Isometric immersions of Euclidean plane into Euclidean 4-space with vanishing normal curvature

Abstract

Every isometric immersion of ${\boldsymbol R}^2$ into ${\boldsymbol R}^4$ with vanishing normal curvature is assosiated with a pair of real-valued functions satisfying a system of second order partial differential equations of hyperbolic type,and vice versa. An isometric immersion with vanishing normal curvature is revealed to be multiple-valued in general as is shown by some concrete examples.