Ricci solitons and harmonic vector fields on $N i l^{3} \times \mathbb{R}$
Gherici Beldjilali
Electronic Research Archive ›› 2026, Vol. 34 ›› Issue (8) : 5367 -5379.
In this paper, we investigated the geometric and variational properties of the four-dimensional Thurston geometry $N i l^{3} \times \mathbb{R}$. First, we provided a complete classification of Ricci soliton structures by solving the associated system of partial differential equations. We proved that all Ricci solitons on this manifold were strictly shrinking with λ = −3/2 and explicitly showed that none of them admitted a gradient formulation. In contrast, by examining the Yamabe flow counterpart, we demonstrated that $N i l^{3} \times \mathbb{R}$ did not admit any nontrivial Yamabe solitons; the only solutions were trivial and generated by Killing vector fields, revealing a stark geometric duality between volume-preserving and conformal deformations. Finally, we shifted to the variational theory of maps and characterized the space of harmonic vector fields. While left-invariant fields were trivially harmonic, we utilized the tension field framework to construct a rich, nontrivial family of harmonic vector fields depending solely on the x-coordinate, providing their precise analytical description.
Lie group $N i l^{3} \times \mathbb{R}$ / Ricci soliton / Yamabe soliton / harmonic vector fields
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