Ricci Solitons and Killing Fields on Generalized Cahen-Wallach Manifolds

We study Ricci solitons and Killing fields on generalized Cahen-Wallach manifolds. The Ricci soliton equation provides a generalization of the Einstein equation on (pseudo-)Riemannian manifolds which is closely connected with Ricci flows. We prove that the Ricci soliton equation is locally solvable with any constant in the Ricci soliton equation on generalized Cahen-Wallach manifolds. Using a Brinkmann coordinate system, we study the Killing fields on these manifolds and give constraints on the dimension of the space of Killing fields. Also, we obtain solutions to the Killing equations for 2-symmetric Lorentzian manifolds in small dimensions.