Tohoku Mathematical Journal
2009
December
SECOND SERIES VOL. 61, NO. 4

Tohoku Math. J.
61 (2009), 523-550

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

Author Hiroshi Mori and Norio Shimakura

(Received August 23, 2007, revised August 3, 2009)
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.

2000 Mathematics Subject Classification. Primary 53C42; Secondary 35L70.
Key words and phrases. Isometric immersions, structure equations, Goursat problem, asymptotic analysis.

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