On a differential equation characterizing Euclidean spheres.
type: Item
instance of: scholarly article
title: On a differential equation characterizing Euclidean spheres.
zbMATH Open document ID: 1058.53027
DOI: 10.1016/S0022-0396(03)00173-6
author: Bülent Ünal
author: Eduardo García-Río
author: Demir N. Kupeli
published in: Journal of Differential Equations
publication date: 2003-11-19T00:00:00Z
review text: Obata showed that a necessary and sufficient condition for a connected, complete Riemannian manifold \((M,g)\) to be isometric with the Euclidean sphere of radius \(1/\sqrt{\lambda},\) \( \lambda>0 \) is the existence of a nonconstant function \(f\) on \(M\) satisfying the differential equation \(H_{f}+\lambda f g =0,\) where \(H_{f}\) is the Hessian of f on \((M,g).\) While, the differential equation \(1/\sqrt{\lambda},\) \(\lambda>0 \) on a connected, complete, Riemannian manifold \((M,g)\) has a nontrivial solution iff its domain \((M,g)\) is the Euclidean sphere of radius \(1/\sqrt{\lambda}.\) In this paper, the authors show that a necessary and sufficient condition for a connected, complete \(n(\geq2)\)-dimensional Riemannian manifold to be isometric with the Euclidean sphere of radius \(1/\sqrt{\lambda},\) \(\lambda>0,\) is the existence of a nonzero vector field \(Z\) on \((M,g)\) satisfying the differential equation \((\nabla \nabla Z) (X,Y) + \lambda g (Z,X) (Y)=0\) for all \(X,Y \in \Gamma(TM),\) where \(\nabla\) is the covariant derivative on \(M\). They also analyze the case \(\lambda \leq 0\) for the differential equation \((\nabla \nabla Z) (X,Y) + \lambda g (Z,X) (Y)=0\) for all \(X,Y \in \Gamma(TM),\) on a connected complete Riemannian manifold \((M,g).\)
reviewed by: Yong Seung Cho
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 2005880
zbMATH Keywords: Second covariant differential
zbMATH Keywords: Laplacian
zbMATH Keywords: Conformal vector field
zbMATH Keywords: Affine
zbMATH Keywords: conformal vector field
zbMATH Keywords: \(k\)-Nullity vector field
zbMATH Keywords: Projective vector field
zbMATH Keywords: Euclidean sphere
MaRDI profile type: Publication
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full work available at URL: s0022-0396(03)00173-6
OpenAlex ID: W1988739256
description: scientific article; zbMATH DE number 2005880