On a Class of Gradient Almost Ricci Solitons

Abstract

In this study, we provide some classifications for half-conformally flat gradient f-almost Ricci solitons, denoted by (M, g, f), in both Lorentzian and neutral signature. First, we prove that if $||\nabla f||$ is a non-zero constant, then (M, g, f) is locally isometric to a warped product of the form $I \times _{\varphi } N$, where $I \subset \mathbb {R}$ and N is of constant sectional curvature. On the other hand, if $||\nabla f|| = 0$, then it is locally a Walker manifold. Then, we construct an example of 4-dimensional steady gradient f-almost Ricci solitons in neutral signature. At the end, we give more physical applications of gradient Ricci solitons endowed with the standard static spacetime metric.