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Ruled Ricci surfaces and curves of constant torsion

Alcides de Carvalho, Iury Domingos, Roney Santos

TL;DR

This work characterizes non-developable ruled Ricci surfaces in $\mathbb{R}^3$ as being generated by curves of constant torsion and their binormal, establishing a tight link between intrinsic Ricci curvature and a simple geometric generator. The authors prove that the Ricci condition forces the metric to have the form $g=(c^2+u^2)dt^2+du^2$ with constant $c$, and they show a dichotomy: either the line of striction lies in a plane (producing a helicoid) or the surface arises from a curve of constant torsion with its binormal as ruling. They provide a constructive approach via canonical parametrizations $X(t,u)=\alpha(t)+uB(t)$, where $\alpha$ has constant torsion $\tau_0$ and $B$ is the binormal, together with explicit examples including parallel circles and anti-Salkowski-type curves. The results yield a complete intrinsic-to-geometric classification of this surface class and identify the helicoid as the unique constant-mean-curvature instance among them.

Abstract

We show that all non-developable ruled surfaces endowed with Ricci metrics in the three-dimensional Euclidean space may be constructed using curves of constant torsion and its binormal. This allows us to give characterizations of the helicoid as the only surface of this kind that admits a parametrization with plane line of striction, and as the only with constant mean curvature.

Ruled Ricci surfaces and curves of constant torsion

TL;DR

This work characterizes non-developable ruled Ricci surfaces in as being generated by curves of constant torsion and their binormal, establishing a tight link between intrinsic Ricci curvature and a simple geometric generator. The authors prove that the Ricci condition forces the metric to have the form with constant , and they show a dichotomy: either the line of striction lies in a plane (producing a helicoid) or the surface arises from a curve of constant torsion with its binormal as ruling. They provide a constructive approach via canonical parametrizations , where has constant torsion and is the binormal, together with explicit examples including parallel circles and anti-Salkowski-type curves. The results yield a complete intrinsic-to-geometric classification of this surface class and identify the helicoid as the unique constant-mean-curvature instance among them.

Abstract

We show that all non-developable ruled surfaces endowed with Ricci metrics in the three-dimensional Euclidean space may be constructed using curves of constant torsion and its binormal. This allows us to give characterizations of the helicoid as the only surface of this kind that admits a parametrization with plane line of striction, and as the only with constant mean curvature.
Paper Structure (6 sections, 6 theorems, 54 equations, 3 figures)

This paper contains 6 sections, 6 theorems, 54 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ruled Ricci surfaces
  4. Metric classification
  5. Ricci condition for ruled surfaces
  6. Constructing ruled Ricci surfaces

Key Result

Lemma 1

Let $g$ be a non-flat Riemannian metric given as where $f, \lambda: I \to \mathbb{R}$ are functions with $f(t)^2 \geq \lambda(t)^2$. Then, $g$ is a Ricci metric if and only if $\lambda(t)^2 = f(t)^2 = c^2$ for some $c>0$ and each $t \in I.$ In particular, we have $g = (c^2+u^2)dt^2 + du^2.$

Figures (3)

  • Figure 3.1: Non-developable ruled Ricci surfaces associated with circles in $\mathbb{S}^2$ of radius $\frac{1}{2}$, $\frac{3}{4}$ and $1$, respectively.
  • Figure 3.2: Spherical curves and non-developable ruled Ricci surfaces for anti-Salkowski curves with $\ell = 0.1$, $\ell = \frac{1}{3}$ and $\ell = 0.57$, respectively.
  • Figure 3.3: Borderline spherical curve and its associated non-developable ruled Ricci surface.

Theorems & Definitions (19)

  • Lemma 1
  • proof
  • Remark 1
  • Corollary 1
  • proof
  • Remark 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • ...and 9 more