξ = α1x1 + α3x3, α2 1 + α2 3 6= 0 and ϕ is one of the following: b.1) ϕ(ξ) = bξ + ξ0, if α2 1 6= α2 3, where ξ0, b ∈ R; b.2) ϕ is any function of ξ, if α2 1 = α2 3; c) l3 = λ3, l2 = λ3 cosh(bξ + ξ0), l1 = λ3 sinh(bξ + ξ0), where ξ = α1x1 + α2x2, α2 1 + α2 2 6= 0 and b, ξ0 ∈ R. 12 J.P. dos Santos and K. Tenenblat 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 = λ1 coshϕ(ξ) and l3 = λ1 sinhϕ(ξ). In order to determi- ne ϕ, we use (3.10) with the following indices h23,x3 + h32,x2 + h21h31 = 0, to obtain α2 3 ( λ1ϕ,ξ sinhϕ λ1 sinhϕ ) ,ξ + α2 2 ( λ1ϕ,ξ coshϕ λ1 coshϕ ) ,ξ = 0. Since l2 and l3 are not constant, we have α2 2 + α2 3 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 that l1 = λ2 cosϕ(ξ) and l3 = λ2 sinϕ(ξ). As in the case a), from equation (3.10) we get( α2 1 − α2 3 ) ϕ,ξξ = 0. Since l1 and l3 are non constant, we have α2 1 + α2 3 6= 0. Then we have two cases to consider: b.1) If α2 1 6= α2 3, then ϕ(ξ) = bξ + ξ0; b.2) If α2 1 = α2 3, then ϕ can be any function of ξ. c) The proof is the same as in a). � 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,ηηxi = ai − biη b1x1 + b2x2 + b3x3 f,η. In order to simplify the computations, we will use the following notation: Ni := ai − biη and β = b1x1 + b2x2 + b3x3. (3.32) Then we have η,xi = Ni β . In order to obtain the solutions of Lamé’s system li(η), which depend on η, 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 Lamé’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 (ak, bk) 6= (0, 0). Then there exists ci ∈ R such that li,η = ci lklj NkNj , i 6= j, k, (3.33) where Nk is given by (3.32). Proof. From (3.7), we have that hij = li,ηNj ljβ . Then, equation (3.9) can be written as[ li,ηNkNj lj ] η − li,ηNkNj lj lk,η lk = 0, The Symmetry Group of Lamé’s System and Conformally Flat Hypersurfaces 13 which implies( li,ηNkNj lklj ) ,η = 0. Since (aj , bj) 6= (0, 0) and (ak, bk) 6= (0, 0), we have that Nj 6= 0, Nk 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 Lamé’s system (3.6)–(3.11). If (ai, bi) = (0, 0), for some i ∈ {1, 2, 3}, then li 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 (ak, bk) 6= (0, 0) for i, j, k distinct and we can use Lemma 3. By considering equation (3.10), we have( cilk βNk ) ,η + ( cilk βNk ) lj,ηNk βlk = 0, which implies ci [ lk,ηNj Nk − lk N2 k (Nkβ),xj + ljlj,ηNk lkNj ] = 0. (3.34) By interchanging j with k, we have analogously ci [ lj,ηNk Nj − lj N2 j (Njβ),xk + lklk,ηNj ljNk ] = 0. (3.35) Suppose by contradiction that ci 6= 0. Then, it follows from (3.34) and (3.35) that l2k N2 k (Nkβ)xj = l2j N2 j (Njβ)xk . If ai = bi = 0, we must have (akbj − bkaj) ( l2k N2 k + l2j N2 j ) = 0, which is a contradiction since (akbj − bkaj) 6= 0. Therefore ci = 0 and li is constant. � Lemma 5. Let l1(η), l2(η), l3(η), with η given by (3.31), be a solution of Lamé’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). Since lj and lk are not constant, for i, j, k distinct, it follows from Lemma 4, that we must have (aj , bj) 6= (0, 0) and (ak, bk) 6= (0, 0). Then, Lemma 3 implies that there are constants ci, cj and ck such that li,η = ci ljlk NjNk , lj,η = cj lkli NkNi and lk,η = ck lkli NkNi . Using equation (3.10) and interchanging the indices we have ck lilj Nj − (akbi − bkai)lk Nk = 0, (3.36) 14 J.P. dos Santos and K. Tenenblat cjck ljlk NjNk − li N2 i [cj(aibk − akbi) + ck(aibj − biaj)] = 0, (3.37) cj lilk Nk − (ajbi − bjai)lj Nj = 0. (3.38) Multiplying equation (3.36) by cj Nk lk , (3.37) by N2 i li and (3.38) by ck Nj lj , the sum will reduce to cjck [ (liljNk) 2 + (lilkNj) 2 + (ljlkNi) 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 (as, bs) are different from zero, then the proof of Lemma 5 shows that the solution li(η) of Lamé’s system is constant. We will now obtain the solutions ls(η), when one pair (as, bs) = (0, 0). Theorem 5. Let li(η), with η given by (3.31), be a solution of Lamé’s system invariant under the 2-dimensional subgroup involving translation and dilations. Suppose that one of the pairs (as, bs) = (0, 0). Then one of the following occur: a) If (a1, b1) = (0, 0) then l1 = λ1, l2 = λ1 coshϕ(η), l3 = λ1 sinhϕ(η), where η = a2x2+a3x3 b2x2+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 = λ2 cosϕ(η), l3 = λ2 sinϕ(η), where η = a1x1+a3x3 b1x1+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 = λ3 coshϕ(η), l1 = λ3 sinhϕ(η), with η = a1x1+a2x2 b1x1+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 Lemma 4 implies that l1 = λ1 and Guichard condition implies that l2 = λ1 coshϕ(η) and l3 = λ1 sinhϕ(η). In order to find ϕ, we use equation (3.10) with the following indices h32,x2 + h23,x3 + h31h21 = 0. The Symmetry Group of Lamé’s System and Conformally Flat Hypersurfaces 15 Since h32 = ϕ,ηN2 β , h23 = ϕ,ηN3 β and h31 = h21 = 0, we rewrite the equation above as( ϕ,ηN2 β ) ,x2 + ( ϕ,ηN3 β ) ,x3 = 0. By substituting the derivatives, we have the following ODE ϕ,ηη [ (N2)2 + (N3)2 ] − 2ϕ,η(b2N2 + b3N3) = 0, whose solution is exactly (3.39). b) If (a2, b2) = (0, 0), then l2 = λ2 and Guichard condition implies that l1 = λ2 cosϕ(η) and l3 = λ2 sinϕ(η). In order to find ϕ, we use equation (3.10) with the following indices h13,x3 + h31,x1 + h12h32 = 0. By using the same arguments as in a), we have the following ODE ϕ,ηη [ (N1)2 − (N3)2 ] − 2ϕ,η(b1N1 − b3N3) = 0, whose solution will depend on b1 and b3. If b1 = b3 we have ϕ given by (3.40) and if b1 6= b3, the solution is given by (3.41). c) The arguments when (a3, b3) = (0, 0) are the same as in a). � Remark 2. Although our calculation of the symmetry group for the Lamé’s system has similar techniques to those used by Tenenblat and Winternitz for the intrinsic generalized wave and sine-Gordon equations in [24], we observe that the solutions invariant under the subgroups are quite different. In fact, when we consider the solutions invariant under the translation subgroup in Theorem 3, the solutions of (3.22) are given by Jacobi elliptic functions that cannot be reduced to elementary functions. Moreover, the only solutions of the Lamé’s system, which are invariant under the action of the subgroup involving translation and dilations, that depend on all three variables are constant, in contrast to the solutions in [24]. The main reason is due to Guichard condition. In the next two sections, we will deal with the geometric properties of the Guichard nets and of the conformally flat hypersurfaces associated to the solutions invariant under the 2- dimensional translation subgroup. As we will see in Section 5, these are the solutions that will provide a new class of conformally flat hypersurfaces. 4 Geometric properties of the Guichard nets In this section, we will study the geometric properties of the Guichard nets associated to locally conformally flat hypersurfaces corresponding to the solutions of the Lamé’s system li(ξ), which are invariant under the translation subgroup. Let l1(ξ), l2(ξ), l3(ξ), with ξ = 3∑ s=1 αsxs be a solution of Lamé’s system. Theorem 1 implies that there is a Guichard net x = (x1, x2, x3) : U ⊂ R3 → R3, with a Riemannian metric g = l21dx 2 1 + l22dx 2 2 + l23dx 2 3, (4.1) where U is an open set, given by U = { (x1, x2, x3) ∈ R3 \| ξ1 < ξ < ξ2 } , where ξ1 and ξ2 are real constants. 16 J.P. dos Santos and K. Tenenblat 4.1 Level surfaces In this subsection, we will show that the Guichard nets are foliated by surfaces ξ = ξ0 which are geodesically parallel. Moreover, we will prove that each such surface has flat Gaussian curvature and constant mean curvature that depends on ξ0. Definition 3. Let Mn be a Riemannian manifold and let f : M → R be a differentiable function. The level submanifolds of f are said to be geodesically parallel if \| grad f \| is a non zero constant, along each level submanifold. We have the following theorem Theorem 6. Let (U, g), U ⊂ R3, be a Riemannian manifold with coordinates (x1, x2, x3) and metric g = 3∑ s=1 l2s(ξ)dx 2 i , where ξ = 3∑ s=1 αsxs. Then the level surfaces Pξ0 = { (x1, x2, x3) ∈ U ; 3∑ s=1 αsxs = ξ0 } , where ξ1 < ξ0 < ξ2, endowed with the induced metric, are geodesically parallel. Moreover, each level surface has flat Gaussian curvature and constant mean curvature (depending on ξ0). Proof. Since at least one αi is non zero, we can suppose that α3 6= 0 and we parametrize Pξ0 as X(x1, x2) = ( x1, x2, ξ0 − α1x1 − α2x2 α3 ) . Then X,x1 = (1, 0,−α1/α3) and X,x2 = (0, 1,−α2/α3). Consequently, the coefficients of the induced metric are constant, since ξ = ξ0 in this surface. Therefore the Gaussian curvature is equal to zero. Consider now the function h(x) = 3∑ i=1 αixi. Then Pξ0 = h−1(ξ0). Since h is constant along Pξ0 , it follows that gradh is normal to Pξ0 . Moreover, g(gradh, gradh) = 3∑ j=1 α2 j l2j , which implies that \| gradh\| is constant along Pξ0 . It follows from Definition 3 that the level surfaces h−1(ξ0) are geodesically parallel. Now we compute the mean curvatures of Pξ0 . Given p ∈ Pξ0 , let A : TpPξ0 → TpPξ0 be the Weingarten operator, i.e., Av = −∇v ( gradh \| gradh\| ) (p), where ∇ is the Riemannian connection on (U, g). Since \| gradh\| is constant along Pξ0 , it follows that Av = − 1 \| gradh\| ∇v gradh(p). Then the mean curvature of Pξ0 is given by H = − ∆h(p) \| gradh\| = 1 \| gradh\| ∑ i,k Γkii(ξ0)αk l2i (ξ0) , where Γkij are the Christoffel for the connection ∇. Therefore, the mean curvature of Pξ0 is a constant depending on ξ0. � The Symmetry Group of Lamé’s System and Conformally Flat Hypersurfaces 17 4.2 Coordinate surfaces In this subsection, we will use the solutions invariant by the group of translations to show that the coordinate surfaces of the corresponding Guichard net (U, g) have constant Gaussian curvature. Moreover, the values of these curvatures satisfy an algebraic relation. Theorem 7. Let (U, g), U ⊂ R3, be a Riemannian manifold, with coordinates (x1, x2, x3) and metric g = 3∑ s=1 l2s(ξ)dx 2 i , with ξ = 3∑ s=1 αsxs. Then each coordinate surface of U ⊂ R3, xi = const, endowed with the induced metric, has constant Gaussian curvature Ki. Moreover, K1 +K2 +K3 = 0. Proof. Since g is given by (4.1), it follows that the metric induced on each coordinate surface, xi = const, is gi = l2j (dxj) 2 + l2k(dxk) 2, i, j, k distinct, and its Gaussian curvature, Ki, is given by Ki = 1 ljlk ( lk,xi lj,xi l2i ) . (4.2) Assume that none of the functions li is constant and ξ = α1x1 + α2x2 + α3x3, with αi 6= 0, for all i. In this case, we have li,ξ = ciljlk, where i, j and k are distinct indices in {1, 2, 3}. Therefore, it follows from (4.2) that the Gaussian curvature of each coordinate surface is given by Ki = cjckα 2 i . Moreover, it follows from (3.21) that K1 +K2 +K3 = α2 1c2c3 + α2 2c3c1 + α3 1c1c2 = 0. If only one of the functions li is constant, it follows from Lemma 2, that, if li is constant, then αi = 0. Then it follows from (4.2) that all the curvatures are equal to zero. In fact, Ki = 0, since the functions ls, for all s, do not depend on xi. Moreover, for j 6= i, Kj = 0, since li is constant. Hence, the sum 3∑ j=1 Ki = 0 trivially. � 5 Conformally flat hypersurfaces In this section, we describe the generic conformally flat hypersurfaces associated to the solutions of the Lamé’s system invariant under the translation group. It is known that, any locally generic conformally flat hypersurface, in a 4-dimensional space form, has a metric induced by the Guichard net of the form (see [10, 21, 22]) g = e2P (x) { sin2 ϕ(x)dx2 1 + dx2 2 + cos2 ϕ(x)dx2 3 } , (5.1) where x = (x1, x2, x3), or g = e2P̃ (x) { sinh2 ϕ̃(x)dx2 1 + cosh2 ϕ̃(x)dx2 2 + dx2 3 } . (5.2) Suyama classified in [22] the hypersurfaces conformal to the products M2× I ⊂ R4 given by Lafontaine in [13], as the hypersurfaces where ϕ depends only on two variables. Hertrich-Jeromin and Suyama classified in [10] the hypersurfaces where ϕ has two vanishing mixed derivatives. These conformally flat hypersurfaces are associated to the so called cyclic Guichard nets, which are characterized by ϕ,x1x2 = ϕ,x2x3 = 0, when g is of the form (5.1) and by ϕ,x1x3 = ϕ,x2x3 = 0, 18 J.P. dos Santos and K. Tenenblat when g is given by (5.2). Moreover, the authors showed that all the known cases of conformally flat hypersurfaces, up to now, are associated to cyclic Guichard nets. We observe that Theorem 5 shows that each solution of the Lamé’s system, which is invariant under the action of the 2-dimensional subgroup involving translations and dilations, depends only on two variables. Therefore, the conformally flat hypersurfaces associated to these solutions are conformal to the products M2 × I ⊂ R4. We now consider the conformally flat hypersurfaces associated to the solutions invariant under the translation subgroup. We analyse each case separately: i) ξ = α1x1 +α2x2. In this case, we have the solutions l1 = λ3 sinh(ξ+ξ0), l2 = λ3 cosh(ξ+ξ0) and l3 = λ3 6= 0 (see Theorem 4). Then the associated conformally flat hypersurface has a Guichard net, where the induced metric is given by g = e2P (x1,x2,x3) { sinh2(ξ + ξ0)dx2 1 + cosh2(ξ + ξ0)dx2 2 + dx2 3 } . (5.3) The hypersurface is conformal to one of the products considered by Lafontaine in [13] that we describe as follows (see [21, 22] for details): Let H3 be the hyperbolic 3-space, considered as the half space model and as a subset of R4, i.e., H3 = {( y1, y2, y3, 0 ) : y3 > 0 } ⊂ R4 = {( y1, y2, y3, y4 ) : yi ∈ R } , with the metric gij = δij y23 . Consider the rotations of the y3-axis given by( y1, y2, y3, 0 ) → ( y1, y2, y3 cos t, y3 sin t ) , then the hypersurface M3 = M2 × I, obtained by the above rotation of a surface of constant curvature M2 ⊂ H3 is a conformally flat hypersurface. One can show that for g given by (5.3), the surface M2 ⊂ H3 is a flat surface, parametrized by lines of curvature whose first and second fundamental forms are given by I = sinh2(ξ + ξ0)dx2 1 + cosh2(ξ + ξ0)dx2 2, II = sinh(ξ + ξ0) cosh(ξ + ξ0) ( dx2 1 + dx2 2 ) . (5.4) In order to describe the flat surfaces M2 ⊂ H3, we mention a classification result obtained by the authors in collaboration with Mart́ınez [16]. It is well known that, on a neighbourhood of a non-umbilical point, a flat surface in H3 can be parametrized by lines of curvature, so that the first and second fundamental forms are given by (for details, see [23, Theorem 2.4, Corollary 2.7]) I = sinh2 φ(u, v)(du)2 + cosh2 φ(u, v)(dv)2, (5.5) II = sinhφ(u, v) coshφ(u, v) ( (du)2 + (dv)2 ) , (5.6) where φ is a harmonic function, i.e. φuu + φvv = 0. The classification result is given as follows: Theorem 8 ([16]). Let Σ be a flat surface in H3 with a local parametrization, in a neighborhood of a nonsingular and nonumbilic point, such that the first and second fundamental forms are diagonal and given by (5.5) and (5.6), where φ is a (Euclidean) harmonic function. Then φ is linear, i.e., φ = au + bv + c if, and only if, Σ is locally congruent to either a helicoidal flat surface (when (a, b, c) 6= (0,±1, 0)) or to a “peach front” (when (a, b, c) = (0,±1, 0)). Helicoidal surfaces arise as a natural generalization of rotational surfaces. They are invariant by a helicoidal group of isometries, i.e., given an axis, we consider a translation along this axis composed with a rotation around it. In the half space model of H3, up to isometries, we can The Symmetry Group of Lamé’s System and Conformally Flat Hypersurfaces 19 consider the y3-axis, which enables us to write the helicoidal group, relative to this axis, as the composition ht = eβt 0 0 0 eβt 0 0 0 eβt cosαt − sinαt 0 sinαt cosαt 0 0 0 1  , of a rotation around the y3-axis with angular pitch α with a hyperbolic translation of ratio β. The “peach front” is a special case of flat surfaces that is not helicoidal. Details about this surface can be found in [11]. The study of flat surfaces in hyperbolic 3-space has received a lot of attention in the last few years, mainly because Galvéz, Mart́ınez and Milán have shown in [5] that flat surfaces in the hyperbolic 3-space admit a Weierstrass representation formula in terms of meromorphic data as in the theory of minimal surfaces in R3. Namely, if ψ : M2 → H3 is a surface in H3, for any p ∈ M2, there exist G(p), G∗(p) ∈ C∞ distinct points in the ideal boundary of H3 such that the oriented normal geodesic at ψ(p) is the geodesic in H3 starting from G∗(p) towards G(p). The maps G,G∗ : Σ → C∞ are called the hyperbolic Gauss maps of ψ. It is proved in [5] that, for flat surfaces, they are holomorphic when one considers C∞ as the Riemann sphere and M2 has a complex structure induced by the second fundamental form. Conversely, given two holomorphic functions G and G∗, with G 6= G∗, one can recover a flat immersion of a surface in H3 (for more details see also [3, 11, 12]). This representation formula was the main tool to obtain Theorem 8. With the previous results we conclude that Theorem 9. Let li(ξ) be solutions of the Lamé’s system, where ξ = α1x1 + α2x2. Then the associated conformally flat hypersurfaces are conformal to the product, M2 × I, where M2 is locally congruent to either a helicoidal flat surface in H3 or the “peach front”. Proof. When ξ = α1x1 + α2x2, it follows from Theorem 4 that the solution of Lamé’s system is l1 = λ3 sinh(ξ + ξ0), l2 = λ3 cosh(ξ + ξ0) and l3 = λ3 6= 0 and the corresponding conformally flat hypersurface M3 has a metric g given by (5.3). Then M3 is conformal to the product M2× I, where M2 is a flat surface in H3 with fundamental forms given by (5.4). It follows from Theorem 8 that M2 is locally congruent to either a helicoidal flat surface in H3 or to the “peach front”. � ii) ξ = α1x1 +α3x3. In this case, we have the solution of Lamé’s system , l1 = λ2 sin(ξ+ ξ0), l2 = λ2 6= 0 and l3 = λ2 cos(ξ + ξ0) (see Theorem 4 b)). Then the associated conformally flat hypersurface M3 has a Guichard net, whose induced metric is given by g = e2P (x1,x2,x3) { sin2(ξ + ξ0)dx2 1 + dx2 2 + cos2(ξ + ξ0)dx2 3 } . The hypersurface M3 is conformal to another class of products M2×I (see [21, 22]). Namely, let S3 ⊂ R4 be the canonical 3-sphere, then M2 × I = { tp : 0 < t <∞, p ∈M2 ⊂ S3 } , is a confor- mally flat hypersurface, where M2 is a surface with constant curvature in S3. In our case, M3 is conformal to the product M2 × I, where the surface M2 ⊂ S3 is a flat surface, parametrized by lines of curvature, whose first and second fundamental forms are given by I = sin2(ξ + ξ0)dx2 1 + cos2(ξ + ξ0)dx2 3, II = sin(ξ + ξ0) cos(ξ + ξ0) ( dx2 1 − dx2 3 ) . (5.7) The geometry of these surfaces in S3 is being studied and it will appear in another paper. iii) ξ = α1x1 +α2x2 +α3x3, αi 6= 0, for all i. In this case, we will show that the solutions li(ξ) of the Lamé’s system give rise to a new class of conformally flat hypersurfaces, according to the following theorem: 20 J.P. dos Santos and K. Tenenblat Theorem 10. Let M3 be a conformally flat hypersurface in a space form M4 K , associated to a solution of Lamé’s system li(x1, x2, x3) = li(ξ), with ξ = 3∑ s=1 αsxs and αs 6= 0, for all s, given in terms of elliptic functions by (3.22)–(3.24). Then its first fundamental form g is given by g = e2P (x) { cos2 ϕ(ξ)(dx1)2 + (dx2)2 + sin2 ϕ(ξ)(dx3)2 } , (5.8) where ϕ satisfies, ϕ2 ,ξ = c(a cos2 ϕ− b), (5.9) or g is given by g = e2P̃ (x) { sinh2 ϕ̃(ξ)(dx1)2 + cosh2 ϕ̃(ξ)(dx2)2 + (dx3)2 } (5.10) where ϕ̃ satisfies ϕ̃2 ,ξ = c ( b cosh2 ϕ̃− b ) . (5.11) where a, b, c ∈ R \ {0}, P (x) and P̃ (x) are differentiable functions that depend on ls and M4 K . In both cases, ξ ∈ I ⊂ R, where I is an open interval such that g is positive definite. Proof. Guichard condition (2.1) implies that we may consider l1 = l2 cosϕ, (5.12) l3 = l2 sinϕ. (5.13) It follows from Theorem 1 that the metric is given by (5.8). In order to obtain the expression for the derivative of ϕ with respect to ξ, we consider l1,ξ = l2,ξ cosϕ− l2ϕ,ξ sinϕ. Since αs 6= 0 for all s, the functions li are given as in Theorem 3, by (3.22)–(3.24). Hence, using (3.19), we have that c1l2l3 = c2l1l3 cosϕ− ϕ,ξl3, for c1, c2 ∈ R \ {0}, which implies ϕ,ξ = l2 ( c2 cos2 ϕ− c1 ) . (5.14) By taking the derivative again, it follows from (5.12)–(5.14) and (3.19) that ϕ,ξξ = l2,ξ ( c2 cos2 ϕ− 1 ) − 2c2l2 cosϕ sinϕϕ,ξ = 1 l2 [ l2,ξϕ,ξ − 2c2(l2 cosϕ)(l2 sinϕ)ϕ,ξ ] = 1 l2 [ l2,ξϕ,ξ − 2c2l1l3ϕ,ξ ] = 1 l2 [ l2,ξϕ,ξ − 2l2,ξϕ,ξ ] = − l2,ξϕ,ξ l2 . Therefore, ϕ,ξl2 = c, (5.15) where c ∈ R \ {0}, since ϕ,ξ 6= 0. Then, multiplying (5.14) by ϕ,ξ and using (5.15) we have that ϕ2 ,ξ = c ( c2 cos2 ϕ − c1 ) , i.e., (5.9) holds. The proof of the second part of the theorem is analogous, when we consider l2 = l3 coshϕ and l1 = l3 sinhϕ. � The Symmetry Group of Lamé’s System and Conformally Flat Hypersurfaces 21 Corollary 2. Let M3 ⊂ M4 be a conformally flat hypersurface associated to the solutions of Lamé’s system li(ξ) with ξ = 3∑ s=1 αsxs and αs 6= 0 for all s, given in terms of elliptic functions by (3.22)–(3.24). Then the Guichard net of M3 is not cyclic. Proof. It follows from Theorem 10 that the first fundamental form of M3 is given by (5.8), where ϕ(ξ) satisfies (5.9) or by (5.10) where ϕ̃(ξ) satisfies (5.11), ξ ∈ I ⊂ R. In the first case, (5.9) implies that ϕ,ξξ = λ sin 2ϕ, λ 6= 0 and in the second case, (5.11) implies that ϕ̃,ξξ = λ sin 2ϕ̃, λ 6= 0. In either case, ϕ,xixj = αiαjϕ,ξξ 6= 0, i 6= j and ξ ∈ I. Therefore, the Guichard net of M3 is not cyclic. � We observe that, as a consequence of the results of Hertrich-Jeromin and Suyama, the sur- faces M3 of Corollary 2 provide a new class of conformally flat hypersurfaces. It is important to observe that Hertrich-Jeromin and Suyama in [9] have independently considered Guichard nets with the ansatz on the function ϕ such that ϕ(x1, x2, x3) = ϕ(ax1 + bx2 + cx3). They investigated the geometric properties of these Guichard nets, that they called Bianchi-type Guichard nets, as well as the new class of associated conformally flat hypersurfaces. A Appendix Proof of Theorem 2. The infinitesimal generator associated to the symmetry group is written as in (3.12). The functions ξi, ηi, φij will be obtained by solving the determining equations that arise when we apply the first prolongation formula pr(1)V = V + ∑ i,k Dk ( ηi ) ∂ ∂li,xk + ∑ i,j,k i 6=j Dk ( φij ) ∂ ∂hij,xk − ∑ i,k,r Dk(ξ r)li,xr ∂ ∂li,xk − ∑ i,j,k,r Dk(ξ r)hij,xr ∂ ∂hij,xk , with Di = ∂ ∂xi + ∑ j lj,xi ∂ ∂li + ∑ j,l hjl,xi ∂ ∂hjl , on each equation of the system, i.e., when we consider the infinitesimal criterion (3.2). In order to avoid any functional dependence, the following substitutions will be considered li,xj = hijlj , i 6= j, (A.1) li,xi = −εiεjhjilj − εiεkhkilk (A.2) hij,xk = hikhkj , (A.3) hij,xj = −hji,xi − hikhjk, i < j, (A.4) hij,xi = −εiεjhji,xj − εiεkhkihkj , i < j. (A.5) Fixing i, j and k, distinct indices, we start applying pr(1)V to equation (3.9). Then the infinitesimal criterion (3.2), gives φij(k)−φ ikhkj −hikφkj = 0, using the prolongation formula, we get φij,xk + ∑ r φij,lr lr,xk + ∑ r,s φij,hrshrs,xk − ∑ t ( ξt,xk + ∑ r ξt,lr lr,xk + ∑ r,s ξt,hrshrs,xk ) hij,xt − φikhkj − hikφkj = 0. (A.6) 22 J.P. dos Santos and K. Tenenblat For i < j, we apply the substitutions (A.1)–(A.5) and we analyse each term of (A.6) as follows∑ r φij,lr lr,xk = ∑ r 6=k φij,lrhrklk − φ ij ,lk (εkεjhjklj + εkεihikli) , (A.7) ∑ r,s φij,hrshrs,xk = ∑ r 6=k, s6=k φij,hrshrkhks + ∑ s<k φij,hkshks,xk − ∑ s>k φij,hks ( εkεshsk,xs + εkεmhmkhms ) − ∑ r<k φij,hrk (hkr,xr + hrnhkn) + ∑ r>k φij,hrkhrk,xk , (A.8) ∑ t ( ξt,xk + ∑ r ξt,lr lr,xk + ∑ r,s ξt,hrshrs,xk ) hij,xt = Ckkhikhkj − C j k ( hji,xi + hikhjk ) − Cik ( εiεjhji,xj + εiεkhkihkj ) , (A.9) where the coefficients Ctk are given by Ctk = ξt,xk + ∑ r 6=k ξt,lrhrklk − ξ t ,lk ( εkεjhjklj + εkεihikli ) + ∑ r 6=k, s6=k ξt,hrshrkhks + ∑ s<k ξt,hkshks,xk − ∑ s>k ξt,hks ( εkεshsk,xs + εkεmhmkhms ) − ∑ r<k ξt,hrk ( hkr,xr + hrnhkn ) + ∑ r>k ξt,hrkhrk,xk , (A.10) and the indices m and n are such that {k, s,m}, s > k and {k, r, n}, r < k are two sets of three distinct numbers. Now we analyse the coefficients of equation (A.6), considering (A.7)–(A.9). By equating to zero the coefficients of the products hji,xjhks,xk , with k > s, we obtain ξi,hks = 0. Analogously, for the coefficients of hji,xjhsk,xs , with k < s, we obtain ξi,hks = 0. This implies that ξi,hks = 0, ∀ s, s 6= k, i.e. ξi,hkj = ξi,hki = 0. Similarly, from the coefficients of hji,xjhkr,xk , r < k and hji,xjhrk,xk , with r > k, we obtain ξi,hrk = 0, ∀ r, r 6= k, i.e. ξi,hjk = ξi,hik = 0, where i, j, k ∈ {1, 2, 3} are distinct and i < j. By analysing the coefficients of hji,xihks,xk with k > s and hji,xihsk,xs with k < s, we obtain ξj,hks = 0, ∀ s, s 6= k, i.e. ξj,hki = ξj,hkj = 0. Similarly, the coefficients of hji,xihkr,xr , with k > r, and hji,xihrk,xk , with k < r, lead to ξj,hrk = 0, ∀ r, r 6= k, i.e. ξj,hik = ξj,hjk = 0. Since i, j, k ∈ {1, 2, 3} are distinct and arbitrary indices, with i < j, we conclude that ξm,hst = 0 for any indices m, s and t, s 6= t, i.e., ξm depends only on x and l. Therefore, the expression of Ctk given in (A.10) reduces to Ctk = ξt,xk + ∑ r 6=k ξt,lrhrklk − ξ t ,lk (εkεjhjklj + εkεihikli), that can be rewritten as Ctk = ξt,xk + ∑ r 6=k ( ξt,lr lk − ξ t ,lk εrεklr ) hrk. (A.11) The Symmetry Group of Lamé’s System and Conformally Flat Hypersurfaces 23 From (A.8), we have that the coefficients of hks,xk , with s < k, and the coefficients of hsk,xs , with s > k, lead to φij,hks = 0, ∀ s 6= k, i.e. φij,hki = φij,hkj = 0. Considering (A.9), the coefficients of hji,xi and hji,xj imply that Cjk = 0 and Cik = 0, respectively. Since i < j and i, j, k ∈ {1, 2, 3} are arbitrary and distinct, we conclude that Cik = Cjk = 0, for all i, j, k distinct indices. Since ξm does not depend on hst, the analysis of (A.11) gives us the following system ξm,xk = 0, ∀m 6= k, ξm,lr lk − εrεkξ m ,lk lr = 0, ∀ r 6= k. The first equation of this system says that ξm depends only on xm and l. By solving the characteristic system for the second equation, we have that ξm depends on xm and a variable ζ = εil 2 i + εjl 2 j + εkl 2 k. However, Guichard condition implies that ζ ≡ 0, hence ξm does not depend on ls, for all s. Summarising the conclusions of this first part of the proof, we have that φst = φst(hst, hts, x, l), and ξm = ξm(xm). We now consider equation (3.7). By applying the prolongation pr(1)V to (3.7), we have that ηi(j) − φ ijlj − hijηj = 0, which implies, ηi,xj + ∑ r ηi,lr lr,xj + ∑ r,s ηi,hrshrs,xj − ξ j ,xj li,xj − φ ijlj − hijηj = 0. (A.12) Observe that by applying the substitution (A.1), we have∑ r ηi,lr lr,xj = ∑ r 6=j ηi,lrhrjlj − η i ,lj (εjεihijli + εjεkhkjlk). Moreover, by applying the substitutions (A.3), (A.4) and (A.5) we have∑ r, s ηi,hrshrs,xj = ∑ r 6=j, s6=j ηi,hrshrjhjs + ∑ s<j ηi,hjshjs,xj − ∑ s>j (εjεshsj,xs + εjεmhmjhms) − ∑ r<j ηi,hrj (hjr,xr + hrnhjn) + ∑ r>j ηi,hrjhrj,xj . Therefore, by considering in (A.12), the coefficients of hjs,xj , with s < j, and hsj,xs , with s > j, we conclude that ηi,hjs = 0. Similarly, the analysis of the coefficients of hjr,xr , with r < j, and hrj,xj , with r > j, imply that ηi,hrj = 0. Hence, we conclude that ηi,hjt = ηi,htj = 0, ∀ t 6= j. Since i and t 6= j are arbitrary, we conclude that ηm does not depend on hst, for any indices, m, s and t with s 6= t. Consequently, (A.12) reduces to ηi,xj + ( ηi,li lj − εiεjη i ,lj li − ξj,xj lj − η j ) hij + ( ηi,lk lj − εjεkη i ,lj lk ) hkj − φijlj = 0. (A.13) Since φij depends only on x, l, hij and hji, we obtain from (A.13) the following system ηi,lk lj − εjεkη i ,lj lk = 0, (A.14) ηi,xj + ( ηi,li lj − εiεjη i ,lj li − ξj,xj lj − η j ) hij − φijlj = 0. (A.15) 24 J.P. dos Santos and K. Tenenblat By solving the characteristic system for (A.14), we have that ni = ni(x, li). By taking derivatives of (A.15) with respect to hji we get φij,hji = 0. (A.16) On the other hand, by taking the derivatives of (A.15) twice with respect to hij , we obtain φij,hijhij = 0. (A.17) Consequently, it follows from (A.16) and (A.17) that φij is given by φij = Aij(x, l)hij +Bij(x, l). (A.18) Therefore, (A.6) reduces to φij,xk + ∑ r φij,lr lr,xk +Aijhij,xk − ξ k ,xk hij,xk − ( Aikhik +Bik ) hkj − ( Akjhkj +Bkj ) hik = 0. By considering the substitutions (A.1)–(A.5), this equation reduces to φij,xk + φij,lihiklk + φij,ljhjklk − φ ij ,lk (εkεjhjklj + εkεihikli) +Aijhikhkj − ξk,xkhikhkj −A ikhikhkj −Bikhkj −Akjhikhkj −Bkjhik = 0, which can be rewritten as Bij ,xk +Aij,xkhij + ( Bij ,li lk − εkεiBij ,lk li −Bkj ) hik + ( Bij ,lj lk − εkεjBij ,lk lj ) hjk −Bikhkj + ( Aij,li lk − εkεiA ij ,lk li ) hikhij + ( Aij,lj lk − εkεjA ij ,lk lj ) hijhjk + ( Aij − ξk,xk −A ik −Akj ) hikhkj = 0. It follows from the coefficients of hkj that Bik = 0. The permutation of the indices i, j and k leads to Bst = 0, ∀ s, t, s 6= t. (A.19) By equating to zero the coefficients of hikhkj and hijhjk, the following system is obtained Aij,li lk − εkεiA ij ,lk li = 0, Aij,lj lk − εkεjA ij ,lk lj = 0, where we solve the characteristic system to conclude that Aij depends only on x. On the other hand, the coefficient of hij implies that Aij does not depend on xk, therefore Aij = Aij(xi, xj). Considering the coefficient of hikhkj , we obtain the following equation Aij − ξk,xk −A ik −Akj = 0. (A.20) Therefore, equation (A.13) reduces to ηi,xj + ( ηi,li lj − ξ j ,xj lj − η j −Aijlj ) hij = 0. Since ηi does not depend on hij , we must have ηi,xj = 0, ηi,li lj − ξ j ,xj lj − η j −Aijlj = 0, (A.21) The Symmetry Group of Lamé’s System and Conformally Flat Hypersurfaces 25 By applying pr(1)V to equation (3.10), we have φij(j) +φji(i) +φikhkj+hikφ kj = 0, which implies that φij,xj + φij,hijhij,xj − ξ j ,xjhij,xj + φji,xi + φji,hjihji,xi − ξ i ,xihji,xi + +φikhkj + hikφ kj = 0. Considering the substitution (A.4), for i < j, we obtain φij,xj + φji,xi + ( ξj,xj −A ij +Aji − ξi,xi ) hji,xi + ( ξj,xj −A ij +Aik +Akj ) hikhjk = 0. Then, the coefficient of hji,xi leads to ξj,xj −A ij +Aji − ξi,xi = 0. (A.22) By applying the prolongation pr(1)V to (3.11) we have εiφ ij (i)+εjφ ji (j)+εkφ kihkj+εkφ kjhki = 0, which implies εi ( φij,xi + φij,hijhij,xi − ξ i ,xihij,xi ) + εj ( φji,xj + φji,hjihji,xj − ξ j ,xjhji,xj ) + εk ( φkihkj + φkjhki ) = 0. Considering the substitution (A.5) with i < j, we obtain εiφ ij ,xj + εjφ ji ,xj + εj ( ξi,xi −A ij +Aji − ξj,xj ) hji,xj + εk ( ξi,xi −A ij +Aki +Akj ) hkihkj = 0. From the coefficient of hji,xj , we get Aij − ξi,xi −A ji + ξj,xj = 0. (A.23) Therefore, it follows from (A.22) and (A.23) that Aji = Aij . (A.24) Consequently, both equations imply that ξi,xi = ξj,xj , which enables us to conclude that ξm = axm + am, ∀ 1 ≤ m ≤ 3, (A.25) where a and am are real constants. Moreover, from (A.25) and (A.20), we have that Aij − a−Aik −Akj = 0 and Aik − a−Aij −Ajk = 0. By taking the sum of these equations and using (A.24), we obtain Akj = −a. Therefore, it follows from (A.18) and (A.19), that φst = −ahst, ∀ s 6= t. (A.26) Moreover, from (A.25) and (A.21), we get ηi,li lj = ηj . (A.27) Since the function ηm depends only on xm and lm, we conclude that, ηi,lili = 0, i.e., ηi = N i(xi)li +M i(xi). (A.28) Hence, it follows from (A.27) and (A.28) that ηi,li = ηj,lj = N(xi). Therefore, N ′(xi) = ηj,ljxi = 0, which implies that, ηi = cli +M i(xi). (A.29) 26 J.P. dos Santos and K. Tenenblat Finally, we apply the prolongation pr(1)V to equation (3.8) to obtain εiη i (i) + εjφ jilj + εjhjiη j + εkφ kilk + εkhkiη k = 0, which implies that εiη i ,xi + εiη i ,li li,xi − εiξi,xi li,xi + εjφ jilj + εjhjiη j + εkφ kilk + εkhkiη k = 0. When we substitute (A.2) for li,xi and we consider equations (A.25), (A.26) and (A.29), we obtain εiM i ,xi + εjhjiM i + εkhkiM k = 0. 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Phys. 34 (1993), 3527–3542. http://dx.doi.org/10.1007/978-1-4684-0274-2 http://dx.doi.org/10.1063/1.530042 1 Introduction 2 Lamé's system and conformally flat hypersurfaces 3 The symmetry group of Lamé's system 3.1 Obtaining the symmetry group of Lamé's system 3.2 Group invariant solutions 4 Geometric properties of the Guichard nets 4.1 Level surfaces 4.2 Coordinate surfaces 5 Conformally flat hypersurfaces A Appendix

The Symmetry Group of Lamé's System and the Associated Guichard Nets for Conformally Flat Hypersurfaces

Institution

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