# BOOST INVARIANT SURFACES WITH POINTWISE 1-TYPE GAUSS MAP IN MINKOWSKI 4-SPACE E41

• Published : 2014.11.30

#### Abstract

In this paper, we study spacelike rotational surfaces which are called boost invariant surfaces in Minkowski 4-space $\mathbb{E}^4_1$. We give necessary and sufficient condition for flat spacelike rotational surface to have pointwise 1-type Gauss map. Also, we obtain a characterization for boost invariant marginally trapped surface with pointwise 1-type Gauss map.

#### References

1. B. Y. Chen, Classification of marginally trapped Lorentzian flat surfaces in $E_2^4$ and its application to biharmonic surfaces, J. Math. Anal. Appl. 340 (2008), no. 2, 861-875. https://doi.org/10.1016/j.jmaa.2007.09.021
2. K. Arslan, B. K. Bayram, B. Bulca, Y. H. Kim, C. Murathan, and G. Ozturk, Rotational embeddings in $E^4$ with pointwise 1-type Gauss map, Turkish J. Math. 35 (2011), no. 3, 493-499.
3. K. Arslan, B. K. Bayram, Y. H. Kim, C. Murathan, and G. Ozturk, Vranceanu surface in $E^4$ with pointwise 1-type Gauss map, Indian J. Pure. Appl. Math. 42 (2011), no. 1, 41-51. https://doi.org/10.1007/s13226-011-0003-y
4. K. Arslan, B. Bulca, B. Kilic, Y. H. Kim, C. Murathan, and G. Ozturk, Tensor product surfaces with pointwise 1-type Gauss map. Bull. Korean Math. Soc. 48 (2011), no. 3, 601-609. https://doi.org/10.4134/BKMS.2011.48.3.601
5. B. Y. Chen, M. Choi, and Y. H. Kim, Surfaces of revolution with pointwise 1-type Gauss map, J. Korean Math. 42 (2005), no. 3, 447-455. https://doi.org/10.4134/JKMS.2005.42.3.447
6. B. Y. Chen and P. Piccinni, Submanifolds with finite type-Gauss map, Bull. Austral. Math. Soc. 35 (1987), no. 2, 161-186. https://doi.org/10.1017/S0004972700013162
7. M. Choi and Y. H. Kim, Characterization of the helicoid as ruled surfaces with pointwise 1-type Gauss map, Bull. Korean Math. Soc. 38 (2001), no. 4, 753-761.
8. M. Choi, D. S. Kim, and Y. H. Kim, Helicoidal surfaces with pointwise 1-type Gauss map, J. Korean Math. Soc. 46 (2009), no. 1, 215-223. https://doi.org/10.4134/JKMS.2009.46.1.215
9. U. Dursun and E. Coskun, Flat surfaces in the Minkowski space $E_1^3$ with pointwise 1-type Gauss map, Turkish J. Math. 36 (2012), no. 4, 613-629.
10. M. Choi, Y. H. Kim, and D. W. Yoon, Classification of ruled surfaces with pointwise 1-type Gauss map in Minkowski 3-space, Taiwanese J. Math. 15 (2011), no. 3, 1141-1161. https://doi.org/10.11650/twjm/1500406291
11. U. Dursun, Hypersurfaces with pointwise 1-type Gauss map in Lorentz-Minkowski space, Proc. Est. Acad. Sci. 58 (2009), no. 3, 146-161. https://doi.org/10.3176/proc.2009.3.02
12. U. Dursun and G. G. Arsan, Surfaces in the Euclidean space $E^4$ with pointwise 1-type Gauss map, Hacet. J. Math. Stat. 40 (2011), no. 5, 617-625.
13. U. Dursun and N. C. Turgay, General rotational surfaces in Euclidean space $E^4$ with pointwise 1-type Gauss map, Math. Commun. 17 (2012), no. 1, 71-81.
14. U. Dursun and N. C. Turgay, On spacelike surfaces in Minkowski 4-space with pointwise 1-type Gauss map of second type, Balkan J. Geom. Appl. 17 (2012), no. 2, 34-45.
15. U. Dursun and N. C. Turgay, Space-like surfaces in Minkowski space $E_1^4$ with pointwise 1-type Gauss map, arXiv:1305.5419.
16. S. Haesen and M. Ortega, Boost invariant marginally trapped surfaces in Minkowski 4-space, Classical Quantum Gravity 24 (2007), no. 22, 5441-5452. https://doi.org/10.1088/0264-9381/24/22/009
17. S. Haesen and M. Ortega, Marginally trapped surfaces in Minkowski 4-space invariant under a rotation subgroup of the Lorentz group, Gen. Relativity Gravitation 41 (2009), no. 8, 1819-1834. https://doi.org/10.1007/s10714-008-0754-x
18. Y. H. Kim and D. W. Yoon, Ruled surfaces with pointwise 1-type Gauss map, J. Geom. Phys. 34 (2000), no. 3-4, 191-205. https://doi.org/10.1016/S0393-0440(99)00063-7
19. Y. H. Kim and D. W. Yoon, Classification of rotation surfaces in pseudo Euclidean space, J. Korean Math. 41 (2004), no. 2, 379-396. https://doi.org/10.4134/JKMS.2004.41.2.379
20. V. Milousheva, Marginally trapped surfaces with pointwise 1-type Gauss map in Minkowski 4-space, Int. J. Geom. 2 (2013), no. 1, 34-43.
21. A. Niang, On rotation surfaces in the Minkowski 3-dimensional space with pointwise 1-type Gauss map, J. Korean Math. Soc. 41 (2004), no. 6, 1007-1021. https://doi.org/10.4134/JKMS.2004.41.6.1007
22. A. Niang, Rotation surfaces with 1-type Gauss map, Bull. Korean Math. Soc. 42 (2005), no. 1, 23-27. https://doi.org/10.4134/BKMS.2005.42.1.023
23. D. W. Yoon, Rotation surfaces with finite type Gauss map in $E^4$, Indian J. Pure. Appl. Math. 32 (2001), no. 12, 1803-1808.
24. D. W. Yoon, Some properties of the Clifford torus as rotation surface, Indian J. Pure. Appl. Math. 34 (2003), no. 6, 907-915.

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