We show that a compact almost Ricci soliton whose soliton vector field is divergence-free is Einstein and its soliton vector field is Killing. Next we show that an almost Ricci soliton reduces to Ricci soliton if and only if the associated vector field is geodesic. Finally, we prove that a contact metric manifold is K-contact if and only if its Reeb vector field is geodesic.
Sharma, Ramesh, "Some Results on Almost Ricci Solitons and Geodesic Vector Fields." (2017). Mathematics Faculty Publications. 14.
Sharma, R. (2017). Some results on almost Ricci solitons and geodesic vector fields. Beiträge zur Algebra und Geometrie/Contributions to Algebra and Geometry, 1-6. doi:10.1007/s13366-017-0367-1
Available for download on Thursday, November 22, 2018