Helicoidal Surfaces in the three dimensional simply isotropic space I₃¹

Murat Kemal Karacan, Dae Won Yoon, Sezai Kiziltug

Abstract


In this paper, we classify helicoidal surfaces in the three dimensional simply isotropic space  I₃¹ satisfying some algebraic equations in terms of the coordinate functions and the Laplacian operators with respect to the first, the second and the third fundamental form of the surface. We also give explicit forms of these surfaces.

Keywords


Simply isotropic space, helicoidal surfaces, Laplacian operator.

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References


L. J. Alias, A. Ferrandez and P. Lucas, Surfaces in the 3-dimensional Lorentz-Minkowski space satisfying $Delta mathbf{x}=mathbf{Ax}+mathbf{B}$ , Pacific J. Math., (1992), 201--208.

M. E. Aydin, Classification results on surfaces in the isotropic 3-space,http://arxiv.org/pdf/1601.03190.pdf.

M. E. Aydin, A generalization of translation surfaces with constant curvature in the isotropic space, J. Geom, DOI 10.1007/s00022-015-0292-0.

C. Baikoussis and L. Verstraelen, On the Gauss map of helicoidal surfaces, Rend. Sem. Math. Messina Ser. II, 2(16)(1993), 31--42.

M. Bekkar, Surfaces of Revolution in the $3$-Dimensional Lorentz-Minkowski Space Satisfying $Delta mathbf{x}^{i}=mathbf{lambda}^{i}mathbf{x}^{i}$, Int. J. Contemp. Math. Sciences, 3(2008), 1173--1185.

S. M. Choi, On the Gauss map of surfaces of revolution in a 3-dimensional Minkowski space, Tsukuba J. Math., 19(1995),351--367.

S. M. Choi, Y. H. Kim and D. W. Yoon, Some classification of surfaces of revolution in Minkowski 3-space, J. Geom., 104(2013),85--106.

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

F. Dillen, J. Pas and L. Vertraelen, On surfaces of finite type in Euclidean 3-space, Kodai Math. J., 13(1990), 10--21.

F. Dillen, J. Pas and L. Vertraelen, On the Gauss map of surfaces of revolution, Bull. Inst. Math. Acad. Sinica, 18(1990), 239--246.

O. J. Garay, An extension of Takahashi's theorem, Geom. Dedicata, 34(1990), 105--112.

Ch. B. Hamed and M. Bekkar, Helicoidal Surfaces in the three-Dimensional Lorentz-Minkowski space satisfying $Delta mathbf{r}_{i}=%mathbf{lambda }_{i}r_{i},$ Int. J. Contemp. Math. Sciences, 4(2009), 311--327.

G. Kaimakamis, B. Papantoniou and K. Petoumenos, Surfaces

of revolution in the 3-dimensional Lorentz-Minkowski space satisfying $%Delta ^{mathbf{III}}mathbf{r}=mathbf{Ar}$, Bull. Greek Math. Soc., 50(2005), 75-90.

B. Senoussi and M. Bekkar, Helicoidal surfaces with $% Delta ^{J}r=Ar$textit{ in $3$-dimensional Euclidean space}, Stud. Univ. Babes-Bolyai Math., 60(3)(2015), 437--448.

Z. M. Sipus, Translation Surfaces of constant curvatures in a simply isotropic space, Period Math. Hung., 68(2014), 160--175.

K. Strubecker, Differentialgeometrie des isotropen Raumes III, Flachentheorie, Math. Zeitsch. 48(1942), 369--427.

T. Takahashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18(1966), 380--385.

D. W. Yoon, Surfaces of Revolution in the three dimensional pseudo-galilean space, Glasnik Matematicki, 48(2013), 415--428.




DOI: http://dx.doi.org/10.5556/j.tkjm.48.2017.2200

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