Seminarium Gamma

A broad class of linear problems (Lax pairs) with values in Clifford algebras will be discussed. Such systems, through a suitable generalization of the Sym formula, give rise to interesting families of multidimensional submanifolds associated with integrable nonlinear partial differential equations. These equations can be interpreted as the Gauss–Codazzi–Ricci equations for the corresponding submanifolds.The development of this subject was stimulated by the discovery of a Lax pair associated with isothermic surfaces (Cieśliński, Goldstein, Sym 1995). Initially, this was formulated in terms of the groups SO(4,1) or Sp(2,2), and later expressed in a more elegant way using the Clifford algebra and the group Spin(4,1). The application of the Sym formula does not directly yield isothermic surfaces; instead, it produces flat submanifolds immersed in a six-dimensional space. Only after projection onto appropriate three-dimensional subspaces does one obtain a pair of isothermic surfaces, which are mutually dual (known as Christoffel transforms). In this way, the classical theory of isothermic surfaces, developed by Bianchi and Darboux, has been reformulated in terms of spectral problems within modern soliton theory.