In this paper, we investigated the geometric and variational properties of the four-dimensional Thurston geometry $ Nil^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 $ \lambda = -3/2 $ and explicitly showed that none of them admitted a gradient formulation. In contrast, by examining the Yamabe flow counterpart, we demonstrated that $ Nil^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.
Citation: Beldjilali Gherici. Ricci solitons and harmonic vector fields on $ Nil^3 \times \mathbb{R} $[J]. Electronic Research Archive, 2026, 34(8): 5367-5379. doi: 10.3934/era.2026239
In this paper, we investigated the geometric and variational properties of the four-dimensional Thurston geometry $ Nil^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 $ \lambda = -3/2 $ and explicitly showed that none of them admitted a gradient formulation. In contrast, by examining the Yamabe flow counterpart, we demonstrated that $ Nil^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.
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Beldjilali Gherici. Ricci solitons and harmonic vector fields on $ Nil^3 \times \mathbb{R} $[J]. Electronic Research Archive, 2026, 34(8): 5367-5379. doi: 10.3934/era.2026239
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