Complete hypersurfaces with two distinct principal curvatures in a locally symmetric Riemannian manifold

We deal with complete hypersurfaces with two distinct principal curvatures in a locally symmetric Riemannian manifold, which is supposed to obey some appropriated curvature constraints. Initially, considering the case that such a hypersurface has constant mean curvature, we apply a Simons type formula jointly with the well known generalized maximum principle of Omori–Yau to show that it must be isometric to an isoparametric hypersurface of the ambient space. Afterwards, we use a Cheng–Yau modified operator in order to obtain a sort of extension of this previously mentioned result for the context of linear Weingarten hypersurfaces, that is, hypersurfaces whose mean and scalar curvatures are linearly related.