Symmetries of Lorentzian Three-Manifolds with Recurrent Curvature

Abstract

Abstract. Locally homogeneous Lorentzian three-manifolds with recurrect curvature are special examples of Walker manifolds, that is, they admit a parallel null vector field. We obtain a full classification of the symmetries of these spaces, with particular regard to symmetries related to their curvature: Ricci and matter collineations, curvature and Weyl collineations. Several results are given for the broader class of three-dimensional Walker manifolds.

On conformal geometry of four-dimensional generalized symmetric spaces

“…Special homogeneous pseudo-Riemannian manifolds were also studied in several cases. Three-dimensional Lorentzian manifolds with recurrent curvature were studied in [19] and [10]. Non-reductive homogeneous manifolds of dimension four were studied in [17], [9] and [7].…”

Recurrent curvature over four-dimensional homogeneous manifolds

“…In recent years, for the Lorentzian setting, several authors have studied threemanifolds with recurrent curvature; for example, E. García-Río et al obtained a complete description of all locally homogeneous Lorentzian manifolds with recurrent curvature [5]. Using this classification, G. Calvaruso and A. Zaeim [1] investigated symmetries on this space and computed Ricci and curvature collineations 244 MILAD BASTAMI, ALI HAJI-BADALI, AND AMIRHESAM ZAEIM on Lorentzian three-manifolds with recurrent curvature. A. Haji-Badali [7] investigated these spaces and showed that a Lorentzian three-manifold with recurrent curvature does not accept non-trivial proper gradient Ricci solitons; he also obtained a classification for Ricci almost solitons on these spaces.…”

On the symmetries of three-dimensional generalized symmetric spaces

“…Symmetries of the Levi-Civita connection ∇ which correspond to the affine vector fields. Since symmetries are more significant from physical aspects, they have been studied on several kinds of space-times (see [11,12], [10,8,9], [14]).…”