Rough isometries, and combinatorial approximations of geometries of non- compact Riemannian manifolds
type: Item
instance of: scholarly article
title: Rough isometries, and combinatorial approximations of geometries of non- compact Riemannian manifolds
zbMATH Open document ID: 0554.53030
published in: Journal of the Mathematical Society of Japan
publication date: 1985-01-01T00:00:00Z
review text: We study a map \(\phi\) : \(X\to Y\), not necessarily continuous, between non-compact Riemannian manifolds X and Y, which satisfies the following two conditions: (i) for some \(\epsilon >0\), the \(\epsilon\)-neighborhood of the image of \(\phi\) in Y coincides with Y; (ii) for all \(x_ 1,x_ 2\in X\), the inequalities \[ a^{-1} d(x_ 1,x_ 2)-b\leq d(\phi (x_ 1),\phi (x_ 2))\leq a d(x_ 1,x_ 2)+b \] hold, where \(a\geq 1\) and \(b\geq 0\) are suitable constants, and d denotes the distance functions on X and Y. We call such a map a rough isometry. The main theorems of the paper suggest that some attributes of non- compact Riemannian manifolds are inherited through rough isometries. More precisely, we show that the volume growth rates and the validity of isoperimetric inequalities of non-compact complete Riemannian manifolds are preserved by rough isometries, under the additional conditions on the manifolds that the Ricci curvatures are bounded from below and the injectivity radii are positive. We also prove the Liouville theorem for positive harmonic functions on a complete Riemannian manifold which is roughly isometric to the Euclidean space and satisfies the above conditions on the curvature and the injectivity radius. In the proofs of these theorems, we frequently employ the method of combinatorial approximation of Riemannian geometry: We approximate ''continuous'' geometry of a non-compact Riemannian manifold by combinatorial geometry of a suitable discrete subset of the manifold.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58E20
zbMATH DE Number: 3883148
zbMATH Keywords: rough isometry
zbMATH Keywords: volume growth rates
zbMATH Keywords: isoperimetric inequalities
zbMATH Keywords: Liouville theorem
zbMATH Keywords: combinatorial approximation
author: Masahiko Kanai
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03730391
OpenAlex ID: W2061159431
Wikidata QID: Q115225839
DOI: 10.2969/JMSJ/03730391
description: scientific article; zbMATH DE number 3883148