Remarks on hypersurfaces with constant higher order mean curvature in Euclidean space

Abstract

In this paper we consider complete oriented hypersurfaces of Euclidean space with constant higher order mean curvature and having two principal curvatures, one of them simple. As an application of the so called principal curvature theorem, a purely geometric result on the principal curvatures of the hypersurface given by Smyth and Xavier (Invent Math 90:443–450, 1987), we characterize those hypersurfaces for which the Gauss–Kronecker curvature does not change sign, extending to the general n-dimensional case a previous result for surfaces due to Klotz and Osserman.