Vectorial Ribaucour Transformations for the Lamé Equations

The vectorial extension of the Ribaucour transformation for the Lamé equations of orthogonal conjugates nets in multidimensions is given. We show that the composition of two vectorial Ribaucour transformations with appropriate transformation data is again a vectorial Ribaucour transformation, from which it follows the permutability of the vectorial Ribaucour transformations. Finally, as an example we apply the vectorial Ribaucour transformation to the Cartesian background. On leave of absence from Beijing Graduate School, CUMT, Beijing 100083, China Supported by Beca para estancias temporales de doctores y tecnólogos extranjeros en España: SB95-A01722297 Partially supported by CICYT: proyecto PB95–0401 1 1. The connection between Soliton Theory and Differential Geometry of Surfaces in Euclidean Space is well established. Many systems considered in Geometry have been analyzed independently in Soliton Theory, as examples we cite the Liouville and sine-Gordon equations which characterize minimal and pseudo-spherical surfaces, respectively. An important case is given by the the Darboux equations for conjugate systems of coordinates that were solved 12 years ago in its matrix generalization, using the ∂̄–dressing, by Zakharov and Manakov [19], and further the Lamé equations for orthogonal conjugate nets were solved only very recently [18] by Zakharov imposing appropriate constraints in the Marchenko integral equation associated with the Darboux equations. In this note we present a vectorial extension of a transformation that preserves the Lamé equations which is known as Ribaucour transformation [15]. This vectorial extension can be thought as the result of the iteration of the standard Ribaucour transformation; i. e. sequences of Ribaucour transformations. The expressions that we found are expressed in terms of multi-Grammian type determinants, as in the fundamental transformation case. The layout of this latter is as follows. In §2 we recall the reader the Darboux system for conjugate nets and its vectorial fundamental transformations, then in §3 we present the Lamé equations for orthogonal conjugate nets and show how the vectorial fundamental transformation reduces to the vectorial Ribaucour transformation. Here we also prove that, given an orthonormal basis of tangent vectors to the orthogonal conjugate coordinate lines, the vectorial Ribaucour transformation preserves this character. Next, in §4 we prove the permutability for the vectorial Ribaucour transformation basing the discussion in a similar existing result for the vectorial fundamental transformation. Finally, in §5 we present an example: we dress the zero background, specifically the Cartesian coordinates. 2. The Darboux equations ∂βij ∂uk = βikβkj, i, j, k = 1, . . . , N, with i, j, k different, (1) for the N(N − 1) functions {βij}i,j=1,...,N i6=j of u := (u1, . . . , uN), characterize N -dimensional submanifolds of R, N ≤ D, parametrized by conjugate coordinate systems [3, 7], and are the compatibility conditions of the following