Generalized constant ratio hypersurfaces in Euclidean spaces

TL;DR

This paper classifies generalized constant ratio hypersurfaces in Euclidean 4-space, focusing on $ ext{delta}(2)$-ideal, constant mean curvature, and vanishing Gauss-Kronecker curvature cases, providing explicit examples.

Contribution

It offers a complete classification of GCR hypersurfaces in $ ext{E}^4$ with specific curvature conditions, extending previous understanding of these geometric structures.

Findings

Classification of $ ext{delta}(2)$-ideal GCR hypersurfaces

Characterization of GCR hypersurfaces with constant mean curvature

Complete classification of GCR hypersurfaces with zero Gauss-Kronecker curvature

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