Isometric deformations of surfaces preserving the third fundamental form

摘要

We study the following problem: To what extend is a surface in the Euclidean space R-4 determined by the third fundamental form? We prove the existence of families of surfaces in R-4 which allow isometric deformations with isometric but not congruent Gaussian images. In particular, we provide a method which gives locally all surfaces in R-4 with conformal Gauss map that allow such deformations. As a consequence, we have a way for constructing non-spherical pseudoumbilical surfaces in R-4.