Classification of hypersurfaces with constant Möbius curvature in \({\mathbb{S}^{m+1}}\)
type: Item
instance of: scholarly article
title: Classification of hypersurfaces with constant Möbius curvature in \({\mathbb{S}^{m+1}}\)
zbMATH Open document ID: 1259.53010
author: Zhen Guo
author: Tongzhu Li
author: Limiao Lin
author: Xiang Ma
author: Changping Wang
published in: Mathematische Zeitschrift
publication date: 2012-06-26T00:00:00Z
review text: Let \(x:M \rightarrow S^{m+1}\) be an \(m\)-dimensional umbilic-free hypersurface \(M\) immerse in \(S^{m+1}\), with the standard metric \(I=dx \cdot dx\). By using the positive function \(\rho=\sqrt{\frac{m}{m-1} \| II - \frac{1}{m} tr(II)I \|}\), where \(II\) is the second fundamental form, the authors define the \((0,2)\) tensor \(g=\rho^2 I\) called Möbius metric, which is invariant under conformal transformations of \(S^{m+1}\). The curvature induced by the metric \(g\) is known as Möbius curvature. The principal result in this paper is the classification of \(m\)-dimensional hypersurfaces \(M\) in \(S^{m+1}\) with constant Möbius sectional curvature by using the Möbius geometry method of submanifolds established by \textit{C. P. Wang} [Manuscr. Math. 96, No. 4, 517--534 (1998; Zbl 0912.53012)].
Mathematics Subject Classification ID: 53A30
Mathematics Subject Classification ID: 53B25
zbMATH DE Number: 6050613
zbMATH Keywords: Möbius metric
zbMATH Keywords: constant sectional curvature
zbMATH Keywords: Möbius flat hypersurfaces
zbMATH Keywords: Möbius deformable hypersurfaces
zbMATH Keywords: curvature-spiral
MaRDI profile type: Publication
full work available at URL: s00209-011-0860-4
OpenAlex ID: W2087204766
cites work: Q4889886
cites work: A conformal differential invariant and the conformal rigidity of hypersurfaces
cites work: Q3322795
cites work: The Möbius characterizations of Willmore tori and Veronese submanifolds in the unit sphere
cites work: The second variational formula for Willmore submanifolds in \(S^n\).
cites work: Submanifolds with constant Möbius scalar curvature in \(S^n\)
cites work: Classification of hypersurfaces with parallel Möbius second fundamental form in \(S^{n+1}\)
cites work: Möbius geometry of surfaces of constant mean curvature 1 in hyperbolic space
cites work: Möbius isoparametric hypersurfaces in \(S^{n+1}\) with two distinct principal curvatures
cites work: Möbius geometry of hypersurfaces with constant mean curvature and scalar curvature
cites work: Surfaces with vanishing Möbius form in \(S^n\).
cites work: A Moebius characterization of Veronese surfaces in \(S^n\)
cites work: Möbius isotropic submanifolds in \(\mathbb{S}^n\)
cites work: Willmore surfaces of constant Möbius curvature
cites work: Q3816661
cites work: Moebius geometry of submanifolds in \(\mathbb{S}^n\)
cites work: Q3966739
cites work: Flat Möbius hypersurfaces in \(S^{n+1}\)
DOI: 10.1007/S00209-011-0860-4
description: scientific article; zbMATH DE number 6050613