## The classification of static spaces and related problems

### Volume 151 / 2018

#### Abstract

We treat the gradient Ricci almost soliton equation, the static equation and the critical point equation in the same way. In all three cases, on a given Riemannian manifold $(M^n,g)$, there exists a smooth solution $f$ satisfying the equation $$f_{,ij}=f^pR_{ij}+\lambda g_{ij},$$ where $p\geq 0$ is a nonnegative integer, $R_{ij}$ denotes the Ricci curvature and $\lambda $ is a $C^1$ function. Using some ideas of Cao and Chen (2012, 2013), we obtain some classifications under the assumption that the Bach tensor vanishes, which generalize some results of Qing and Yuan (2013) and Cao and Chen (2013). In particular, for $n=3$, we prove that $(M^3, g)$ is locally conformally flat for any $p$ if the Bach tensor is divergence-free.