Concircular hypersurfaces and concircular helices in space forms

TL;DR

This paper characterizes concircular hypersurfaces and helices in space forms, providing a full description of these geometric structures and their relationships through differential equations and geodesic properties.

Contribution

It offers a complete description of concircular hypersurfaces as ruled hypersurfaces and characterizes concircular helices via differential equations in space forms.

Findings

Concircular hypersurfaces are fully described as a special family of ruled hypersurfaces.

Concircular helices are characterized by a specific differential equation involving curvature and torsion.

Concircular helices are exactly the geodesics of concircular surfaces.

Abstract

In this paper, we find a full description of concircular hypersurfaces in space forms as a special family of ruled hypersurfaces. We also characterize concircular helices in 3-dimensional space forms by means of a differential equation involving the concircular factor and their curvature and torsion, and we show that the concircular helices are precisely the geodesics of the concircular surfaces.