Geodesic Chord Property and Hypersurfaces of Space Forms

Dong-Soo Kim,Dae Won Yoon,Young Ho Kim

doi:10.3390/sym11081052

Abstract

In the Euclidean space E n , hyperplanes, hyperspheres and hypercylinders are the only isoparametric hypersurfaces. These hypersurfaces are also the only ones with chord property, that is, the chord connecting two points on them meets the hypersurfaces at the same angle at the two points. In this paper, we investigate hypersurfaces in nonflat space forms with the so-called geodesic chord property and classify such hypersurfaces completely.

Highlights

A circle in the plane E2 is characterized as a closed curve with the chord property that the chord connecting any two points on it meets the curve at the same angle at the two end points ([1], pp. 160–162).For space curves, B.-Y
Chen et al showed that a W-curve is characterized as a curve in the Euclidean space En with the property that the chord joining any two points on the curve meets the curve at the same angle at the two points, that is, as a curve in the Euclidean space En with the chord property ([2])
The hypersurface M is said to satisfy geodesic chord property if the geodesic chord in the ambient space Mn (c) joining any two points on M meets the hypersurface at the same angle at the two points

Summary

Introduction

A circle in the plane E2 is characterized as a closed curve with the chord property that the chord connecting any two points on it meets the curve at the same angle at the two end points ([1], pp. 160–162). For hypersurfaces in the Euclidean n-space En which satisfies the chord property, D.-S. Let us consider a hypersurface M in the Euclidean space En. the following are equivalent: 2. The hypersurface M is said to satisfy geodesic chord property if the geodesic chord in the ambient space Mn (c) joining any two points on M meets the hypersurface at the same angle at the two points. Symmetry 2019, 11, 1052 we study and classify completely the hypersurfaces in the n-dimensional hyperbolic space H n with the geodesic chord property, which is imbedded in the Minkowski space E1n+1. Throughout this article, we assume that all objects are smooth and connected unless otherwise mentioned

Spherical Hypersurfaces

Hypersurfaces in the Hyperbolic Space

Conclusions