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.
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Acknowledgments
The third author would like to thank the IMPA-Brazil for its support, where part of the work was started. He would like to extend his special thank to Professor F. C. Marques for very helpful conversations. Finally, the authors want to thank the referees for their careful reading and helpful suggestions.
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Communicated by A. Cap.
A. Barros, R. Batista: partially supported by CNPq-Brazil. E. Ribeiro: partially supported by Postdoctoral program of IMPA-Brazil and FUNCAP-Brazil.
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Barros, A., Batista, R. & Ribeiro, E. Compact almost Ricci solitons with constant scalar curvature are gradient. Monatsh Math 174, 29–39 (2014). https://doi.org/10.1007/s00605-013-0581-3
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DOI: https://doi.org/10.1007/s00605-013-0581-3