Tohoku Mathematical Journal
2022
June
SECOND SERIES VOL. 74, NO. 2

Tohoku Math. J.
74 (2022), 215-227

Title CLAIRAUT SURFACES IN EUCLIDEAN THREE-SPACE

Author Andrew D. Hwang and Xinyi Wang

(Received August 16, 2019, revised December 4, 2020)
Abstract. In this note, we investigate local isometric immersion of Clairaut metrics in Euclidean three-space. A Clairaut metric is determined up to isometry by a single function of one variable. We show that an isometric immersion is formally determined by two functions of one variable, uniquely up to coordinate reflection and ambient Euclidean motions. Further, if the Clairaut metric and these two functions are real-analytic, there exists a local isometric immersion realizing these data. We give a more explicit description for a finite-dimensional family of examples.

Mathematics Subject Classification. Primary 53A05; Secondary 53B25, 53D99.
Key words and phrases. Surfaces, Clairaut metric, isometric immersion, symplectic form

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