Locally Homogeneous Four-Dimensional Manifolds of Signature (2,2)

Abstract

In this paper we consider 4-dimensional neutral-signature curvature models and obtain the complete classification of the Ricci operator. We then consider the property of curvature homogeneity for the above manifolds and prove that every complete, connected and simply connected 1-curvature homogeneous 4-dimensional manifold of signature (2,2) with a non-degenerate Ricci operator is isometric to a four-dimensional Lie group equipped with a left invariant neutral metric. We also classify Ricci-parallel curvature homogeneous 4-dimensional manifolds of signature (2,2).