Classification of minimal Lorentzian surfaces in ${\mathbb{S}}_2^4(1)$ with constant Gaussian and normal curvatures

Abstract

In this paper we consider Lorentzian surfaces in the 4-dimensional pseudo-Riemannian sphere ${\mathbb{S}}_2^4(1)$ with index 2 of curvature one. We obtain the complete classification of minimal Lorentzian surfaces ${\mathbb{S}}_2^4(1)$ 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.

arXiv commentKeywords. Gaussian curvature, minimal submanifolds, Lorentzian surfaces, normal curvature