Compact almost Ricci solitons with constant scalar curvature are gradient

Abstract

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.

Gradient Ricci Almost Solitons on Two Classes of Almost Kenmotsu Manifolds

“…By relation h ′2 = −φ 2 , we shall denote by [1] ′ and [−1] ′ the distributions of the eigenvectors of h ′ orthogonal to ξ with eigenvalues 1 and −1, respectively. Also, from h ′2 = −φ 2 we may consider a local orthonormal φ-frame {ξ, e i , φe i } for 1 ≤ i ≤ n with e i ∈ [1] ′ and φe i ∈ [−1] ′ .…”

“…Also, from h ′2 = −φ 2 we may consider a local orthonormal φ-frame {ξ, e i , φe i } for 1 ≤ i ≤ n with e i ∈ [1] ′ and φe i ∈ [−1] ′ . From (3.11), we see that Df has no components on the distribution [1] ′ . Thus, we write Df = n i=1 β i φe i + ξ(f )ξ, where β i , 1 ≤ i ≤ n , are smooth functions on M 2n+1 .…”

“…Otherwise, it will be called indefinite. It is worth pointing out that Barros et al [1] proved that a Ricci almost soliton on a compact manifold is a gradient Ricci almost soliton provided that the scalar curvature is a constant.…”

On generalized m-quasi-Einstein manifolds with constant scalar curvature

“…[8]) under the condition that the manifold is Einstein. When m approaches infinite and λ is a function, there are also results about the so-called Ricci almost solitons under some curvature conditions in [5,7,11,22].…”

f– Kenmotsu Metric as Conformal Ricci Soliton

“…Yildiz, et al [21] generated some interesting results on Ricci solitons in 3-dimensional f -Kenmotsu manifolds. Recently, Basu, Bhattacharya and Dutta [1,2] established some results on conformal Ricci solitons in Kenmotsu manifolds and trans-Sasakian manifolds.…”