Classification of Minimal Lorentzian Surfaces in $\mathbb{S}^4_2(1)$ with Constant Gaussian and Normal Curvatures

Abstract

In this paper we consider Lorentzian surfaces in the $4$-dimensional pseudo-Riemannian sphere $\mathbb{S}^4_2(1)$ with index $2$ and curvature one. We obtain the complete classification of minimal Lorentzian surfaces $\mathbb{S}^4_2(1)$ whose Gaussian and normal curvatures are constants. We conclude that such surfaces have the Gaussian curvature $1/3$ and the absolute value of normal curvature $2/3$. We also give some explicit examples.