Rigidity results on gradient Schouten solitons

In this paper, we consider gradient Schouten solitons, conformal to an n-dimensional pseudo-Euclidean space and invariant under the action of an $(n-1)$-dimensional translation group. We provide all such solutions and further, we prove that, in the Riemannian case, there does not exist a complete gradient Schouten Soliton that is conformal to Euclidean space and invariant to translation. Furthermore, we consider ρ-Einstein solitons of type $M=(B^n,g^{⁎})\times (F^m,g_F)$, where $(B^n,g^{⁎})$ is conformal to a pseudo-Euclidean space and invariant under the action of the pseudo-orthogonal group, and $(F^m,g_F)$ is an Einstein manifold. We provide all the solutions for the gradient Schouten soliton case, in the Riemannian case, we prove that if $M=(B^n,g^{⁎})\times (F^m,g_F)$ is a complete gradient Schouten soliton, then $(B^n,g^{⁎})$ is isometric to $S^{n-1}\times R$ and $F^m$ is a compact Einstein manifold.