Some characterizations for compact almost Ricci solitons (Q2880664)

A Riemannian manifold \((M^n,g)\) is an almost Ricci soliton if there exist a vector field \(X\) and a soliton function \(\lambda: M^n\rightarrow{\mathbb R}\) such that \(\operatorname {Ric} + \frac{1}{2}{\mathcal L}_X g=\lambda g\), where \({\mathcal L}\) is the Lie derivative. When the vector field \(X\) is the gradient of a function \(f: M^n\rightarrow{\mathbb R}\), then \((M^n,g)\) is a gradient almost Ricci soliton with potential \(f\).NEWLINENEWLINENEWLINEThe vector field \(X\) on a compact oriented Riemannian manifold can be decomposed as the gradient of a function \(h\) and a divergence free vector field \(Y\), hence \(X=\nabla h + Y\), where \({\mathrm{div}}Y=0\).NEWLINENEWLINEIn the paper, this decomposition is used to prove that for a compact almost Ricci gradient soliton with potential \(f\), the potential \(f\) agrees with the Hodge-de Rham potential \(h\), up to a constant. Further, for \(n\geq 3\) a compact almost Ricci gradient soliton with a nontrivial conformal vector field \(X\) is isometric to an Euclidean sphere. Integral formulas for an almost Ricci soliton which are generalizations of similar formulas for a Ricci soliton are also obtained. As a consequence, it is proved that a compact nontrivial almost Ricci soliton is isometric to a sphere provided either it has constant scalar curvature or its associated vector field is conformal.