Special ball-homogeneous spaces (Q1381101)

A ball-homogeneous space is a Riemannian manifold \((M,g)\) on which the volumes of all sufficiently small geodesic balls only depend on the radius. In dimensions greater than two, it is an open problem whether each ball-homogeneous Riemannian manifold is locally homogeneous. In the present paper a positive answer is given in the following special situations: (a) \(M\) is three-dimensional with at most two distinct Ricci eigenvalues; (b) \(M\) is three-dimensional and its Ricci tensor is either a Codazzi tensor, or it is cyclic parallel; (c) \(M\) is conformally flat with at most three distinct Ricci eigenvalues (and of arbitrary dimension).