On Non-degenerate Null Normal Sections of Codimension Two Spacelike Surfaces
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In this paper, we develop a formula for spacelike surfaces in a four-dimensional Lorentzian space form which involves its mean curvature vector field, the Gauss curvature of the induced metric and the Gauss curvature of the second fundamental form associated to a non-degenerate null normal section. By means of this formula, we establish several sufficient conditions for a compact spacelike surface in a four-dimensional Lorentzian space form which has a null umbilical normal direction. As another application, we give a new proof of Liebmann rigidity theorems in Euclidean, hemispherical, hyperbolic spaces and in the De Sitter spacetime.
KeywordsSpacelike surface Gauss curvature Liebmann theorem Lorentzian space forms
Mathematics Subject Classification53C24 53C50 53C42
We would like to thank the Referees of the paper, specially to Referee \(\#2\) for his/her careful reading and suggestions to improve the previous version of the manuscript.
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