# Normal forms of two-dimensional metrics admitting exactly one essential projective vector field

## TL;DR

This paper classifies all two-dimensional metrics with exactly one essential projective vector field, providing a complete list of normal forms and revising previous results to solve a problem posed by Sophus Lie in 1882.

## Contribution

It offers a comprehensive classification of such metrics, extending prior work and correcting earlier classifications in the literature.

## Key findings

- Complete list of non-diffeomorphic normal forms
- Revised classification results from previous studies
- Solved a historical problem posed by Sophus Lie

## Abstract

We give a complete list of mutually non-diffeomorphic normal forms for the two-dimensional metrics that admit one essential (i.e., non-homothetic) projective vector field. This revises a result from the literature and extends the results of two papers, by R.L. Bryant & G. Manno & V.S. Matveev (2008) and V.S. Matveev (2012) respectively, solving a problem posed by Sophus Lie in 1882.