The equi-affine curvatures of curves in 3-dimensional pseudo-Riemannian manifolds

TL;DR

This paper explores the relationship between Cartan frames, equi-affine curvatures, and Frenet curvatures of curves in 3D pseudo-Riemannian manifolds, including null curves in Lorentzian spaces, revealing key geometric properties.

Contribution

It establishes the connection between Frenet and equi-affine curvatures for non-null curves and analyzes equi-affine curvatures of null curves in Lorentzian manifolds.

Findings

Constancy of Frenet curvatures implies constancy of equi-affine curvatures.

The converse does not necessarily hold.

Equi-affine curvatures of null curves relate to their pseudo-torsion.

Abstract

In this paper, the Cartan frames and the equi-affine curvatures are described with the help of the Frenet frames and the Frenet curvatures of a non-null and non-degenerate curve in a 3-dimensional pseudo-Riemannian manifold. The constancy of the Frenet curvatures of such a curve always implies the constancy of the equi-affine curvatures. We show that the converse statement does not hold in general. Finally, we study the equi-affine curvatures of null curves in 3-dimensional Lorentzian manifolds, and prove that they are related to their pseudo-torsion.