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

TL;DR

This paper investigates L1-2-type surfaces in 3D De Sitter and anti De Sitter spaces, establishing equivalences among constant curvature conditions and classifying such surfaces into specific geometric types.

Contribution

It proves the equivalence of constant principal, mean, and second mean curvatures for L1-2-type surfaces and classifies these surfaces into standard products, B-scrolls, or non-constant curvature cases.

Findings

Equivalence of constant curvature conditions for L1-2-type surfaces.

Classification of L1-2-type surfaces into specific geometric types.

Identification of non-constant curvature cases.

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.