3-dimensional Levi-Civita metrics with projective vector fields

TL;DR

This paper extends the classification of 2D metrics with projective symmetries to 3D Levi-Civita metrics of any signature, identifying conditions for the existence of such vector fields.

Contribution

It provides a comprehensive solution to the 3D Levi-Civita metrics admitting non-trivial projective vector fields, generalizing previous 2D results.

Findings

Classification of 3D Levi-Civita metrics with projective vector fields

Conditions for existence of projective symmetries in arbitrary signature

Extension of Lie's problem from 2D to 3D metrics

Abstract

Projective vector fields are the infinitesimal transformations whose local flow preserves geodesics up to reparametrisation. In 1882 Sophus Lie posed the problem of describing 2-dimensional metrics admitting a non-trivial projective vector field, which was solved in recent years. In the present paper, we solve the analog of Lie's problem in dimension 3, for Riemannian metrics and, more generally, for Levi-Civita metrics of arbitrary signature.