Induced surfaces and their integrable dynamics II. Generalized Weierstrassrepresentations in 4-D spaces and deformations via DS hierarchy

Extensions of the generalized Weierstrass representation to generic surface s in 4-D Euclidean and pseudo-Euclidean spaces are given. Geometric charact eristics of surfaces are calculated. It is shown that integrable deformatio ns of such induced surfaces are generated by the Davey-Stewartson hierarchy . Geometrically, these deformations are characterized by the invariance of an infinite set of functionals over surface. The Willmore functional (the t otal squared mean curvature) is the simplest of them. Various particular cl asses of surfaces and their integrable deformations are considered.