The k-Yamabe solitons and the quotient Yamabe solitons

In this paper, we consider the solitons to the fully nonlinear type Yamabe flows. We prove that any k-Yamabe soliton with positive k-curvature must have constant σk-curvature on closed locally conformally flat manifolds. We also prove that any quotient Yamabe soliton with positive k and l curvatures must have constantσkσl-curvature on closed locally conformally flat manifolds. Similar results are also proved on compact locally conformally flat manifolds with totally geodesic boundary.