On Bäcklund and Ribaucour Transformations for Hyperbolic Linear Weingarten Surfaces

abstract: We consider Bäcklund transformations for hyperbolic linear Weingarten surfaces in Euclidean 3-space. The composition of these transformations is obtained in the Permutability Theorem that generates a 4-parameter family of surfaces of the same type. The analytic interpretation of the geometric results is given in terms of solutions of the sine-Gordon equation. Since a Ribaucour transformation of a hyperbolic linear Weingarten surface also gives a 4-parameter family of such surfaces, one has the following natural question. Are these two methods equivalent, as it occurs with surfaces of constant positive Gaussian curvature or constant mean curvature? In this paper, we obtain necessary and sufficient conditions for the surfaces given by the two procedures to be congruent.


Introduction
A surface M contained in the Euclidean space R 3 whose mean and Gaussian curvatures, H and K, satisfy a relation of the form α + 2βH + γK = 0, α, β, γ ∈ R, is called a linear Weingarten surface.The development of the theory of these surfaces started in the early 19 hundreds.More recent results obtained by several authors can be found in [1], [8], [10], [15]- [17].If the linear Weingarten surface satisfies β 2 − αγ < 0, then it is said to be hyperbolic.In this case, without loss of generality, we may assume that α = 1.Moreover, if β = 0, and γ = 1 then M is a pseudo-spherical surface, i.e., K = −1, and there is a well known theory on Bäcklund transformations for pseudo-spherical surfaces studied by Bäcklund [2,3] and on composition of such transformations called Permutability theorem obtained by Bianchi [4] .
In this paper, we study an extension of the concept of pseudo-spherical line congruence, called a hyperbolic linear Weingarten congruence.Namely, we consider a diffeomorphism between surfaces M and M ′ such that at corresponding points p ∈ M and p ′ ∈ M ′ , the straight line determined by these points has a constant angle φ with the normal N p and a constant angle ρ with the normal N ′ p ′ .Moreover, we assume that the segment pp ′ has constant length r and N p has a constant angle θ with N ′ p ′ .Then M and M ′ are hyperbolic linear Weingarten surfaces satisfying, respectively, 1 + 2βH + γK = 0 and 1 + 2β ′ H ′ + γ ′ K ′ = 0, where β 2 − γ = (β ′ ) 2 − γ ′ < 0. We observe that whenever φ = ρ = π/2, then the theory coincides with the classical results for pseudo-spherical surfaces.
The Integrability Theorem shows that given such a surface M there exists a 3-parameter family of surfaces M ′ , satisfying 1 + 2β ′ H ′ + γ ′ K ′ = 0, associated to M by a hyperbolic linear Weingarten congruence.The surfaces M ′ are said to be associated to M by a Bäcklund transformation for hyperbolic linear Weingarten surfaces.
The Permutability theorem shows that the composition of such transformations is commutative when one chooses the parameters apropriately.In this case, starting with a hyperbolic linear Weingarten surface M satisfying 1 + 2βH + γK = 0, one gets a 4-parameter family of surfaces M * , satisfying 1 + 2βH * + γK * = 0, with the same constants β, γ of the surface M .
Therefore, starting with a hyperbolic linear Weingaten surface M in R 3 , satisfying 1 + 2βH + γK = 0, one gets a 4-parameter family of hyperbolic linear Weingarten surfaces with the same constants β, γ, either by the composition of Bäcklund transformations or by Ribaucour transformations.Hence, it is natural to ask if these two methods are equivalent, i.e., if the surfaces obtained by these two methods are congruent.In this paper, we will show that in general the families obtained by these procedures are distinct, in contrast with what happens in the case of constant positive Gaussian curvature (see for example Tenenblat [18]) and surfaces of nonzero constant mean curvature (Jeromin-Pedit [12]).In the particular case K = −1, necessary and sufficient conditions were established by Goulart-Tenenblat [11], for a composition of Bäcklund transformations to be congruent to a Ribaucour transformation.
The analytic interpretation of the Bäcklund transformation gives an integrable system of equations, in terms of ψ and 2 parameters, whose solutions ψ ′ give new solutions of the sine-Gordon equation.By considering ψ ′ and ψ ′′ two distinct such solutions, the analytic permutability theorem gives a superposition formula that provides an algebraic expression for new solutions ψ * , which depend on 4 parameters.Moreover, the Ribaucour transformation gives an integrable linear system in terms of ψ and a constant C R , whose solutions ψ depend also in 4parameters and satisfy the sine-Gordon equation.The solutions ψ * and ψ obtained by these procedures are distinct.
The paper is organized as follows: In Section 2, we introduce the hyperbolic linear Weingarten congruence and we prove Bäcklund Theorem for hyperbolic linear Weingaten surfaces, the Geometric Integrability Theorem and the Geometric Permutability Theorem.In Section 3, considering the correspondence between such surfaces and solutions of the sine-Gordon equation, we prove the Analytic Integrability Theorem and we state the Analytic Permutability Theorem, whose proof is given in the Appendix.In Section 4, we start recalling some results on Ribaucour transformation.Then we obtain necessary and sufficient conditions for the hyperbolic linear Weingarten surfaces, obtained by the composition of Bäcklund transformations, to be congruent to those obtained by Ribaucour transformations.These conditions are given in terms of the first fundamental forms i.e., in terms of the corresponding solutions of the sine-Gordon equation.

Bäcklund transformations for hyperbolic linear Weingarten surfaces in R 3 -Geometric Theory
In this section, we introduce the concept of hyperbolic linear Weingarten congruence and we study a Bäcklund transformation for hyperbolic linear Weingarten surfaces in R 3 .Moreover, we also prove the integrability and the permutability theorems for these transformations.
For each p ∈ M and p ′ = l(p) ∈ M ′ with p ′ = p, denote by v = v(p) the unit vector in the direction of the straight line passing through p and p ′ .Let N p (resp.N ′ p ′ ) be the unit vector normal to M (resp.M ′ ) in p (resp.p ′ ).We say that l is a hyperbolic linear Weingarten congruence with constants (r, θ, φ, ρ), where r > 0, 0 < θ < π, 0 < φ, ρ ≤ π 2 , if the distance between p and p ′ is constant equal to r, the angle between N p and N ′ p ′ is θ, the angle between N p and v is φ and the angle between N ′ p ′ and (−v) is equal to ρ. Remark 2.4.When φ = ρ = π/2, then the direction of the line congruence is tangent to both surfaces M and M ′ and it reduces to the so called pseudo-spherical line congruence of surfaces in R 3 .
The following theorem justifies the definition of a hyperbolic linear Weingarten congruence, for a diffeomorphism l as in Definition 2.3.Moreover, it reduces to the classical Bäcklund Theorem between pseudo-spherical surfaces when φ = ρ = π/2.Theorem 2.5.(Bäcklund Theorem for hyperbolic linear Weingarten surfaces) Let M and M ′ be two surfaces imersed in R 3 .Suppose there exists a hyperbolic linear Weingarten congruence l : M −→ M ′ with constant (r, θ, φ, ρ) as in Definition 2.3.For any p ∈ M and p ′ = l(p) ∈ M ′ , suppose that the normal vectors N p and N ′ p ′ and the vector v = v(p) are not coplanar.Then M and M ′ are hyperbolic linear Weingarten surfaces.More precisely, the Gaussian curvature K (resp.K ′ ) and mean curvature H (resp. H ′ ) of M (resp.M ′ ) satisfy the relation (2.5) Since the vectors e ′ 3 , e 3 , v are not coplanar then a 32 = 0. Using (2.4), we obtain 1 a 32 {a 13 ω 1 + r sin ρω 13 }.
Therefore, it follows from the second equation of (2.5) that where (2.7) Differentiating (2.6) it follows from the structure equations, the definition of mean and Gaussian curvatures and the Gauss equation that On the other hand, we kwon that dω 12 = −K(ω 1 ∧ ω 2 ).Therefore, the mean and Gaussian curvature of M satisfy The constants c 1 , c 2 , c 3 , c 4 defined in (2.7) imply that (2.9) In other words, M is a linear Weingarten surface satisfying 1 + 2βH + γK = 0, where β and γ are given by (2.1).Interchanging φ and ρ in the previous computations, we obtain that M ′ is also a linear Weingarten surface satisfying 1 + 2β ′ H ′ + γ ′ K ′ = 0, where β ′ and γ ′ are given by (2.2).Moreover, using the constants a 31 and a 32 defined by (2.4), we have We conclude this section by establishing some notation and some identities that will be used throughout this paper.

The Geometric Integrability Theorem
The Geometric Integrability Theorem, that we prove below, shows that given a hyperbolic linear Weingarten surface M satisfying (2.6) there exists a family of surfaces M ′ associated to M by a hyperbolic linear Weingarten congruence.
Theorem 2.8.(Geometric Integrability Theorem) Let M ⊂ R 3 be a hyperbolic linear Weingarten surface with Gaussian curvature K and mean curvature H satisfying 1 + 2βH + γK = 0. We consider real numbers r > 0, 0 < θ < π and 0 < φ, ρ ≤ π 2 satisfying (2.1).Let p 0 ∈ M and let v 0 ∈ R 3 be a unit vector whose angle with N p0 (normal to M at p 0 ) is φ.Suppose that v T 0 , the tangential component of v 0 , is not a principal direction.Then there exists a linear Weingarten 2) and a hyperbolic linear Weingarten congruence l with constants (r, θ, φ, ρ) between neighborhoods of p 0 in M and l(p 0 ) in M ′ , such that the straight line connecting p 0 to l(p 0 ) is in the direction of v 0 .
Proof: Since M is a hyperbolic linear Weingarten surface satisfying 1 + 2βH + γK = 0 then, taking real numbers r > 0, 0 < θ < π e 0 < φ, ρ ≤ π 2 such that (2.1) is verified, we have Thus, we can consider the real constants b 1 , b 2 , b 3 and c 1 , c 2 , c 3 , c 4 defined by (2.11) and (2.13), respectively.The idea is to apply Frobenius theorem to construct an orthonormal frame {e 1 , e 2 , e 3 } adapted to M , defined in a neighborhood of p 0 , whose dual and connection forms satisfy such that e . Let ℑ be the ideal generated by the 1-form Differentiating and using the structure equations we have By hypothesis, the constants r, θ, φ, ρ satisfy (2.1) and M is a hyperbolic linear Weingarten surface such that 1 + 2βH + γK = 0. Thus, dζ = ζ ∧ µ, i.e., ℑ is closed under exterior differentiation.By Frobenius theorem, the equation ζ = 0 is integrable.Therefore, there exists an adapted frame {e 1 , e 2 , e 3 } such that (2.20) holds in a neighborhood of p 0 , with initial condition e 1 (p . Since the angle between v 0 and N p0 = e 3 (p 0 ) is equal to φ and the unit vectors e 3 (p 0 ), e 1 (p 0 ) and v 0 are coplanar then v 0 = sin φe 1 (p 0 ) + cos φe 3 (p 0 ).Define, in this neighborhood, the vector function v = sin φe 1 + cos φe 3 .
By hypothesis, e 1 (p 0 ) is not a principal direction hence we can assume, by continuity, that e 1 is not a principal direction on an open subset V of this neighborhood.
We consider V parametrized by X : Differentiating and using the structure equations, we obtain Since e 1 is not a principal direction and r sin φ = 0 we conclude that M ′ = X ′ (U ) is a regular surface and z 1 , z 2 are tangent to M ′ .Moreover, e ′ 3 = b 1 e 1 +b 2 e 2 + cos θe 3 is a unit vector normal to M ′ .Consequently, M ′ is related to X(U ) by a hyperbolic linear Weingarten congruence, l with constants r, θ, φ, ρ.Using Theorem 2.5, we conclude that M ′ is a hyperbolic linear Weingarten surface satisfying 1 + 2β ′ H ′ + γ ′ K ′ = 0, where β ′ , γ ′ are given by (2.2). ✷ Observe that Theorem 2.8 shows that given a hyperbolic linear Weingarten surface M in R 3 there exists a 3-parameter family of surfaces M ′ associated to M by a hyperbolic linear Weingarten congruence.The three parameters are determined by the unit vector v 0 and the four constants (r, θ, φ, ρ) satisfying two conditions given by (2.1).

The Geometric Permutability Theorem
In this section, we consider the composition of Bäcklund transformations for hyperbolic linear Weingarten surfaces in R 3 .We observe that applying a Bäcklund transformation to a surface in R 3 satisfying 1 + 2βH + γK = 0, we obtain new surfaces of the same type but with different constants β and γ.We will now consider a composition of such transformations so that the surface obtained by this composition has the same constants as the surface we started with.This is obtained by imposing certain conditions on the parameters and in this case, the composition is commutative.
Proof: Let X be a local parametrization of M in a neighborhood of p. Since l 1 : M → M ′ and l 2 : M → M ′′ are hyperbolic linear Weingarten congruences, we have that and M ′′ at p 1 and p 2 , respectively.By hypothesis, Observe that finding hyperbolic linear Weingarten congruences l * 1 and l * 2 as required by the theorem is equivalent to obtaining unit vector fields u 1 , u 2 satisfying (2.33) We consider new orthonormal frames {e ′ 1 , e ′ 2 , e ′ 3 } adapted to M ′ and {e ′′ 1 , e ′′ 2 , e ′′ 3 } adapted to M ′′ given by (2.34) Define the vector fields (2.35) The idea is to show that these vectors u 1 , u 2 satisfy equation (2.33).Initially, using (2.27), we observe that Similarly, using (2.34), (2.35), (2.28), (2.21) and the constant δ given by (2.22), we have Moreover Therefore, equation (2.33) is equivalent to the following linear system where δ is given by (2.22), E 11 and E 12 are given by (2.26) and the real numbers a ′ ij and a ′′ ij defined by (2.28) are given by (2.4), taking We observe that, as a consequence of (2.4), a ′ 33 = cos θ 1 and a ′′ 33 = cos θ 2 .Then using (2.25) we conclude that the third equation of the linear system (2.36) is satisfied.Substituting the expressions of F 11 and F 12 given by (2.31) and using equations (2.21)-(2.28),(2.4) and (2.30), we conclude that the first and the second equations of this linear system are also satisfied.

Analytic interpretation of Bäcklund transformation
In this section we will present an analytic interpretation of the Geometric Integrability Theorem (Theorem 2.8) and of the Geometric Permutability Theorem (Theorem 2.9) given in the previous section.We start recalling that given a hyperbolic linear Weingarten surface in R 3 satisfying 1 + 2βH + γK = 0, then D = γ − β 2 > 0 and there exists a solution ψ of the sine-Gordon equation where C βγ is a real constant defined by sin Conversely, given a solution ψ of equation (3.1),where C βγ is a real constant defined by (3.2), there exists a hyperbolic linear Weingarten surface in R 3 satisfying 1 + 2βH + γK = 0, parametrized by lines of curvature, whose first and second fundamental forms are given by ) with For more details, see Tenenblat [19].

C. Goulart
Let ψ be a solution of the sine-Gordon equation (3.1),where C βγ is a real constant defined by (3.2).We consider the hyperbolic linear Weingarten surface M ⊂ R 3 satisfying 1+2βH +γK = 0. Let r > 0, 0 < θ < π and 0 < φ, ρ ≤ π 2 be real numbers satisfying (2.1) and (2.12).Using the Geometric Integrability Theorem, we can construct an orthonormal frame {e 1 , e 2 , e 3 } tangent to M , locally defined, with dual forms ω 1 , ω 2 and connection forms ω 12 , ω 13 , ω 23 associated to this frame satisfying the Bäcklund transformation (2.6), where c 1 , c 2 , c 3 and c 4 are given by (2.13).Moreover, the correspondence between hyperbolic linear Weingarten surfaces and solutions of the sine-Gordon equation allows us to conclude that the Bäcklund transformation (2.6) is equivalent to the system of partial differential equations where and c 1 , c 2 , c 3 , c 4 are given by (2.13).Using these real numbers we define the following constants In fact, in this case, On Bäcklund and Ribaucour Transformations 23

✷
The following theorem provides an analytic interpretation of the Geometric Integrability Theorem (Theorem 2.8).
Proof: Differentiating the first equation of the system (3.6) with respect to x 2 and subtracting from the derivative of the second equation with respect to x 1 , we obtain

25
where we used the fact that ψ is a solution of the sine-Gordon equation (3.1).Thus, using (3.6) and the relations given by (3.8) and (3.14), we have Similarly, differentiating the first equation of (3.6) with respect to x 1 and subtracting from the derivative of the second equation with respect to x 2 , we obtain where we used the fact that ψ is differentiable.Therefore, using (3.6) and the relations given by (3.9) and (3.16), we have ie, ψ ′ is a solution of the sine-Gordon equation (3.12).The functions ψ ′ obtained by integrating (3.6) depend on 3-parameters, namely the initial condition ψ ′ (x 0 1 , x 0 2 ), and four constants (r, θ, φ, ρ) satisfying two equations given by (2.1).✷ Definition 3.4.Let ψ be a solution of the sine-Gordon equation (3.1).We say that a function ψ ′ is associated to ψ by a Bäcklund transformation BT (r, θ, φ, ρ) if ψ ′ is a solution of the system (3.6).

Analytic Interpretation of the Permutability Theorem
Let ψ be a solution of the sine-Gordon equation (3.1),where C βγ is the constant given by (3.2) and β, γ are constants such that γ − β 2 > 0. The Geometric Permutability Theorem (Theorem 2.9) and the correspondence between hyperbolic linear Weingarten surfaces and solutions of the sine-Gordon equation allows us to construct a new solution ψ * of the sine-Gordon equation (3.1).The analytic interpretation of the Permutability Theorem (Theorem 3.5) will allow us to obtain ψ * algebraically.This is the content of our next result.However, the proof of this theorem is highly technical and, therefore, it will be presented in the Appendix.Theorem 3.5.(Analytic Permutability Theorem) Let ψ be a solution of the sine-Gordon equation (3.1),where C βγ is the real constant given by (3.2) and the real numbers β, γ are such that γ − β 2 > 0. We consider real numbers r i > 0, 0 < φ i , ρ i ≤ π 2 and 0 < θ i < π (i = 1, 2) with θ 1 = θ 2 , satisfying (2.1) and (2.23).Let ψ i , i = 1, 2 be solutions of equation (3.12), associated to ψ by the Bäcklund transformations BT (r i , θ i , φ i , ρ i ), where C β ′ γ ′ is the constant given by (3.13) and β ′ , γ ′ are given by (2.2), when r = r i , θ = θ i , φ = φ i and ρ = ρ i .Then there exists a unique solution ψ * of the sine-Gordon equation (3.1) associated to ψ i by BT (r j , θ j , ρ j , φ j ), 1 ≤ i = j ≤ 2.Moreover, ψ * is determined algebraically by Observe that in Theorem 3.5 the constants β ′ and γ ′ defined by (2.2) are independent of i since (2.23) is satisfied.

The composition of Bäcklund transformations and the Ribaucour transformation for hyperbolic linear Weingarten surfaces in R 3
We consider a hyberbolic linear Weingarten surface in R 3 parametrized by orthogonal lines of curvatures X(x 1 , x 2 ) satisfying 1 + 2βH + γK = 0, where β and γ are real constants such that β 2 − γ < 0. There are two methods which provide 4-parameter families of linear Weingarten surfaces, with the same constants β and γ, associated to the surface X(x 1 , x 2 ).Namely, the composition of Bäcklund transformations, as we have seen in the previous sections and the Ribaucour transformation.In general the surfaces obtained by these two methods are not congruent.In fact, by starting with the pseudo-sphere, Goulart-Tenenblat [11] proved, with an explicit example, that a composition of Bäcklund transformations is not a Ribaucour transformation.In this section, we will determine necessary and sufficient conditions for the hyberbolic linear Weingarten surfaces constructed by using these two methods, to be congruent.

Ribaucour Transformation
We state the main concepts and results of the theory of Ribaucour transformations for surfaces in R 3 , in particular for linear Weingarten surfaces, that will be used in the following subsections.More details of the theory can be found in [6] or [8].Definition 4.1.Let M and M be orientable surfaces in R 3 and let N and Ñ be their Gauss maps.We say that M is associated to M by a Ribaucour transformation if, and only if, there exists a differentiable function h defined on M and a diffeomorphism l : 3 and the diffeomorphism l preserves lines of curvature.
We say that M and M are locally associated by a Ribaucour transformation if for all p ∈ M there exists a neighborhood of p in M that is associated to a open subset of M by a Ribaucour transformation.Similarly, we define parametrized surfaces associated by such transformations.
The Ribaucour transformation is characterized in terms of a differential equation which must be satisfied by map h of the definition (see [6] or [8]).
Theorem 4.2.Let M be an orientable surface in R 3 , without umbilic points and let N be its Gauss map.We consider {e i }, i = 1, 2, orthonormal principal direction vector fields and −λ i the corresponding principal curvatures, ie, dN (e i ) = λ i e i .A surface M is locally associated to M by a Ribaucour transformation if, and only if, there exist parametrizations X : and Ñ is a Gauss map of M given by where and h satisfies the differential equation where ω ij are the connection forms associated to {e i }.
We observe that the differential equation (4.2) is of second order and highly non linear.The proposition below shows how the problem of obtaining the function h can be linearized.
Conversely, if Ω and W satisfy (4.3) and Observe that Ω i i = 1, 2 are the covariant derivatives of Ω.Moreover, considering Z i defined by (4.1), one can show that Z i = Ω i /W (see [8]).
Next theorem shows that, by imposing an additional condition, the Ribaucour transformation of a linear Weingarten surface, satisfying α+2βH +γK = 0 provides a family of surface this same type, with the same constants α, β, γ.

C. Goulart
Theorem 4.4.(Corro-Ferreira-Tenenblat [8]) Let M be a surface of R 3 , without umbilic points and let M be associated to M by a Ribaucour transformation, such that the normal lines at corresponding points intersect at a distance h.Assume that h = Ω W is not constant along the lines of curvature and suppose that the functions Ω and W satisfy the additional condition where α, β, γ, C R = 0 are real constants.Then M is a linear Weingarten surface satisfying α + 2β H + γ K = 0 if, and only if, M satisfies α + 2βH + γK = 0, where H and K (resp.H and K) are, respectively, the mean and Gaussian curvatures of M (resp.M ).
Observe that we are denoting by C R the constant of the Ribaucour transformation.
Theorem 4.5.( Corro-Ferreira-Tenenblat [8]) Let M ⊂ R 3 be a linear Weingarten surface satisfying α + 2βH + γK = 0, with no umbilic points.Let e i , i = 1, 2 be orthonormal principal direction vector fields.Let ω i , ω ij and ω i3 be the dual and the connection forms.Then the system is integrable, for any constant C R = 0. On a simply connected domain, any solution, whose initial conditions satisfy (4.4), satisfies (4.4) identically.If M is locally parametrized by X : U ⊂ R 2 → M and Ω, W is a non trivial solution of (4.5) satisfying (4.4), then each surface of the family is a linear Weingarten surface, locally associated to X by a Ribaucour transformation, satisfying α + 2β H + γ K = 0, where H and K are the mean and Gaussian curvatures of X.
Remark 4.6.Considering Z i given by (4.1), since Z i = Ω i /W , we can rewrite condition (4.4) as where i, j = 1, 2, g i = |X xi |, −λ i are the principal curvatures of M and C R = 0 is a real constant.
and h = Ω W .

Necessary and sufficient conditions
Given a hyberbolic linear Weingarten surface M in R 3 , satisfying 1 + 2βH + γK = 0, one can consider the surfaces M associated to M by Ribaucour transformations as in Theorem 4.5 and the surfaces M * associated to M by composition of Bäcklund transformations as in Theorem 2.9.We will determine necessary and sufficient conditions for M and M * to be congruent.
Substituting (4.10) into (4.11) and using (4.9), we obtain that the first fundamental form of X is given by Ĩ = g2 We obseve that the first fundamental form of X * is given by we define the functions Using the Analytic Permutability Theorem (Theorem 3.5), we observe that ϕ = tan ψ * −ψ 4 .Therefore, Considering a hyperbolic linear Weingarten surface M immersed in R 3 , our next theorem establishes the necessary and sufficient conditions for a composition of Bäcklund transformations and a Ribaucour transformation of M to be congruent.Theorem 4.10.Let M ⊂ R 3 be a linear hyperbolic Weingarten surface satisfying 1 + 2βH + γK = 0, parametrized by lines of curvature X(x 1 , x 2 ).Let X * (x 1 , x 2 ) be a surface associated to X by a composition of Bäcklund transformations as in Theorem 2.9.Let X(x 1 , x 2 ) be a hyperbolic linear Weingarten surface associated to X by a Ribaucour transformation as in Theorem 4.5, such that the normal lines at corresponding points intersect at a distance h(x 1 , x 2 ).Then, with the notation of Remark 4.9, X and X * are congruent if, and only if, h is one of the following functions where ϕ and Λ are given by (4.14) and D = γ − β 2 .
Proof: We observe that the first fundamental form of a linear Weingarten surface determines its second fundamental form.Considering the notation established in Remark 4.9, let g1 , g2 and g * 1 , g * 2 given by (4.12) and (4.15), respectively.Since the fundamental forms of X * are determined by the solution ψ * of the sine-Gordon equation (3.1) given by (3.17), then X and X * are congruent if, and only if, g1 = ±g * 1 and g2 = ±g * 2 .Observe that the equality gi = ±g * i (i = 1, 2) is a quadratic equation for h in terms of ϕ.
Corollary 4.11.Under the same conditions as in Theorem 4.10, if β = 0 and γ = 1, i.e. if the surfaces X, X * and X have Gaussian curvature equal to -1, then X and X * are congruent if, and only if, h is one of the following functions where ϕ and Λ are given by (4.14).

Appendix
We now prove the analytic version of the permutability theorem (Theorem 3.5), for the Bäcklund transformations BT (r i , θ i , φ i , ρ i ), i = 1, 2.
In order to achieve our goal, we need to prove some lemmas.We define the real numbers L ℓ (1 ≤ ℓ ≤ 6) below, Applying some trigonometric identities, we obtain (5.5).Analogously, we prove that △ 3 = −△ 2 and △ 4 = △ 1 .✷ With the aid of the lemmas above, we will obtain the analytic interpretation of the permutability theorem for linear Weingarten hyperbolic surfaces in R 3 .

Proposition 4 . 3 .
If h is a nonvanishing function, defined on a simply connected domain, which satisfies equation (4.2) then h = Ω W , where Ω and W are nonvanishing functions satisfying

Proposition 4 . 8 .
) Let M ⊂ R 3 be a linear Weingarten surface satisfying α + 2βH + γK = 0.If M is associated to M by a Ribaucour transformation as in Theorem 4.5, then the first fundamental form of M is given by Ĩ ) and D = γ − β 2 (see (3.2)-(3.5)).Remark 4.9.For later use, let us establish the following notation C. Goulart 2.1.Bäcklund Theorem for hyperbolic linear Weingarten surfaces Definition 2.1.We say that M ⊂ R 3 is a Weingarten surface if there exists a differentiable function relating the mean and Gaussian curvatures H and K of M .A surface M is said to be linear Weingarten if H and K satisfy a linear relation, i.e., there exist real constants α, β, γ such that α 2 , e 3 } and {e ′ 1 , e ′ 2 , e ′ 3 } be orthonormal frames adapted to M and M ′ , respectively such that, for every p ∈ M , e 3 (p) = N p , e ′ 3 (p ′ ) = N ′ p ′ and the sets {v, e 1 , e 3 } and {v, e ′ , it follows from (2.26), (2.27) and (2.22), that r 2 v 2 = δr 1 sin φ 1 E 11 e 1 + δr 1 sin φ 1 E 12 e 2 + r 2 cos φ 2 e 3 .