Ricci solitons on four-dimensional Lorentzian Walker manifolds (Q1707408)

The authors prove some classification results on four-dimensional Lorentzian manifolds which are Walker and they describe a Ricci soliton as well. More precisely, recall that an \(m\)-dimensional pseudo-Riemannian manifold \((M,g)\) is called a \textit{Walker manifold} if it admits a parallel null distribution of maximal dimension. For example, in the case of a Lorentzian \(4\)-manifold every Walker manifold has the form \((M,g_f)\) where \(f:M\rightarrow{\mathbb R}\) is an arbitrary smooth function such that the metric in a suitable local coordinate system \((U,t,x,y,z)\) looks like \[ g_f|_U=\mathrm ds^2=2\mathrm dt\mathrm dz+ \mathrm dx^2+\mathrm dy^2+ f(y)\mathrm dz^2. \] In order to make the structure more rigid one can require a general Walker manifold to be a \textit{Ricci soliton} as well, that is to satisfy the PDE \[ L_Xg+\mathrm{Ric}_g=\lambda g \] with respect to a vector field \(X\) on \(M\) and a real constant \(\lambda\). The main results of the paper can be summarized as follows. A Ricci soliton is called \textit{expanding, steady} or \textit{shrinking}, respectively if \(\lambda >0\), \(\lambda =0\) or \(\lambda <0\), respectively; the authors determine in the case of Lorentzian Walker \(4\)-manifolds conditions under which these cases can occur (see Theorems 3.1, 3.2 as well as Corollary 3.3). Likewise, a Ricci soliton is called \textit{gradient} if the vector field \(X\) in its definition is the gradient of a function on \(M\); the authors also characterize gradient Ricci solitons in the case of Lorentzian Walker \(4\)-manifolds (see Theorems 4.1, 4.2 and Corollary 4.3).