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Null hypersurfaces in generalized Robertson-Walker spacetimes. (English) Zbl 1341.53041

Summary: We study the geometry of null hypersurfaces \(M\) in generalized Robertson-Walker spacetimes. First we characterize such null hypersurfaces as graphs of generalized eikonal functions over the fiber and use this characterization to show that such hypersurfaces are parallel if and only if their fibers are also parallel. We further use this technique to construct several examples of null hypersurfaces in both de Sitter and anti de Sitter spaces. Then we characterize all totally umbilical null hypersurfaces \(M\) in a Lorentzian space form (viewed as a quadric in a semi-Euclidean ambient space) as intersections of the space form with a hyperplane. Finally we study the totally umbilical spacelike hypersurfaces of null hypersurfaces in space forms and characterize them as planar sections of \(M\).

MSC:

53B25 Local submanifolds
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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