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

TL;DR

This paper introduces the concept of strong curvature homogeneity of type (1,3) in manifolds, characterizes it via model spaces, and provides examples illustrating its properties and distinctions from local homogeneity.

Contribution

It extends the definition of curvature homogeneity to include homotheties preserving derivatives, characterizes these properties, and presents explicit Lorentzian metric examples.

Findings

Existence of manifolds that are curvature homogeneous but not locally homogeneous.

Strong curvature homogeneity of order 1 implies local homogeneity in certain cases.

Strong curvature homogeneity of order 2 guarantees local homogeneity.

Abstract

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.