Some characterizations for compact almost Ricci solitons

Abstract

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.

Conformal Vector Fields on Doubly Warped Product Manifolds and Applications

“…Proof. From the above proof we have where L ζ denotes the Lie-derivative of the metric tensor g, Ric is the Ricci curvature and λ is a constant [5,6,19,25,30].…”

Gradient Ricci almost soliton warped product

“…Moreover, some basic tools from the weighted manifold theory such as general weighted volume comparisons and maximum principles at infinity for diffusion operators were also discussed. The structure equations for Ricci almost solitons were published shortly after by Barros and Ribeiro [3]. As a consequence of these equations they showed that a compact nontrivial Ricci almost soliton is isometric to a sphere provided it is either of constant scalar curvature or its associated vector field is conformal.…”

Ricci almost solitons and contact geometry

“…Barros and Ribeiro Jr. proved (cf. [2]) that a compact almost Ricci soliton with constant scalar curvature is isometric to a Euclidean sphere. An analogous result has also been proved by Wang-Gomes-Xia [20] for the case of k-almost Ricci soliton.…”