Homogeneity of Lorentzian three-manifolds with recurrent curvature

Abstract

k-Curvature homogeneous three-dimensional Walker metrics are described for k=0,1,2. This allows a complete description of locally homogeneous three-dimensional Walker metrics, showing that there exist exactly three isometry classes of such manifolds. As an application one obtains a complete description of all locally homogeneous Lorentzian manifolds with recurrent curvature. Moreover, potential functions are constructed in all the locally homogeneous manifolds resulting in steady gradient Ricci and Cotton solitons.

We revised the paper in version 2 to correct a minor mistake in Remark 4.5; the essential mathematical content is unchanged but the original argument was slightly garbled