Compact almost Ricci solitons with constant scalar curvature are gradient
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In this article, it was shown that any compact non-trivial almost Ricci soliton with constant scalar curvature is isometric to a Euclidean sphere.Abstract:
The aim of this note is to prove that any compact non-trivial almost Ricci soliton $$\big (M^n,\,g,\,X,\,\lambda \big )$$
with constant scalar curvature is isometric to a Euclidean sphere $$\mathbb {S}^{n}$$
. 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.read more
Citations
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On conformal solutions of the Yamabe flow
Ezequiel Barbosa,Ernani Ribeiro +1 more
TL;DR: In this article, the authors define almost Yamabe solitons as special conformal solutions of the Yamabe flow and obtain some rigidity results concerning Yamabe almost-solitons.
Journal ArticleDOI
Almost ricci solitons and k-contact geometry
TL;DR: In this article, the authors give a short Lie-derivative theoretic proof of the following recent result of Barros et al. that a compact almost Ricci soliton with constant scalar curvature is gradient, and isometric to a Euclidean sphere.
Journal ArticleDOI
Conformal $ \eta $-Ricci solitons within the framework of indefinite Kenmotsu manifolds
TL;DR: In this article , the authors consider the class of Ricci tensor tensors that admit conformal Ricci solitons on $ \epsilon $-Kenmotsu manifolds and present a characterization of the potential function.
Journal ArticleDOI
Certain Contact Metrics as Ricci Almost Solitons
TL;DR: In this paper, it was shown that if a compact K-contact metric is a gradient Ricci almost soliton, then it is isometric to a unit sphere S 2n+1.
References
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Book
Foundations of Differentiable Manifolds and Lie Groups
TL;DR: Foundations of Differentiable Manifolds and Lie Groups as discussed by the authors provides a clear, detailed, and careful development of the basic facts on manifold theory and Lie groups, including differentiable manifolds, tensors and differentiable forms.
Journal ArticleDOI
Complete Riemannian manifolds and some vector fields
TL;DR: In this article, a nonconstant scalar field p in an n-dimensional Riemannian manifold with metric tensor field (1) g is defined as a concircular scalar fields.
Journal ArticleDOI
Ricci almost solitons
TL;DR: The Ricci almost soliton as discussed by the authors is a natural extension of the concept of gradient Ricci soliton, and it has been shown to be a generalization of the Ricci Almost Soliton.