Iterated Darboux Transformation for Isothermic Surfaces in Terms of Clifford Numbers

: Isothermic surfaces are deﬁned as immersions with the curvture lines admitting conformal parameterization. We present and discuss the reconstruction of the iterated Darboux transformation using Clifford numbers instead of matrices. In particulalr, we derive a symmetric formula for the two-fold Darboux transformation, explicitly showing Bianchi’s permutability theorem. In algebraic calculations an important role is played by the main anti-automorphism (reversion) of the Clifford algebra C ( 4,1 ) and the spinorial norm in the corresponding Spin group.


Introduction
Isothermic surfaces have a very long history. They have been first introduced by Lamé in studies on stationary heat flows (described by the Laplace equation), in the broader context of triply ortogonal systems of coordinates [1]. Then, the main progress towards the theory of isothermic surfaces was done by Bertrand [2], who was first to notice that "in any triply isothermic (in physical sense) orthogonal system in E 3 any coordinate surface admits conformal curvature parameterization" [3]. Transformations of isothemic surfaces, studied by Darboux and Bianchi [4,5], strongly suggested that the related system of nonlinear partial differential equations (see (2) below) is integrable in the sense of the soliton theory [6] and, indeed, such modern formulation of this problem was found [7], which started new developments in this field [8][9][10][11][12]. It is worthwhile to mention that isothermic immersions are invariant with respect to conformal transformations of the ambient space and can be naturally described in terms of conformal geometry (then Darboux transformations correspond to Ribaucour congruences [13]). Studies on isothermic surfaces are still active, see, e.g., [14][15][16][17][18][19]. In this paper we develop an approach based on using Clifford algebras and Spin groups [20,21] (different from the approach of [12,22]). We re-derive the construction of "multisoliton" surfaces by iterated Darboux transformation. In particular, we present detailed computation of the two-fold Darboux transform.

Isothermic Surfaces in R 3
Isothermic surfaces (or, more precisely, isothermic immersions) are characterized as surfaces immersed in E 3 with curvature lines admitting conformal parameterization. It means that there exist coordinates (u, v) in which the isothermic immersion has the following fundamental forms: I = e 2ϑ (du 2 + dv 2 ) , where ϑ, k 1 , k 2 are functions of u, v, which have to satisfy the following system of nonlinear partial differential equations known as Gauss-Mainardi-Codazzi equations: ϑ, uu +ϑ, vv +k 1 k 2 e 2ϑ = 0 , where comma denotes partial derivtive. Geometrically, k 1 and k 2 are principal curvatures, and their product k 1 k 2 yields the Gaussian curvature. The above nonlinear system can be obtained (see [7]) as compatibility conditions for the following linear problem (or Lax pair): where e 1 , . . . , e 5 are 4 × 4 complex matrices (for their exact form see [7] or [23]) that satisfy the relations We point out that using a suitable extension of the famous Sym formula (see, e.g., [24]) we can reconstruct the radius vector of the isothermic surface implicitly determined by the fundamental forms (1), for more details see Section 4 and Theorem 1.

Clifford Algebras
The matrices e 1 , . . . , e 5 satisfying (4) can be interpreted as elements of a Clifford algebra, see below. Their exact matrix form is not needed. From technical point of view, it is even easier to use Clifford numbers instead of particular matrix representations [21].
We recall the definition of a Clifford algebra generated by vectors of a Euclidean or pseudo-Euclidean inner product space. Let V be a vector space endowed with a non-degenerate (but not necessarily positive definite) quadratic form Q, see, e.g., [25,26]. If Q is positive definite (the Euclidean case), then Q(v) is the square of the length of v. The associated bilinear form (scalar product) will be denoted by brackets. In particular, The Clifford algebra C(V, Q) is generated by products ("Clifford products") of vectors (elements of V). The Clifford product is defined by the following relation: where 1 denotes the unit of the Clifford algebra. Let {e 1 , . . . , e n } be an orthonormal basis of V, i.e., e j | e k = 0 for j = k, and e k | e k = ±1. The dimension of the Clifford algebra is 2 n . Its standard basis consists of 1, e k , e jk (j < k), e i e j e k (i < j < k), . . .
If the signature of Q is (m, p) (i.e., among e 1 , . . . , e n there are m vectors such that e 2 j = 1 and p vectors such that e 2 j = −1, and m + p = n), then we denote C(V, Q) ≡ C m,p . The Clifford group (or Lipschitz group) Γ(V, Q) is the multiplicative group (with respect to the Clifford product) generated by the non-isotropic vectors (we recall that w ∈ V is isotropic (or null) vector if w | w = 0 ). The spinor norm of an element where β is a reversion, i.e., One can easily see that the spinor norm of a vector is its scalar square, N(v) = v | v , and for any element of Γ(V, Q) we have The group Pin(V, Q) is a subgroup of Γ(V, Q) consisting of elements X such that N 2 (X) = ±1 (i.e., Γ(V, Q) contains products of unit vectors), and the group Spin(V, Q) (a subgroup of Pin(V, Q)) consists of products of even number of unit vectors [26].

Spin-Valued Lax Pairs
Our approach to the construction of Darboux transformation consists of two steps. First, we characterize the structure of the Lax pair. Second, we are looking for a transformation preserving the structure [27].
The structure of the Lax pair is characterized by the dependence on λ (e.g., divisor of poles) [6,28], reduction group (loop group) [29] and, possibly, by other invariants of Darboux transformations, like linear and multilinear constraints on coefficients of the Laurent expansion around poles [30].
In this section we present the characterization of the Lax pair (3), see [20,31]. First of all, we consider Clifford numbers instead of matrices. Then, we notice that both matrices of this Lax pair are Clifford bi-vectors linear in λ such that W are real vector spaces, V is spanned by e 1 , e 2 and e 3 , and W is spanned by e 4 and e 5 . We assume relations (4), which means that form now on the quadratic form Q, defined on V ⊕ W, is assumed to have signature (4, 1). The compatibility conditions for the linear system (8) imply that a 1 and a 2 form an orthogonal basis in W. We can confine ourselves to the particular case (3) without loss of the generality. Indeed, both linear problems are equivalent up to a re-parameterization of independent variables and a discrete transformation in the space W.
The form (8) of the spectral problem can be described in terms of some group constraints ("reduction group", compare [27,29]). First, U µ are linear combinations of Clifford bi-vectors. In other words, U µ take values in the Lie algebra of the group Spin(V ⊕ W, Q). In principle, Ψ could be a spinor, but here and in the sequel we assume that it is an element of the Clifford algebra. Without loss of the generality we can confine ourselves to solutions Proof. It is sufficient to differentiate N(Ψ): where one has to remember that N(Ψ) is a scalar, so it commutes with any elements.
Therefore, Ψ ∈ Spin(V ⊕ W, Q) (for any u, v) provided that Ψ is Spin-valued at some initial point (u 0 , v 0 ). In an analogous way one can show the following loop group conditions: Ψ(−λ) = e 4 e 5 Ψ(λ)e 4 e 5 , (where the bar denotes complex conjugate and, by definition, e j = e j ). They follow from The properties β β β(U µ ) = −U µ and (12) hold for the Lax pair (8), which can be easily verified using commutation relations (4).
Here we present some details of calculations involving Clifford numbers instead of matrices. They are closely related but not identical to the approach of our earlier papers [20,23,32].
We use the projection P : C(V ⊕ W) → C(V) defined as a homomorphism of Clifford algebras such that P(e 4 ) = P(e 5 ) = 1 .
Note that This projection yields an original isothermic surface (the dual surface is a result of another homomorphism P , defined by P (e 4 ) = −P (e 5 ) = 1).

Theorem 1.
We assume that Ψ satisfies the linear system (3) and Then r := P(F) yields the original isothermic immersion (up to a Euclidean motion), provided that we identify span{e 1 , e 2 , e 3 } with the space E 3 .
Proof. The crucial property of the Sym-Tafel formula (15) is a compact form of its derivative (compare [24]): where Ψ 0 := Ψ(0) (i.e., Ψ evaluated at λ = 0), and we use the slightly more general form (8) of the Lax pair (3). Then r, µ = e θ Ψ −1 0 e µ Ψ 0 . Therefore (for µ = 1, 2) form an orthonormal basis in the tangent space and, therefore, the corresponding metric is given by the first equation of (1). Obviously, E 3 (defined by (18) for µ = 3) is orthogonal to E 1 and E 2 ; hence, it can be identified with the normal vector. In order to derive the second fundamental form we differentiate (17). Taking into account Therefore, coefficients of the second fundamental form (given by r, ij | E 3 ) yield the second formula of (1). The proof is completed by applying the Bonnet theorem.

The Darboux-Bäcklund Transformation in the Case of Spin Groups
The Darboux transformation is a gauge-like transformation using the Darboux matrix D (we will keep using the name "matrix" even for D given in terms of Clifford numbers without referring to any matrix representation): provided thatŨ µ has the same dependence on dependent variables as U µ , see, e.g., [6,33]. In Section 4 we have shown that the form (3) can be derived by imposing a set of constraints on a general linear problem (U µ are Clifford bi-vectors, linear in λ and belong to the appropriate loop algebra). Then the Darboux transformation has to preserve this structure, which means, in particular, that D should belong to the same group as Ψ. Different methods of constructing the Darboux matrix need different form of λ-dependence of D (these forms are equivalent up to a λ-dependent scalar factor [30]). In particular, one can assume D as polynomial in λ (eigenvalues, corresponding to solitons, are zeros of det D) [34], sum of simple fractions (eigenvalues: poles of D and D −1 ) [6,29], or a "realization" (D = N + F(λ − A) −1 G) [35,36].
Generalization of this approach on Spin-valued linear problems is quite natural. Instead of multiplying both sides of (20) by D 2 we multiply them by Dβ β β(D): Note that β β β(D) = D if D is a Clifford vector (which has been usually assumed in earlier papers, like [31]), and in this case Equation (22) assumes the form (21). (22), and G is any Clifford number G (not necessarily constant), then D = GD satisfies Equation (22) as well.

Proof. Lemma can be shown by straightforward calculation. First, we have
so the right-hand side of (22) vanishes. Then which ends the proof.
In this paper we confine ourselves to iterations of the simplest Darboux transformations (defined by D linear in λ). Then we can use the results of [31], where the case of the Clifford vector (here denoted by D ) was considered, and the following form of the Darboux transformation was derived: and D (λ ± ) can be expressed as (26) where ρ ± are (arbitrary) scalar functions and d ± are constant elements such that d 2 ± = 0. Reductions (12) impose constraints on λ ± , ρ ± , and d ± (see [31]): Moreover we denote (compare [21]) We assume p 2 0 = n 2 0 = 1 (in the Clifford algebra p 2 = p | p etc.). Therefore computing the Clifford square of both sides of (28) and taking into account that p, p 0 anticommute with n, n 0 we get p 2 = n 2 .

Remark 1.
It is important to remember that the obtained Darboux matrix D depends on the function Ψ (an exact solution of the linear problem (3)) and constant parameters: κ, p 0 , n 0 . The notation D = D [Ψ,κ,p 0 ,n 0 ] would be very awkward, so in the sequel we omit the dependence on p 0 and n 0 , writing D = D [Ψ,κ] .
We omit the proof, which consists in splitting the equation D, µ +DU µ =Ũ µ D into a system of equations by equating coefficients by powers of λ and basis elements of the Clifford algebra.
In the context of soliton surfaces the Darboux transformation is often called the Darboux-Bäcklund transformation [21,31] or the Darboux-Bianchi transformation [23].
Therefore, substituting (45), which can be rewritten as (compare (29)) which is a similarity transformation and can be interpreted as an orthogonal transformation in the (complexified) Clifford algebra. Note that for any Clifford vectors v, w we have Therefore p + in = p 2 2 e 4 2(κ 1 p 1 |p 2 − κ 2 n 1 |n 2 )

The Trivial Background (Plane)
The data ϑ = 0, k 1 = k 2 = 0 correspond to the trivial background, i.e., to the plane. The linear system (3) assumes the form Hence Ψ = e Finally Performing the projection (13) we get

Conclusions
We constructed an iterated Darboux transformation for isothermic surfaces using the Clifford algebra approach. Our main result is a symmetric representation of two-fold Darboux transformation (Theorem 4). Thus we made some progress in the direction of constructing symmetric compact formulas for "multi-soliton" isothermic surfaces, what reduces to transforming Ψ [K] (given by (38)) into a form that is explicitly invariant with respect to permutations of real eigenvalues κ 1 , . . . , κ K . Another open problem, more chal-lenging, is to find analogous formulas in a direct way and with more general set of complex eigenvalues. We also expect to extend our approach on related multidimensional problems (see, e.g., [32]).