We first classify space-like surfaces in the Minkowski space , de Sitter space , and hyperbolic space ℍ3 with harmonic Gauss maps. Then we characterize and present a classification of the space-like surfaces with pointwise 1-type Gauss maps of the first kind. We also give some explicit examples.
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K. Arslan, B. K. Bayram, B. Bulca, Y. H. Kim, C. Murathan, and G. Öztürk, Rotational embeddings in E4 with pointwise 1-type Gauss map,” Turkish J. Math., 35, 493–499 (2011).
C. Baikoussis, “Ruled submanifolds with finite-type Gauss map,” J. Geom., 49, 42–45 (1994).
C. Baikoussis and D. E. Blair, “On the Gauss map of ruled surfaces,” Glasgow Math. J., 34, 355–359 (1992).
C. Baikoussis, B. Y. Chen, and L. Verstraelen, Ruled surfaces, tubes with finite-type Gauss map,” Tokyo J. Math., 16, 341–348 (1993).
C. Baikoussis and L. Verstraelen, “The Chen type of the spiral surfaces,” Results Math., 28, 214–223 (1995).
B. Y. Chen, Total Mean Curvature, Submanifolds of Finite Type, World Scientific, Singapore, etc. (1984).
B. Y. Chen, “Some classification theorems for submanifolds in Minkowski space-time,” Arch. Math. (Basel), 62, 177–182 (1994).
B. Y. Chen, “Complete classification of spatial surfaces with parallel mean curvature vector in arbitrary non-flat pseudo-Riemannian space forms,” Centr. Europ. J. Math., 7, 400–428 (2009).
B. Y. Chen, “Classification of spatial surfaces with parallel mean curvature vector in pseudo-Euclidean spaces of arbitrary dimension,” J. Math. Phys., 50, 14 p. (2009)
B. Y. Chen, M. Choi, and Y. H. Kim, “Surfaces of revolution with pointwise 1-type Gauss map,” J. Korean Math. Soc., 42, 447–455 (2005).
B. Y. Chen and S. Ishikawa, “Biharmonic surfaces in pseudo-Euclidean spaces,” Kyushu J. Math., 45, 323–347 (1991).
B. Y. Chen and P. Piccinni, “Submanifolds with finite-type Gauss map,” Bull. Aust. Math. Soc., 35, 161–186 (1987).
B. Y. Chen and Van Der J. Veken, “Complete classification of parallel surfaces in 4-dimensional Lorentzian space forms,” Tohoku Math. J. (2), 61, 1–40 (2009).
B. Y. Chen and Van Der J. Veken, “Classification of marginally trapped surfaces with parallel mean curvature vector in Lorentzian space forms,” Houston J. Math., 36, 421–449 (2010).
M. Choi and Y. H. Kim, “Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map,” Bull. Korean Math. Soc., 38, 753–761 (2001).
U. Dursun, “Hypersurfaces with pointwise 1-type Gauss map,” Taiwan. J. Math., 11, 1407–1416 (2007).
U. Dursun, “Hypersurfaces with pointwise 1-type Gauss map in Lorentz–Minkowski space,” Proc. Est. Acad. Sci., 58, 146–161 (2009).
U. Dursun, “Flat surfaces in the Euclidean space E 3 with pointwise 1-type Gauss map,” Bull. Malays. Math. Sci. Soc., 33, 469–478 (2010).
U. Dursun and G. G. Arsan, “Surfaces in the Euclidean space 𝔼4 with pointwise 1-type Gauss map,” Hacet. J. Math. Stat., 40, 617–625 (2011).
U. H. Ki, D. S. Kim, Y. H. Kim, and Y. M. Roh, “Surfaces of revolution with pointwise 1-type Gauss map in Minkowski 3-space,” Taiwan. J. Math., 13, 317–338 (2009).
Y. H. Kim and Y. W. Yoon, “Ruled surfaces with pointwise 1-type Gauss map,” J. Geom. Phys., 34, 191–205 (2000).
Y. H. Kim and Y. W. Yoon, “Classification of rotation surfaces in pseudo-Euclidean space,” J. Korean Math. Soc., 41, 379–396 (2004).
Y. H. Kim and Y. W. Yoon, “On the Gauss map of ruled surfaces in Minkowski space,” Rocky Mountain J. Math., 35, 1555–1581 (2005).
Y. W. Yoon, “Rotation surfaces with finite-type Gauss map in E 4,” Indian J. Pure. Appl. Math., 32, 1803–1808 (2001).
Y. W. Yoon, “On the Gauss map of translation surfaces in Minkowski 3-spaces,” Taiwan. J. Math., 6, 389–398 (2002).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 71, No. 1, pp. 59–72, January, 2019.
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Dursun, U., Turgay, N.C. Space-Like Surfaces in the Minkowski Space with Pointwise 1-Type Gauss Maps. Ukr Math J 71, 64–80 (2019). https://doi.org/10.1007/s11253-019-01625-8
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DOI: https://doi.org/10.1007/s11253-019-01625-8