A Loop Group Formulation for Constant Curvature Submanifolds of Pseudo-euclidean Space

We give a loop group formulation for the problem of isometric immersions with flat normal bundle of a simply connected pseudo-Riemannian manifold M c,r, of dimension m, constant sectional curvature c 6= 0, and signature r, into the pseudo-Euclidean space R s , of signature s ≥ r. In fact these immersions are obtained canonically from the loop group maps corresponding to isometric immersions of the same manifold into a pseudo-Riemannian sphere or hyperbolic space S s or H m+k s , which have previously been studied. A simple formula is given for obtaining these immersions from those loop group maps.