Minimal 2-tori in \(S^ 4\) whose Gauss images have constant curvature
type: Item
instance of: scholarly article
title: Minimal 2-tori in \(S^ 4\) whose Gauss images have constant curvature
zbMATH Open document ID: 0809.53067
DOI: 10.1007/BF01196439
author: Makoto Sakaki
published in: Archiv der Mathematik
publication date: 1995-03-30T00:00:00Z
review text: Let \(M\) be a minimal surface immersed in the \(n\)-dimensional sphere \(S^ n\) of curvature 1 and let \(K\) and \(\widetilde {K}\) denote the Gaussian curvatures of the induced metrics by the immersion and the Obata's Gauss map respectively. In this paper it is given a characterization of the Clifford torus by the Gauss map, among minimal 2-tori in \(S^ 4\). Namely, it is proved the following: Let \(M\) be a minimal 2-torus in \(S^ 4\) with constant \(\widetilde{K}\) except, possibly, at isolated points where \(K = 1\). Then \(M\) is flat and the Clifford torus in a totally geodesic \(S^ 3\).
reviewed by: R. Tribuzy
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 590078
zbMATH Keywords: minimal surface
zbMATH Keywords: Gauss map
zbMATH Keywords: 2-torus
zbMATH Keywords: totally geodesic
MaRDI profile type: Publication
cites work: Minimal Surfaces of Constant Curvature in S n
cites work: Q5790826
cites work: Constant mean curvature surfaces in 4-space forms
cites work: On the Gauss map of minimal surfaces immersed in \(R^ n\)
cites work: Minimal immersions of surfaces into 4-dimensional space forms
cites work: The geometry of the generalized Gauss map
cites work: On compact minimal surfaces with non-negative Gaussian curvature in a space of constant curvature. I
cites work: Minimal Surfaces with Constant Curvature in 4-Dimensional Space Forms
cites work: Complete minimal surfaces in \(S^ 3\)
cites work: Some intrinsic characterizations of minimal surfaces
cites work: The Gauss map of immersions of Riemannian manifolds in spaces of constant curvature
cites work: Exceptional minimal surfaces whose Gauss images have constant curvature
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description: scientific article; zbMATH DE number 590078