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A property that characterizes the Enneper surface and helix surfaces

Pascual Lucas, José Antonio Ortega-Yagües

TL;DR

This work classifies surfaces in $\mathbb{R}^3$ whose isogonal lines are simultaneously generalized helices and pseudo-geodesic lines. It develops by analyzing the isogonal flow, deriving differential equations for pseudo-geodesic lines, and linking these concepts to curvature conditions (e.g., $\kappa_n$ and $\tau_g$) along isogonal curves, which in turn leads to CRPC- and CSkC-surface characterizations. The main result shows that a nonplanar connected surface must be a helix surface or an open piece of the Enneper surface to have all isogonal lines with these properties. The paper also provides an explicit Enneper-surface construction demonstrating the connection to generalized helices, underscoring the geometric rigidity and completeness of the classification.

Abstract

The main goal of this paper is to show that helix surfaces and the Enneper surface are the only surfaces in the 3-dimensional Euclidean space $R^3$ whose isogonal lines are generalized helices and pseudo-geodesic lines.

A property that characterizes the Enneper surface and helix surfaces

TL;DR

This work classifies surfaces in whose isogonal lines are simultaneously generalized helices and pseudo-geodesic lines. It develops by analyzing the isogonal flow, deriving differential equations for pseudo-geodesic lines, and linking these concepts to curvature conditions (e.g., and ) along isogonal curves, which in turn leads to CRPC- and CSkC-surface characterizations. The main result shows that a nonplanar connected surface must be a helix surface or an open piece of the Enneper surface to have all isogonal lines with these properties. The paper also provides an explicit Enneper-surface construction demonstrating the connection to generalized helices, underscoring the geometric rigidity and completeness of the classification.

Abstract

The main goal of this paper is to show that helix surfaces and the Enneper surface are the only surfaces in the 3-dimensional Euclidean space whose isogonal lines are generalized helices and pseudo-geodesic lines.
Paper Structure (6 sections, 15 theorems, 37 equations, 1 figure)

This paper contains 6 sections, 15 theorems, 37 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Some results on isogonal lines
  3. Pseudo-geodesic lines
  4. Differential equations of pseudo-geodesic lines
  5. The main result
  6. Appendix: A special family of geodesics in the Enneper surface

Key Result

Proposition 1

Given a point $p\in M$ and a vector $v\in T_pM$, $v\neq0$, there exist an $\varepsilon>0$ and a unique isogonal line $\gamma:(-\varepsilon,\varepsilon)\to M$ such that $\gamma(0)=p$, $\gamma'(0)=v$ and $|\gamma'(t)|=|v|$ for all $t\in(-\varepsilon,\varepsilon)$.

Figures (1)

  • Figure 1: The Enneper surface and two generalized helices passing through the origin.

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Proposition 7
  • ...and 12 more