Ricci solitons and geometry of four-dimensional non-reductive homogeneous spaces (Q2903305)

A Ricci soliton is a pseudo-Riemannian manifold \((M,g)\) admitting a smooth vector field \(V\) such that NEWLINE\[NEWLINE{\mathcal{L}}_Vg+\varrho=\lambda g, NEWLINE\]NEWLINE where \({\mathcal{L}}_V\) denotes the Lie derivative in the direction of \(V\), \(\varrho\) is the Ricci tensor, and \(\lambda\) is a real number. A Ricci soliton is said to be shrinking, steady or expanding according to whether \(\lambda>0\), \(\lambda=0\) or \(\lambda<0\), respectively. Ricci solitons are the self-similar solutions of the Ricci flow and are important in understanding its singularities.NEWLINENEWLINEThis paper is devoted to the geometry of non-reductive four-dimensional homogeneous spaces. After describing their Levi-Civita connection and curvature properties, the authors classify Einstein-like metrics and homogeneous Ricci solitons on these spaces, proving the existence of shrinking, expanding, and steady examples. Moreover, for all the non-trivial examples given, the Ricci operator is diagonalizable. Finally, the authors study invariant symplectic and complex structures on four-dimensional, non-reductive homogeneous, pseudo-Riemannian manifolds.