# HELICOIDAL SURFACES OF THE THIRD FUNDAMENTAL FORM IN MINKOWSKI 3-SPACE

• CHOI, MIEKYUNG (DEPARTMENT OF MATHEMATICS EDUCATION GYEONGSANG NATIONAL UNIVERSITY) ;
• YOON, DAE WON (DEPARTMENT OF MATHEMATICS EDUCATION AND RINS GYEONGSANG NATIONAL UNIVERSITY)
• Published : 2015.09.30

#### Abstract

We study helicoidal surfaces with the non-degenerate third fundamental form in Minkowski 3-space. In particular, we mainly focus on the study of helicoidal surfaces with light-like axis in Minkowski 3-space. As a result, we classify helicoidal surfaces satisfying an equation in terms of the position vector field and the Laplace operator with respect to the third fundamental form on the surface.

#### Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

#### References

1. M. Choi, Y. H. Kim, H. Liu, and D. W. Yoon, Helicoidal surfaces and their Gauss map in Minkowski 3-space, Bull. Korean Math. Soc. 47 (2010), no. 4, 859-881. https://doi.org/10.4134/BKMS.2010.47.4.859
2. M. Choi, Y. H. Kim, and G. Park, Helicoidal surfaces and their Gauss map in Minkowski 3-space II, Bull. Korean Math. Soc. 46 (2009), no. 3, 567-576. https://doi.org/10.4134/BKMS.2009.46.3.567
3. M. Choi, Y. H. Kim, and D. W. Yoon, Some classification of surfaces of revolution in Minkowski 3-space, J. Geom. 104 (2013), no. 1, 85-106. https://doi.org/10.1007/s00022-013-0149-3
4. O. J. Garay, An extension of Takahashi's Theorem, Geom. Dedicata 34 (1990), no. 2, 105-112. https://doi.org/10.1007/BF00147319
5. G. Kaimakamis and B. Papantoniou, Surfaces of revolution in the 3-dimensional Lorentz-Minkowski space satisfying ${\Delta}^{{II}^{\rightarrow}_r}$ = $A^{\rightarrow}_r$, J. Geom. 81 (2004), no. 1-2, 81-92. https://doi.org/10.1007/s00022-004-1675-9
6. G. Kaimakamis and B. Papantoniou, Surfaces of revolution in the 3-dimensional Lorentz-Minkowski space ${\mathbb{E}}^3_1$ satisfying ${\Delta}^{{III}^{\rightarrow}_r}$ = $A^{\rightarrow}_r$, Bull. Greek Math. Soc. 50 (2005), 75-90.
7. O. Kobayashi, Maximal surfaces in the 3-dimensional Minkowski space ${\mathbb{L}}^3$, Tokyo J. Math. 6 (1983), no. 2, 297-309. https://doi.org/10.3836/tjm/1270213872
8. C. W. Lee, Y. H. Kim, and D. W. Yoon, Ruled surfaces of non-degenerate third fundamental forms in Minkowski 3-space, Appl. Math. Comput. 216 (2010), no. 11, 3200-3208. https://doi.org/10.1016/j.amc.2010.04.043
9. B. O'Neill, Semi-Riemannian Geometry and its applications to Relativity, Academic Press, New York, 1983.
10. B. Senoussi and M. Bekkar, Helicoidal surfaces in the 3-dimensional Lorentz-Minkowski space ${\mathbb{E}}^3_1$ satisfying ${\Delta}^{III}r$ = Ar, Tsukuba J. Math. 37 (2013), no 2, 339-353. https://doi.org/10.21099/tkbjm/1389972033
11. T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan 18 (1966), 380-385. https://doi.org/10.2969/jmsj/01840380

#### Cited by

1. Isometric Deformation of (m,n)-Type Helicoidal Surface in the Three Dimensional Euclidean Space vol.6, pp.11, 2018, https://doi.org/10.3390/math6110226
2. The Gauss Map and the Third Laplace-Beltrami Operator of the Rotational Hypersurface in 4-Space vol.10, pp.9, 2018, https://doi.org/10.3390/sym10090398