Abstract : In this article, we apply the weak maximumprinciple in order to obtain a suitable characterization of the complete linear Weingarten hypersurfaces immersed in locally symmetric Riemannian manifold $N^{n+1}$. Under the assumption that the mean curvature attains its maximum and supposing an appropriated restriction on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or hypersurface is an isoparametric hypersurface with two distinct principal curvatures one of which is simple.
Keywords : locally symmetric, linear Weingarten hypersurfaces, totally umbilical
