Homogeneous Ricci almost solitons (Q2408018)

A Ricci almost soliton is a Riemannian manifold \((M,g)\) on which there exist a vector field \(X\) and a smooth function \(\lambda\) satisfying the equation \[ \mathcal{L}_X g+\rho=\lambda\rho \] where \(\mathcal{L}\) denotes the Lie derivative and \(\rho\) the Ricci tensor. A Ricci almost soliton \((M,g,X,\lambda)\) is called a Ricci soliton if the soliton function \(\lambda\) is constant. Otherwise, \((M,g,X,\lambda)\) is named proper. On the other hand, \((M,g,X,\lambda)\) is called a gradient Ricci almost soliton if \(X\) is the gradient of a function \(f\). In this paper, the authors study Ricci almost solitons under symmetry conditions. They start with the locally homogeneity and the curvature homogeneity conditions. As a main result, they prove that a locally homogeneous proper Ricci almost soliton is either a space of constant sectional curvature or locally isometric to a product \(\mathbb{R}\times N(c)\), where \(N(c)\) is a space of constant curvature. As a corollary, they get in the gradient case that \((M,g)\) is a space of non-zero constant sectional curvature. Moreover, they prove that the local homogeneity condition in the corollary can be changed for the more general condition of curvature-homogeneity. Thus, every curvature-homogeneous proper gradient almost Ricci soliton is a space of non-zero constant sectional curvature. They finish by considering the Codazzi Ricci tensor property on gradient Ricci almost solitons.