The aim of this paper is to find some equations of structure for almost Ricci solitons which generalize the equivalent for Ricci solitons. As a consequence of these equations we derive an integral formula for the compact case which enables us to show 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. Moreover, we also use the Hodge-de Rham decomposition theorem to make a link with the associated vector field of an almost Ricci soliton.

The purpose of this article is to investigate Bach-flat critical metrics of the volume functional on a compact manifold M with boundary ∂ M. Here, we prove that a Bach-flat critical metric of the volume functional on a simply connected 4-dimensional manifold with boundary isometric to a standard sphere must be isometric to a geodesic ball in a simply connected space form R 4 , H 4 or S 4 . Moreover, we show that in dimension three the result even is true replacing the Bach-flat condition by the weaker assumption that M has divergence-free Bach tensor.

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Abstract. We study the space of smooth Riemannian structures on compact three-manifolds with boundary that satisfies a critical point equation associated with a boundary value problem, for simplicity, Miao-Tam critical metrics. We provide an estimate to the area of the boundary of Miao-Tam critical metrics on compact three-manifolds. In addition, we obtain a Böchner type formula which enables us to show that a Miao-Tam critical metric on a compact three-manifold with positive scalar curvature must be isometric to a geodesic ball in S 3 .

The aim of this note is to prove that any compact non-trivial almost Ricci soliton M n , g, X, λ with constant scalar curvature is isometric to a Euclidean sphere S n . As a consequence we obtain that every compact non-trivial almost Ricci soliton with constant scalar curvature is gradient. Moreover, the vector field X decomposes as the sum of a Killing vector field Y and the gradient of a suitable function.

We provide a general Böchner type formula which enables us to prove some rigidity results for V -static spaces. In particular, we show that an n-dimensional positive static triple with connected boundary and positive scalar curvature must be isometric to the standard hemisphere, provided that the metric has zero radial Weyl curvature and satisfies a suitable pinching condition. Moreover, we classify V -static spaces with non-negative sectional curvature.

The aim of this article is to study the space of metrics with constant scalar curvature of volume 1 that satisfies the critical point equation frakturLg*(f)=Ric˚, for simplicity CPE metrics. It has been conjectured that every CPE metric must be Einstein. Here, we shall focus our attention for 4‐dimensional half conformally flat manifolds M4. In fact, we shall show that for a nontrivial f,M4 must be isometric to a sphere double-struckS4 and f is some height function on S4.

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