Space-like isothermic surfaces and Grassmannian systems

Abstract

We show that space-like isothennic surfaces in the pseudo-riemannian space $\bf{R}^{n-j,j}$ are associated to $O(n - j + 1,j + 1)/O(n - j,j) x 0(1, I)$-system, and that the action of a rational map with two simple poles on the space of local solutions of $O(n - j + I,j + l)/O(n - j,j) \times 0(1, I)$-system correspond to Ribaucour and Darboux transformations to space-like surfaces in $\bf{R}^{n-j,i}$.