Homogeneity of Lorentzian three-manifolds with recurrent curvature

TL;DR

This paper classifies all locally homogeneous Lorentzian three-manifolds with recurrent curvature, identifying three isometry classes and constructing potential functions for Ricci and Cotton solitons.

Contribution

It provides a complete classification of locally homogeneous Lorentzian three-manifolds with recurrent curvature, including explicit potential functions for solitons.

Findings

Exactly three isometry classes of such manifolds exist.

Complete description of all locally homogeneous Lorentzian manifolds with recurrent curvature.

Construction of potential functions leading to steady gradient Ricci and Cotton solitons.

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.