DOI: 10.33773/jum.1906429 ISSN: 2618-5660

RICCI–YAMABE SOLITONS ON THE LIE GROUP Sol3

Abdou Bousso, Ameth Ndiaye
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We study Ricci–Yamabe solitons on the three-dimensional solvable Lie group $\mathrm{Sol}_3$, one of Thurston's eight model geometries. After computing the Levi-Civita connection and the Ricci tensor of the canonical left-invariant metric, we derive necessary and sufficient conditions for the existence of such solitons and give an explicit classification of the associated vector fields. We further prove that $\mathrm{Sol}_3$ admits no non-trivial gradient Ricci–Yamabe soliton, and we characterize the conditions under which the dual $1$-form of the soliton vector field defines a contact structure. As an application, we determine when the quintuple $(\mathrm{Sol}_3, g, X, \mu_1, \mu_2)$ constitutes a hyperbolic Ricci soliton.

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