Ricci Solitons on the Lie Group $So{l}_{m,n}^3$ and Applications

In this paper, we study Ricci solitons on the three-dimensional solvable Lie group ${\mathrm{S}\mathrm{o}\mathrm{l}}_{m,n}^3$ equipped with a left-invariant Riemannian metric, viewed as a generalization of the classical ${\mathrm{S}\mathrm{o}\mathrm{l}}^3$ geometry. We investigate harmonic maps, harmonic sections, and geodesic curves, including the geodesic properties of the integral curves of the Ricci soliton vector field. We also characterize harmonic linear maps from ${\mathrm{S}\mathrm{o}\mathrm{l}}_{m,n}^3$ into Euclidean spaces.