Singer invariants and various types of curvature homogeneity

Abstract

We study the concept of curvature homogeneity of type (1,3) introduced by Kowalski and Vanžurova in the context of the standard curvature homogeneity introduced by Singer and studied by many authors. We study a specialization of this property known as homothety \(r\)-curvature homogeneity. We characterize these concepts in terms of model spaces. As the main result, we present two families of three-dimensional Lorentzian metrics on Euclidean space to exhibit various relationships between all introduced concepts of curvature homogeneity and local homogeneity.