Submanifolds in \(S^{n+p}\) with parallel Möbius form
type: Item
instance of: scholarly article
title: Submanifolds in \(S^{n+p}\) with parallel Möbius form
zbMATH Open document ID: 1086.53020
author: Qiaoling Xia
published in: Applied Mathematics. Series B (English Edition)
publication date: 2005-04-04T00:00:00Z
review text: Let \(x: M\to S^{n+p}\) (\(n\geq 2\), \(p\geq 1\)) be an \(n\)-dimensional submanifold in the unit sphere \(S^{n+p}\) without umbilic points and with parallel Möbius form \(\Phi\); let the normalized scalar curvature \(\kappa\) of the Möbius invariant metric \(G\) be constant, and denote by \(\widetilde A\) the traceless Blaschke form. Then the following 3 rigidity theorems are valid: Theorem 1.1: Assume, additionally: \(p\geq 2\), \(\|\widetilde A\|\leq n^{1/2}(n- 1)^{1/2}(n- 2)^{-1}(\kappa- f(n))\) with \(f(n)= (3n- 5)/(2n^2)\), and \(M\) be compact. Then \(M\) is Möbius equivalent to the Veronese surface in \(S^4\). Theorem 1.2: Assume, additionally: \(n\geq 3\), \(p=1\), and \(\|\widetilde A\|\leq n^{1/2}(n-1)^{1/2}(n- 2)^{-1}(\kappa- f(n))\) with \(f(n)= (n- 2)/(n^2)\). Then \(M\) is Möbius equivalent either to the Clifford minimal hypersurface in \(S^{n+1}\) (case: \(\kappa= f(n)\)) or to a torus \(S^1(a)\times S^{n-1}(b)\to S^{n+1}\) (case: \(f(n)< \kappa< n\cdot f(n)\)) with certain radii \(a= a(n,\kappa)\), \(b= b(n,\kappa)\). Theorem 1.3: Assume, additionally: \(n=2\), \(p=1\), \(\kappa\geq 0\) (and \(\kappa=\text{const}\)), and let \(M\) be compact. Then \(M\) is Möbius equivalent to a flat torus in \(S^3\to\mathbb{R}^4\).
Mathematics Subject Classification ID: 53A30
Mathematics Subject Classification ID: 53B25
zbMATH DE Number: 2151012
zbMATH Keywords: Möbius submanifold
zbMATH Keywords: Möbius form
zbMATH Keywords: Blaschke form
zbMATH Keywords: constant scalar curvature
zbMATH Keywords: torus
zbMATH Keywords: Clifford minimal hypersurface
zbMATH Keywords: Veronese surface
MaRDI profile type: Publication
full work available at URL: s11766-004-0007-z
OpenAlex ID: W2086660255
cites work: Q3729588
cites work: Submanifolds with constant Möbius scalar curvature in \(S^n\)
cites work: An intrinsic rigidity theorem for minimal submanifolds in a sphere
cites work: Möbius isoparametric hypersurfaces in \(S^{n+1}\) with two distinct principal curvatures
cites work: Möbius isotropic submanifolds in \(\mathbb{S}^n\)
cites work: Submanifolds with parallel mean curvature vector in spheres
cites work: Moebius geometry of submanifolds in \(\mathbb{S}^n\)
cites work: Q4807897
DOI: 10.1007/S11766-004-0007-Z
description: scientific article; zbMATH DE number 2151012