Geodesibility of algebrizable three-dimensional vector fields

TL;DR

This paper extends the concept of geodesibility from planar to three-dimensional algebrizable vector fields, providing rectifications, Riemannian metrics, and first integrals to describe their integral curves.

Contribution

It introduces a method to analyze three-dimensional algebrizable vector fields for geodesibility, including explicit rectifications, metrics, and integrals, advancing the understanding of their geometric structure.

Findings

Three-dimensional algebrizable vector fields can be made geodesible with appropriate metrics.

Explicit rectifications and Riemannian metrics are constructed for these vector fields.

Two first integrals are identified for each vector field, describing their integral curves.

Abstract

Recently, the geodesibility of planar vector fields, which are algebrizable (differentiable in the sense of Lorch for some associative and commutative unital algebra), has been established. In this paper, we consider algebrizable three-dimensional vector fields, for which we give rectifications and Riemannian metrics under which they are geodesible. Furthermore, for each of these vector fields $F$ we give two first integrals $h_1$ and $h_2$ such that the integral curves of $F$ are locally defined by the intersections of the level surfaces of $h_1$ and $h_2$.