Minimal flat Lorentzian surfaces in Lorentzian complex space forms

TL;DR

This paper classifies minimal flat Lorentzian surfaces in various Lorentzian complex space forms, showing Ricci equations follow from Gauss and Codazzi equations, and provides explicit classifications in specific spaces.

Contribution

It demonstrates that Ricci equations are consequences of Gauss and Codazzi equations for these surfaces and offers classifications in ${f C}^2_1$, $CP^2_1$, and $CH^2_1$.

Findings

Ricci equations follow from Gauss and Codazzi equations.

Classification of minimal flat Lorentzian surfaces in ${f C}^2_1$.

Classification of minimal flat slant surfaces in $CP^2_1$ and $CH^2_1$.

Abstract

In this article we study minimal flat Lorentzian surfaces in Lorentzian complex space forms. First we prove that, for minimal flat Lorentzian surfaces in a Lorentzian complex form, the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Then we classify minimal flat Lorentzian surfaces in the Lorentzian complex plane ${\bf C}^2_1$. Finally, we classify minimal flat slant surfaces in Lorentzian complex projective plane $CP^2_1$ and in Lorentzian complex hyperbolic plane $CH^2_1$.