Generalized constant ratio hypersurfaces in Euclidean spaces

Abstract

In this paper, we study generalized constant ratio (GCR) hypersurfaces in Euclidean spaces. We mainly focus on the hypersurfaces in $\mathbb E^4$. First, we deal with $\delta(2)$-ideal GCR hypersurfaces. Then, we study on hypersurfaces with constant (first) mean curvature. Finally, we obtain the complete classification of GCR hypersurfaces with vanishing Gauss-Kronecker curvature. We also give some explicit examples. Keywords: Generalized constant ratio submanifolds, $\delta(r)$-invariant hypersurfaces, constant mean curvature, Gauss-Kronecker curvature