Some Non-Trivial and Non-Gradient Closed Pseudo-Riemannian Steady Ricci SolitonsMaryam Jamreh School of Mathematics, Iran University of Science and Technology, Narmak, Tehran
16846-13114, Iran Mehdi Nadjafikhah School of Mathematics, Iran University of Science and Technology, Narmak, Tehran
16846-13114, Iran Received October 10, 2018, revised May 13, 2019. Abstract In this paper, we study the Ricci soliton equation on compact indecomposable Lorentzian 3-manifolds that admit a parallel light-like vector field with closed orbits. These compact structures that are geodesically complete, admit non-trivial, i.e., non-Einstein and non-gradient steady Lorentzian Ricci solitons with zero scalar curvature which show the difference between closed Riemannian and pseudo-Riemannian Ricci solitons. The associated potential vector field of a Ricci soliton structure in all the cases that we construct on these manifolds is a space-like vector field. However, we show that there are examples of closed pseudo-Riemannian steady Ricci solitons in the neutral signature (2, 2) with zero scalar curvature such that the associated potential vector field can be time-like or null. These compact manifolds are also geodesically complete and they cannot admit a conformal-Killing vector field.
Mathematics Subject Classification 2000: 53C50,58J99,35R01. |