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

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).