Ruled Ricci surfaces and curves of constant torsion

TL;DR

This paper characterizes ruled Ricci surfaces in Euclidean space, showing they can be constructed from constant torsion curves, and identifies the helicoid as unique among such surfaces with specific geometric properties.

Contribution

It provides a new construction method for ruled Ricci surfaces using constant torsion curves and characterizes the helicoid uniquely by its parametrization and constant mean curvature.

Findings

Ruled Ricci surfaces are constructed from constant torsion curves.

The helicoid is the only such surface with a plane line of striction.

The helicoid is uniquely characterized by constant mean curvature.

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.