Extrinsic diameter of immersed flat tori in \(S^3\)
type: Item
instance of: scholarly article
title: Extrinsic diameter of immersed flat tori in \(S^3\)
zbMATH Open document ID: 1262.53049
author: Yoshihisa Kitagawa
author: Masaaki Umehara
published in: Geometriae Dedicata
publication date: 2012-11-02T00:00:00Z
review text: The Clifford torus in \(S^{3}\) is defined as \(M_{\theta}=\{(z,w)\in \mathbb{C}^{2}:|z|^{2}=\cos^{2}\theta,|w|^{2}=\sin^{2}\theta\},0<\theta<\frac{\pi}{2}\). It is a flat embedded torus of constant mean curvature. Denote by \(i_{\theta}:M_{\theta}\rightarrow S^{3}\) the canonical inclusion and by \(ds_{\theta}^{2}\) the induced metric on \(M_{\theta}\). This paper proves the following. If \(f:(M_{\theta},ds_{\theta}^{2})\rightarrow S^{3}\) is an isometric immersion, and if the mean curvature function \(H\) of \(f\) does not change sign on \(M_{\theta}\) , then \(f\) is congruent to \(i_{\theta}\) . The proof of this depends on the technique obtained by \textit{M. Umehara} [Nagoya Math. J. 134, 75--89 (1994; Zbl 0806.53002)]. A \(C^{\infty}\) map \(f:M^{n}\rightarrow N^{n+1}\) is called a wave front or front if \(f\) lifts to a Legendrian immersion \(L_{f}:M^{n}\rightarrow P(T^{\ast} N^{n+1})\), where \(P(T^{\ast} N^{n+1})\) is the projectified cotangent bundle of \(N^{n+1}\). A representation formula was obtained for flat surfaces in \(S^{3}\), regarded as wave fronts and this generalizes the first author's correspondence result [J. Math. Soc. Japan 40, No.3, 457--476 (1988; Zbl 0642.53059)]. Also, they provide two applications, one for the existence of a flat torus, a wave front with extrinsic diameter less than \(\pi\) and the other one for the sufficient and necessary condition for when two given tori can be deformed into each other within the class of flat fronts.
reviewed by: Yun Myung Oh
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 6101878
zbMATH Keywords: rigidity
zbMATH Keywords: flat torus
zbMATH Keywords: Clifford torus
zbMATH Keywords: wave front
zbMATH Keywords: 3-sphere
zbMATH Keywords: mean curvature
zbMATH Keywords: Gaussian curvature
zbMATH Keywords: extrinsic diameter
MaRDI profile type: Publication
full work available at URL: s10711-011-9580-5
OpenAlex ID: W2052926499
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DOI: 10.1007/S10711-011-9580-5
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description: scientific article; zbMATH DE number 6101878