Classification Theorems for Space-Like Surfaces in 4-Dimensional Indefinite Space Forms with Index 2

Abstract

Surfaces in 4D Riemannian space forms have been investigated extensively. In contrast, only few results are known for surfaces in 4D neutral indefinite space forms $R^4_2(c)$. Thus, in this paper we study space-like surfaces in $R^4_2(c)$ satisfying certain simple geometric properties. In particular, we classify space-like surfaces in $\mathbb E^4_2$ with constant mean and Gauss curvatures and null normal curvature. We also classify Wintgen ideal surfaces in $R^4_2(c)$ whose Gauss and normal curvatures satisfy $K^D = 2K$.