Notes on a conformal characterization of $2$-dimensional Lorentzian manifolds with constant Ricci scalar curvature

TL;DR

This paper characterizes 2D Lorentzian manifolds with constant Ricci scalar curvature by expressing the curvature in terms of the conformal factor and analyzing the resulting differential equations, providing notable examples.

Contribution

It offers a conformal characterization of 2D Lorentzian manifolds with constant Ricci scalar, including explicit solutions and examples, enhancing understanding of their geometric structure.

Findings

Reformulation of Ricci scalar in terms of conformal factor

Solutions to the differential equations for specific cases

Examples illustrating the conformal characterization

Abstract

We present a characterization of $2$-dimensional Lorentzian manifolds with constant Ricci scalar curvature. It is well known that every $2$-dimensional Lorentzian manifolds is conformally flat, so we rewrite the Ricci scalar curvature in terms of the conformal factor and we study the solutions of the corresponding differential equations. Several remarkable examples are provided.