Weakly Disk-Homogeneous Spaces

Naturally reductive Riemannian homogeneous spaces and more generally, g.o. spaces, have the property that the volume of a geodesic disk normal to a geodesic and with center on that geodesic remains constant when the center moves along that central geodesic. Riemannian manifolds having that property for arbitrary geodesics and all sufficiently small geodesic disks are called weakly disk-homogeneous. Since, up to local isometries, there are no other examples known for dimension > 2, we investigate whether a possible converse holds or not. Besides some general results, we give a positive answer for three-dimensional and for several classes of four-dimensional manifolds. Some related results are discussed, in particular about four-dimensional Einstein C-spaces.