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Second order geometry of spacelike surfaces in de Sitter 5-space. (English) Zbl 1327.53015

Summary: 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 space-like surface. In this work we investigate properties of the second order geometry of space-like surfaces in de Sitter space \(S_1^5\) by using the action of \(\mathrm{GL}(2,\mathbb R)\times \mathrm{SO}(1,2)\) 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 \(LMN\)-ellipse. This ellipse gives information on the light-like binormal directions and consequently about their associated asymptotic directions.

MSC:

53A35 Non-Euclidean differential geometry
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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Full Text: DOI Euclid