Abstract
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.
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Notes
In this section all the manifolds are assumed to be orientable.
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Acknowledgements
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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Communicated by Young Jin Suh.
The Daniel de la Fuente and Alfonso Romero are partially supported by Spanish MINECO and ERDF project MTM2016-78807-C2-1-P. The Francisco J. Palomo by Spanish MINECO and ERDF project MTM2016-78807-C2-2-P.
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de la Fuente, D., Palomo, F.J. & Romero, A. On Non-degenerate Null Normal Sections of Codimension Two Spacelike Surfaces. Bull. Malays. Math. Sci. Soc. 42, 1451–1467 (2019). https://doi.org/10.1007/s40840-017-0557-x
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DOI: https://doi.org/10.1007/s40840-017-0557-x