Geometric flows by parallel hypersurfaces

In a recent work it was proved that any immersed hypersurface Mn of a space form evolves through the mean curvature flow (MCF) by parallel hypersurfaces if and only if Mn is an isoparametric hypersurface. The goal of this article is to extend the quoted work to immersed hypersurface Mn of Qcn×R and Qcn×S1 under a geometric condition of the tangential component of ∂ ∂t. More exactly, supposing that this tangential component is a principal direction of the second fundamental form, we will show that such a hypersurface evolves through the MCF by parallel hypersurfaces if and only if Mn is also an isoparametric hypersurface. Moreover, we will prove that any embedded convex hypersurface Mn of a sphere Sn+1 evolves through the inverse mean curvature flow (IMCF) by parallel hypersurfaces if and only if Mn is an umbilic hypersurface.