A solution of a problem of Sophus Lie: Normal forms of 2-dim metrics admitting two projective vector fields

TL;DR

This paper classifies all 2-dimensional metrics that admit a large group of symmetries preserving geodesics, providing a complete list of their normal forms and confirming their uniqueness up to isometry.

Contribution

It offers a comprehensive classification of 2D metrics with specific symmetry properties, solving a longstanding problem posed by Sophus Lie.

Findings

Complete list of normal forms for 2D metrics with projective symmetries

Proof that these normal forms are mutually non-isometric

Resolution of a problem posed by Sophus Lie

Abstract

We give a complete list of normal forms for the 2-dimensional metrics that admit a transitive Lie pseudogroup of geodesic-preserving transformations and we show that these normal forms are mutually non-isometric. This solves a problem posed by Sophus Lie.