Curvature Homogeneous Lorentzian and Higher Signature

Various notions of symmetry are important in mathematical physics. Sometimes this symmetry arises from a transitive isometric group action. But there are other notions of interest. One says a manifold is curvature homogeneous if there is an isometry between the tangent spaces of any two points of the manifolds preserving the curvature tensor. We exhibit Lorentz manifolds which are curvature homogeneous but not homogeneous and discuss their geometric properties. As pseudo-Riemannian manifolds which have dimension greater than $4$ and signatures other than Riemannian or Lorentzian are important in many physical applications (Kaluza-Klein gravity and brane world cosmology, we shall also discuss higher signature examples with interesting geometrical properties.