L1-2-type surfaces in 3-dimensional De Sitter and anti De Sitter spaces

Abstract

Let $M$ be an orientable surface immersed in the De Sitter space $S_1^3$ in $R^4_1$ or anti de Sitter space $H_1^3$ in $R^4_2$. In the case that $M$ is of $L_1$-2-type we prove that the following conditions are equivalent to each other: $M$ has a constant principal curvature; $M$ has constant mean curvature; $M$ has constant second mean curvature. As a consequence, we also show that an $L_1$-2-type surface is either an open portion of a standard pseudo-Riemannian product, or a $B$-scroll over a null curve, or else its mean curvature, its Gaussian curvature and its principal curvatures are all non-constant.