Riemannian manifolds for which the identity is an \(\varepsilon\)-equilibrium map
type: Item
instance of: scholarly article
title: Riemannian manifolds for which the identity is an \(\varepsilon\)-equilibrium map
zbMATH Open document ID: 1001.53035
DOI: 10.1016/S0926-2245(00)00029-2
author: L. Todjihounde
published in: Differential Geometry and its Applications
publication date: 2002-12-16T00:00:00Z
review text: The concept of \(\varepsilon\)-equilibrium of maps between metric spaces was defined by \textit{J. Jost} by using the notion of center of mass and considering measures with supports contained in convex balls, or maps with images in simply connected and nonpositively curved manifolds [Calc. Var. Partial Differ. Equ. 2, No. 2, 173-204 (1994; Zbl 0798.58021)]. It is well-known that an \(n\)-dimensional Riemannian manifold \((M,g)\) gives rise to a metric space and the volume element \(\mu\) defines a measure on \(M\). Further, we may define a family of measures \(\mu^\varepsilon_x\) \((x\in M)\) such that \(\mu^\varepsilon_x =\mu\) on the \(\varepsilon\)-ball \(B(x,\varepsilon)\) centered at \(x\) and zero on \(M\setminus B(x,\varepsilon)\) for sufficiently small \(\varepsilon >0\). In the present paper, the author studies geometric aspects of Riemannian manifolds \((M,g)\) for which the identity is an \(\varepsilon\)-equilibrium map for sufficiently small \(\varepsilon>0\) with respect to the family \(\mu^\varepsilon_x (x\in M)\) (namely, each point \(x\in M\) is the center of mass with respect to the family \(\mu^\varepsilon_x (x\in M))\), and mainly proves that compact connected Riemannian manifolds for which the identity is an \(\varepsilon\)-equilibrium map for sufficiently small \(\varepsilon>0\) are ball-homogeneous (theorem 2.1). Further, the author derives a sequence of necessary conditions on the volume density function of these manifolds, for example, obtains a result on the Lie derivative of the volume density function (theorem 2.2). Riemannian manifolds with volume-preserving geodesic symmetries are called d'Atri spaces. The author proves that for a compact d'Atri space the identity is an \(\varepsilon\)-equilibrium map with respect to the family \(\mu_x^\varepsilon (x\in M)\) for sufficiently small \(\varepsilon>0\) (proposition 2.5). The result of theorem 2.1 in combination with proposition 2.5 gives a progress to the well-known problem concerned with lo…
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C25
zbMATH DE Number: 1547181
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: volume element
zbMATH Keywords: \(\varepsilon\)-equilibrium map
zbMATH Keywords: d'Atri space
zbMATH Keywords: ball-homogeneous
Wikidata QID: Q115338379
reviewed by: Kouei Sekigawa
MaRDI profile type: Publication
full work available at URL: s0926-2245(00)00029-2
OpenAlex ID: W2048838698
description: scientific article; zbMATH DE number 1547181