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

Abstract

AbstractLet $$M_s^2$$ M s 2 be an orientable surface immersed in the De Sitter space $$\mathbb {S}_1^3\subset \mathbb {R}^4_1$$ S 1 3 ⊂ R 1 4 or anti de Sitter space $$\mathbb {H}_1^3\subset \mathbb {R}^4_2$$ H 1 3 ⊂ R 2 4 . In the case that $$M_s^2$$ M s 2 is of $$L_1$$ L 1 -2-type we prove that the following conditions are equivalent to each other: $$M_s^2$$ M s 2 has a constant principal curvature; $$M_s^2$$ M s 2 has constant mean curvature; $$M_s^2$$ M s 2 has constant second mean curvature. As a consequence, we also show that an $$L_1$$ 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.