Characterizations of complete hypersurfaces in locally symmetric Riemannian manifolds
Volume 153 / 2018
Colloquium Mathematicum 153 (2018), 149-162 MSC: Primary 53C42; Secondary 53A10, 53C20, 53C50. DOI: 10.4064/cm7047-5-2017 Published online: 16 April 2018
We deal with complete linear Weingarten hypersurfaces immersed in a locally symmetric Riemannian manifold whose sectional curvature obeys certain standard constraints. Under an appropriate restriction on the norm of the traceless part of the second fundamental form, we show that such a hypersurface must be either totally umbilical or an isoparametric hypersurface with two distinct principal curvatures, one of which is simple. Our approach is based on the use of a suitable Simons type formula combined with applications of some generalized maximum principles.