Riemannian manifolds for which the identity is an \(\varepsilon\)-equilibrium map (Q1591523)

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 local homogeneity of d'Atri spaces.