Characterizing specific Riemannian manifolds by differential equations
type: Item
instance of: scholarly article
title: Characterizing specific Riemannian manifolds by differential equations
zbMATH Open document ID: 1033.53033
DOI: 10.1023/A:1022987819448
author: Fazilet Erkekoğlu
author: Eduardo García-Río
author: Demir N. Kupeli
author: Bülent Ünal
published in: Acta Applicandae Mathematicae
publication date: 2003-06-09T00:00:00Z
review text: Let \((M,g)\) be a connected complete Riemannian manifold of dimension \(m\geq2\). Obata showed the standard sphere of constant sectional curvature \(k>0\) is characterized as being the only manifold where there exists a non-constant function satisfying the equation \(H_f+kfg=0\). The authors present an elegant survey of the field discussing characterizations of spheres, flat spaces, hyperbolic spaces, complex projective spaces, etc. as domains where certain natural equations on certain tensors have solutions analogous to that given above by Obata. The paper contains a lovely bibliography of the field.
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 1924876
zbMATH Keywords: Hessian tensor
zbMATH Keywords: second covariant differential
zbMATH Keywords: Laplacian
zbMATH Keywords: affinity tensor
zbMATH Keywords: tension field
zbMATH Keywords: eigenvalue
zbMATH Keywords: eigenvector field
zbMATH Keywords: conformal vector field
zbMATH Keywords: projective vector field
zbMATH Keywords: \(k\)-nullity vector field
zbMATH Keywords: Euclidean sphere
zbMATH Keywords: complex projective space
zbMATH Keywords: quaternionic projective space
MaRDI profile type: Publication
full work available at URL: a:1022987819448
OpenAlex ID: W3103301
description: scientific article; zbMATH DE number 1924876