On the Quasi-Minimal Surfaces in the 4-Dimensional de Sitter Space With 1-Type Gauss Map

Nurettin Cenk Turgay

doi:10.5644/SJM.11.1.09

Authors

Nurettin Cenk Turgay Istanbul Technical University, Faculty of Science and Letters Department of Mathematics, 34469 Maslak, Istanbul, Turkey

DOI:

https://doi.org/10.5644/SJM.11.1.09

Keywords:

Quasi-minimal surface, finite type Gauss map, de Sitter spacetime

Abstract

In this paper, we study the Gauss map of the surfaces in the de Sitter space-time $\mathbb S^4_1(1)$. First, we prove that a space-like surface lying in the de Sitter space-time has pointwise 1-type Gauss map if and only if it has parallel mean curvature vector. Then, we obtain the complete classification of the quasi-minimal surfaces with 1-type Gauss map.

2010 Mathematics Subject Classification. 53B25, 53C50.

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References

B. Y. Chen, A report on submanifolds of finite type, Soochow J. Math., 22 (1996), 117–337.

B. Y. Chen, J. M. Morvan and T. Nore, Energy, tension and finite type maps, Kodai Math. J., 9 (1986), 406–418.

B. Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, World Scientific, London, (1984).

B. Y. Chen, Finite type pseudo-riemannian submanifolds, Tamkang J. of Math., 17 (1986), 137–151.

B. Y. Chen and P. Piccini, Submanifolds with finite type Gauss map, Bull. Austral. Math. Soc., 35 (1987), 161–186.

B. Y. Chen, Pseudo-Riemannian geometry, δ-invariants and applications, World Scientific, Hackensack, NJ , (2011).

M. P. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New Jersey, (1983).

N. C. Turgay, On the marginally trapped surfaces in 4-dimensional space-times with finite type Gauss map, (accepted).

B. Y. Chen and J. van der Veken, Classification of marginally trapped surfaces with parallel mean curvature vector in lorentzian space forms, Houston J. Math., 36 (2010), 421–449.

S. M. Choi, U. H. Ki and Y. J. Suh, Classification of rotation surfaces in pseudoeuclidean space, J. Korean Math. Soc., 35 (1998), 315–330.