Singer invariants and strongly curvature homogeneous manifolds of type (1,3)

We extend the definition of curvature homogeneity of type (1,3) to include the possibility that there is a homothety between any two points of a manifold preserving the first r covariant derivatives of the curvature operator simultaneously; we call this strong curvature homogeneity of type (1,3) up to order r. We characterize these properties in terms of model spaces. In addition, we also present two families of three-dimensional Lorentzian metrics on Euclidean space to exhibit the behavior of this property. The first example is curvature homogeneous of type (1,3) of all orders, but is not locally homogeneous. Within this first family, being strongly curvature homogeneous of type (1,3) up to order 1 implies local homogeneity. The second example is strongly curvature homogeneous of type (1,3) up to order one, and is not locally homogeneous, showing that this new definition is not a trivial one. Within this family, being strongly curvature homogeneous of type (1,3) up to order 2 implies local homogeneity.