Geodesic
Some non-trivial and non-gradient closed pseudo-Riemannian steady Ricci solitons
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 15 (2019), pp. 526-542

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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. 
@article{JMAG_2019_15_a4,
     author = {Maryam Jamreh and Mehdi Nadjafikhah},
     title = {Some non-trivial and non-gradient closed {pseudo-Riemannian} steady {Ricci} solitons},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {526--542},
     publisher = {mathdoc},
     volume = {15},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2019_15_a4/}
}
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Maryam Jamreh; Mehdi Nadjafikhah. Some non-trivial and non-gradient closed pseudo-Riemannian steady Ricci solitons. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 15 (2019), pp. 526-542. http://geodesic.mathdoc.fr/item/JMAG_2019_15_a4/