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.