Second order geometry of spacelike surfaces in de Sitter 5-space

Abstract

The de Sitter space is known as a Lorentz space with positive constant curvature in the Minkowski space. A surface with a Riemannian metric is called a spacelike surface. In this work we investigate properties of the second order geometry of spacelike surfaces in de Sitter space $S_1^5$ by using the action of $GL(2,\mathbb R)\times SO(1,2)$ on the system of conics defined by the second fundamental form. The main results are the classification of the second fundamental mapping and the description of the possible configurations of the $\mathit{LMN}$-ellipse. This ellipse gives information on the lightlike binormal directions and consequently about their associated asymptotic directions.