Some Results on Ricci Almost Solitons

We find three necessary and sufficient conditions for an n-dimensional compact Ricci almost soliton (M,g,w,σ) to be a trivial Ricci soliton under the assumption that the soliton vector field w is a geodesic vector field (a vector field with integral curves geodesics). The first result uses condition...

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Bibliographic Details
Published in:Symmetry (Basel) 2021-03, Vol.13 (3), p.430
Main Authors: Deshmukh, Sharief, Alsodais, Hana, Bin Turki, Nasser
Format: Article
Language:English
Subjects:
Einstein manifolds
Ricci almost soliton
Ricci soliton
trivial Ricci soliton
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Summary:We find three necessary and sufficient conditions for an n-dimensional compact Ricci almost soliton (M,g,w,σ) to be a trivial Ricci soliton under the assumption that the soliton vector field w is a geodesic vector field (a vector field with integral curves geodesics). The first result uses condition r2≤nσr on a nonzero scalar curvature r; the second result uses the condition that the soliton vector field w is an eigen vector of the Ricci operator with constant eigenvalue λ satisfying n2λ2≥r2; the third result uses a suitable lower bound on the Ricci curvature S(w,w). Finally, we show that an n-dimensional connected Ricci almost soliton (M,g,w,σ) with soliton vector field w is a geodesic vector field with a trivial Ricci soliton, if and only if, nσ−r is a constant along integral curves of w and the Ricci curvature S(w,w) has a suitable lower bound.
ISSN:2073-8994
2073-8994
DOI:10.3390/sym13030430