Rigidity and Conformal Characterizations of Noncompact Gradient Schouten Solitons

This paper studies the conformal geometry of complete gradient Schouten solitons (GSSs) admitting closed conformal vector fields (CVFs). We establish rigidity and characterization results for nonparallel, homothetic closed CVFs under the assumption that the gradient of the scalar curvature is parallel to the CVF. It is shown that such manifolds are isometric to Euclidean space. Moreover, complete noncompact GSSs with constant scalar curvature are locally conformally flat in dimension four and have harmonic Weyl curvature in higher dimensions. Finally, we prove that these manifolds are totally umbilical if and only if their scalar curvature is constant, and they form warped products with space forms.