The Gauss maps of Demoulin surfaces with conformal coordinates
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https://doi.org/10.1007/s11425-020-1738-0
. ARTICLES .
The Gauss maps of Demoulin surfaces with conformal coordinates In Memory of Professor Zhengguo Bai (1916–2015)
Jun-ichi Inoguchi1,∗ & Shimpei Kobayashi2 1Institute
of Mathematics, University of Tsukuba, Tsukuba 305-8571, Japan; of Mathematics, Hokkaido University, Sapporo 060-0810, Japan
2Department
Email: [email protected], [email protected] Received February 29, 2020; accepted July 8, 2020
Abstract
Demoulin surfaces in the real projective 3-space are investigated. Our result enables us to establish
a generalized Weierstrass type representation for definite Demoulin surfaces by virtue of primitive maps into a certain semi-Riemannian 6-symmetric space. Keywords MSC(2010)
Demoulin surface, Wilczynski frame, Gauss map 53A20, 53C43, 37K10
Citation: Inoguchi J, Kobayashi S. The Gauss maps of Demoulin surfaces with conformal coordinates. Sci China Math, 2021, 64, https://doi.org/10.1007/s11425-020-1738-0
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Introduction
Professor Zhengguo Bai has done great contributions in projective differential geometry. For example, he solved the so-called Fubini’s problem [22] (see also [29]). Projective differential geometry of surfaces is a treasure box of infinite dimensional integrable systems. For example, harmonic maps of Riemann surfaces into the complex projective space CPn (the CPn -sigma models in particle physics) are typical examples of 2-dimensional integrable systems. One of the key clue of the study of harmonic maps into the complex projective space is the use of harmonic sequences introduced by Chern and Wolfson [7]. It should be emphasized that the basic idea of the harmonic sequence goes back to the Laplace sequence in classical projective differential geometry (see [3]). From the modern point of view, the Laplace sequence produces the 2-dimensional Toda field equation of type A∞ (see [9, 10, 26]). In particular, the periodic Laplace sequence produces the 2-dimensional periodic Toda field equations. For example, the Laplace sequences of period 2 produce the sinh-Gordon equation. The T ¸ it¸eica equation is obtained as the Laplace sequence of period 3, and it is a structure equation of affine spheres [13]. The Laplace sequences of period 4 were studied by Su [27, 28]. Hu [15] gave a Darboux matrix, i.e., the simple type dressing for such a sequence. This article addresses the Laplace sequences of period 6. The Toda field equation derived from those sequences is a structure equation of Demoulin surfaces in the real projective 3-space RP3 [11]. * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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Sci China Math
Godeaux gave a method for studying projective surfaces through their Pl¨ uker images in real projective 5-space RP5 . His method relies on the consideration of the Laplace sequence associated with the Pl¨ uker image, called the Godeaux sequence. For a characterization of Demoulin surfaces in terms of Godeaux seq
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