Heterogeneous Riemannian manifolds
type: Item
instance of: scholarly article
title: Heterogeneous Riemannian manifolds
zbMATH Open document ID: 1202.53037
DOI: 10.1155/2010/187232
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2010-04-20T00:00:00Z
full work available at URL: https://eudml.org/doc/228201
review text: A Riemannian metric on a manifold \(M\) is called heterogeneous if no two distinct points of \(M\) have isometric neighborhoods. Intuitively, such metrics are as far as possible from being homogeneous. The author proves that the set of heterogeneous metrics on a smooth manifold \(M\) of dimension \(n \geq 2\) is generic, more precisely, it is residual in the space of Riemannian metrics on \(M\) with the strong \(C^\infty\) topology. (This result was known before for compact manifolds \(M\) by \textit{T. Sunada}, [Ann. Math. (2) 121, 169--186 (1985; Zbl 0585.58047)]). The second result of the paper is a solution of Ambrose's problem for complete, connected, simply connected, heterogeneous manifolds, which shows that such a manifold is determined up to a global isometry by the behavior of the curvature tensor along geodesics emanating from a given point.
reviewed by: Mikhail Belolipetsky
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58D17
zbMATH DE Number: 5695240
zbMATH Keywords: set of heterogeneous metrics
zbMATH Keywords: generic
zbMATH Keywords: residual
Wikidata QID: Q58651704
author: James Hebda
MaRDI profile type: Publication
OpenAlex ID: W2045853713
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description: scientific article; zbMATH DE number 5695240