{L_1}-2-Type Surfaces in 3-Dimensional De Sitter and Anti De Sitter Spaces

Abstract

Let M_s^2 be an orientable surface immersed in the De Sitter space mathbb {S}_1^3subset mathbb {R}^4_1 or anti de Sitter space mathbb {H}_1^3subset mathbb {R}^4_2. In the case that M_s^2 is of L_1-2-type we prove that the following conditions are equivalent to each other: M_s^2 has a constant principal curvature; M_s^2 has constant mean curvature; M_s^2 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.