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Concircular hypersurfaces and concircular helices in space forms

Pascual Lucas, José Antonio Ortega-Yagües

TL;DR

This work extends the LO21 framework to space forms by proving that concircular vector fields on M^n(C) arise as tangential parts of constant ambient vectors, and by fully describing nontrivial concircular hypersurfaces as locally ruled surfaces over a totally umbilical base, with conical surfaces corresponding to vanishing concircular factor. In M^3(C), concircular helices are characterized by a differential system linking the concircular factor with the curve's curvature and torsion, with a dichotomy based on the rectifying slope; moreover, such helices are shown to be geodesics of concircular surfaces. The results unify and extend classical concepts of generalized helices, rectifying curves, and ruled surface geodesics to spaces of constant curvature, providing explicit constructions and a clear geometric correspondence between curves and the ambient concircular geometry. Overall, the paper delivers a comprehensive structural description of concircular submanifolds in space forms and clarifies the relationship between concircular helices and surface geodesics in this setting.

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.

Concircular hypersurfaces and concircular helices in space forms

TL;DR

This work extends the LO21 framework to space forms by proving that concircular vector fields on M^n(C) arise as tangential parts of constant ambient vectors, and by fully describing nontrivial concircular hypersurfaces as locally ruled surfaces over a totally umbilical base, with conical surfaces corresponding to vanishing concircular factor. In M^3(C), concircular helices are characterized by a differential system linking the concircular factor with the curve's curvature and torsion, with a dichotomy based on the rectifying slope; moreover, such helices are shown to be geodesics of concircular surfaces. The results unify and extend classical concepts of generalized helices, rectifying curves, and ruled surface geodesics to spaces of constant curvature, providing explicit constructions and a clear geometric correspondence between curves and the ambient concircular geometry. Overall, the paper delivers a comprehensive structural description of concircular submanifolds in space forms and clarifies the relationship between concircular helices and surface geodesics in this setting.

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.
Paper Structure (5 sections, 12 theorems, 43 equations)

This paper contains 5 sections, 12 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Concircular vector fields in space forms
  3. Concircular hypersurfaces
  4. Concircular helices in $\mathbb{M}^3(C)$
  5. Geodesics of concircular surfaces

Key Result

Theorem 1

A vector field $V\in\frak X(\mathbb{M}^n(C))$ is concircular if and only if $V$ is the tangential part of a constant vector field $p_0$ in $\mathbb{R}^{n+1}_\nu$. Moreover, if $\mu$ is the concircular factor of $V$, then $V= p_0+\mu\phi$, where $\mu=-C\left<p_0,\phi\right>$.

Theorems & Definitions (17)

  • Definition 1
  • Theorem 1
  • Corollary 2
  • Proposition 3
  • Example 1
  • Example 2
  • Proposition 4
  • Proposition 5
  • Theorem 6
  • Proposition 7
  • ...and 7 more