Ricci almost solitons with associated projective vector field

Abstract

A Ricci almost soliton whose associated vector field is projective is shown to have vanishing Cotton tensor, divergence-free Bach tensor and Ricci tensor as conformal Killing. For the compact case, a sharp inequality is obtained in terms of scalar curvature.We show that every complete gradient Ricci soliton is isometric to the Riemannian product of a Euclidean space and an Einstein space. A complete K-contact Ricci almost soliton whose associated vector field is projective is compact Einstein and Sasakian.

Some Results on Ricci Almost Solitons

“…[3]) in an attempt to generalize Ricci solitons, by replacing the soliton constant with the smooth function σ. Geometry of Ricci solitons and Ricci almost solitons has been subject of immense interest due to their elegant geometry as well as applications (cf. [1,[3][4][5][6][7][8][9][10][11][12][13][14][15]). Given a Ricci almost soliton (M, g, w, σ), we call w the soliton vector field and the smooth function σ the potential function.…”

Ricci Solitons on Multiply Warped Product Manifolds

“…Ricci solitons and their some generalizations have been studied by many geometers in the recent years. For example see ( [2], [3], [7], [8], [9], [13], [15], [17], [19], [27], [31]) and the references therein.…”