On isometry of a complete Riemannian manifold to a sphere
type: Item
instance of: scholarly article
title: On isometry of a complete Riemannian manifold to a sphere
zbMATH Open document ID: 0812.53042
author: Yongfan Zheng
published in: Tsukuba Journal of Mathematics
publication date: 1994-12-19T00:00:00Z
review text: The purpose of this article is to obtain sufficient conditions in terms of conformal vector fields and curvatures for a complete Riemannian manifold to be isometric to an ordinary \(n\)-sphere. Among others, he proves the following theorem. Let \(M\) be a complete Riemannian manifold of dimension \(>2\) with non-zero constant scalar curvature. Let \(R\) and \(K\) denote the Riemannian curvature tensor and the Ricci tensor of \(M\), respectively. If \(M\) admits a non-isometric conformal vector field \(X\), \(\mathcal L_ Xg = 2\rho g\), such that \({\mathcal L}_ X| R|^ 2 = 0\) (or \({\mathcal L}_ X| K|^ 2 = 0\)) and if the \(\rho\) has first derivative in \(L^ 2(M)\), then \(M\) is isometric to an ordinary \(n\)- sphere.
reviewed by: Bang-Yen Chen
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 701553
zbMATH Keywords: conformal vector field
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496162460
OpenAlex ID: W1527874893
Wikidata QID: Q115233564
DOI: 10.21099/TKBJM/1496162460
description: scientific article; zbMATH DE number 701553