A local characterization for constant curvature metrics in 2-dimensional Lorentz manifolds

TL;DR

This paper introduces Fermi-type coordinates in 2D Lorentz manifolds to locally characterize constant curvature metrics and demonstrates specific isometric immersions into Lorentzian ambient spaces.

Contribution

It provides a novel local characterization of constant curvature metrics in 2D Lorentz manifolds using Fermi-type coordinates, extending classical Riemannian results.

Findings

Characterization of constant Gaussian curvature metrics in 2D Lorentz manifolds.

Explicit isometric immersions into Lorentzian ambient spaces.

Development of Fermi-type coordinates for Lorentzian geometry.

Abstract

In this paper we define Fermi-type coordinates in a 2-dimensional Lorentz manifold, and use this coordinate system to provide a local characterization of constant Gaussian curvature metrics for such manifolds, following a classical result from Riemann. We then exhibit particular isometric immersions of such metrics in the pseudo-Riemannian ambients L^3 (i.e., usual Lorentz-Minkowski space) and R^3_2 (i.e., R^3 endowed with an index 2 flat metric).