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.
Hypersurface Higher order mean curvature Gauss–Kronecker curvature Principal curvature theorem
Mathematics Subject Classification
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The authors would like to thank the referee for reading the manuscript in great detail and giving several valuable suggestions and useful comments which improved the paper. This work was finished while the two authors were visiting the Laboratoire de Mathématiques et Physique Théorique (LMPT) of the Université Francois Rabelais at Tours (France). They would like to thank this institution for its cordial hospitality during this visit.
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