On the Gauss map of minimal Lorentzian surfaces in 4-dimensional semi-Riemannian space forms with index 2

In this paper, we study minimal Lorentzian surfaces with finite type Gauss map in 4-dimensional semi-Riemannian space forms with index of 2. First, we give the complete classification of Lorentzian surfaces in the semi-Euclidean space E⁴₂ with pointwise 1-type Gauss map. Then, we study all Lorentzian minimal surfaces in S⁴₂(1) regarding their Gauss map. In particular, we proved that a Lorentzian minimal surface in S⁴₂(1) has 2-type Gauss map if and only if it has constant Gaussian curvature and non-zero constant normal curvature.