AFFINE CURVATURE HOMOGENEOUS 3-DIMENSIONAL LORENTZ MANIFOLDS
International Journal of Geometric Methods in Modern Physics (IJGMMP)
We study a family of 3-dimensional Lorentz manifolds. Some members of the family are 0-curvature homogeneous, 1-affine curvature homogeneous, but not 1-curvature homogeneous. Some are 1-curvature homogeneous but not 2-curvature homogeneous. All are 0-modeled on indecomposible local symmetric spaces. Some of the members of the family are geodesically complete, others are not. All have vanishing scalar invariants. 1 2 P. GILKEY AND S. NIKČEVIĆ If, however, m is bounded, one has the following
... the following result due to Singer  in the Riemannian (p = 0) setting and to Podesta and Spiro  in the general setting: Theorem 1.1. There exists an integer k p,q so that if M is a geodesically complete simply connected pseudo-Riemannian manifold of signature (p, q) which is k p,q -curvature homogeneous, then M is homogeneous. We refer to Opozoda  for a similar result in the affine setting; there is an additional technical hypothesis which must be imposed. 1.3. Vanishing scalar invariants. Adopt the Einstein convention and sum over repeated indices. We can construct scalar invariants by contracting indices. For example, the scalar curvature τ , the norm |ρ| 2 of the Ricci tensor, and the norm |R| 2 of the full curvature tensor are scalar invariants defined by: By Weyl's theorem  , all universal polynomial scalar invariants of the covariant derivatives of the curvature tensor arise in this way; thus such invariants are called Weyl scalar invariants. We say that a pseudo-Riemannian manifold is VSI if all the scalar Weyl invariants vanish. This is not possible for non-flat manifolds in the Riemannian setting but is possible in the higher signature setting, see, for example, the discussion in [8, 12] .