ISOMETRIC IMMERSIONS OF EUCLIDEAN PLANE INTO EUCLIDEAN 4-SPACE WITH VANISHING NORMAL CURVATURE

Abstract

Every isometric immersion of $\mathbf{R}^2$ into $\mathbf{R}^4$ 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.