Metallic Ricci soliton on the Sol 3 manifold
In this paper, we study the manifold Sol 3 endowed with its standard left-invariant
Riemannian metric together with a polynomial structure. First, we recall the Lie group structure
of Sol 3 and compute the main geometric objects associated with the standard metric, including
an orthonormal frame, Lie brackets, the Levi-Civita connection, and the Ricci tensor. Then we
introduce a polynomial structure satisfying a quadratic relation and investigate its interaction with
the Ricci tensor. In particular, we prove that Sol 3 is not an Einstein manifold and that the Ricci
operator commutes with the polynomial structure. Finally, we study a metallic Ricci soliton type
equation associated with the polynomial structure and obtain an explicit family of solutions for the
corresponding vector field. These results provide new examples of curvature structures compatible
with polynomial tensor fields on solvable Lie groups.
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