Journal of Mathematical Physics, Analysis, Geometry
2019, vol. 15, No 4, pp. 526-542   https://doi.org/10.15407/mag15.04.526     ( to contents , go back )
https://doi.org/10.15407/mag15.04.526

Some Non-Trivial and Non-Gradient Closed Pseudo-Riemannian Steady Ricci Solitons

Maryam Jamreh

School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran
E-mail: m jamreh@iust.ac.ir, maryamjamreh@gmail.com

Mehdi Nadjafikhah

School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran
E-mail: m nadjafikhah@iust.ac.ir

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.
Key words: Ricci solitons, closed pseudo-Riemannian manifolds, parallel light-like vector field.

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