Constant Curvature Translation Surfaces in Galilean 3-Space

ALYAMAÇ KÜLAHCI, MİHRİBAN; ÖĞRENMİŞ, Alper Osman; Aydin, Muhittin Evren

Constant Curvature Translation Surfaces in Galilean 3-Space

Muhittin Evren AYDIN, (Fırat Üniversitesi)

MİHRİBAN ALYAMAÇ KÜLAHCI, (Fırat Üniversitesi)

Alper Osman ÖĞRENMİŞ (Fırat Üniversitesi)

International Electronic Journal of Geometry

0 0

Yıl: 2019 Cilt: 12 Sayı: 1 Sayfa Aralığı: 9 - 19 Metin Dili: İngilizce İndeks Tarihi: 02-04-2020

Constant Curvature Translation Surfaces in Galilean 3-Space

Öz:

There are five different types of translation surfaces in a Galilean 3-space based upon planarity ofgenerating curves and absolute figure. We obtain these surfaces with arbitrary constant Gaussianand mean curvature, except the type that both of generating curves are non-planar .

Anahtar Kelime:

Konular: Matematik

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık

[1] Abdel-Baky, R.A. and Unluturk, Y., A study on classification of translation surfaces in pseudo-Galilean 3-space. J. Coupl. Syst. Multi. Dynm. 6 (2018), no. 3, 233-240.
[2] Aydin, M.E., Mihai, A., Ogrenmis, A.O. and Ergut, M., Geometry of the solutions of localized induction equation in the pseudo-Galilean space. Adv. Math. Phys. vol. 2015, Article ID 905978, 7 pages, 2015. doi:10.1155/2015/905978.
[3] Aydin, M.E., Ogrenmis, A.O. and Ergut, M., Classification of factorable surfaces in the pseudo-Galilean space. Glas. Mat. Ser. III, 50(70), 441-451, 2015.
[4] Cakmak, A., Karacan, M.K., Kiziltug, S. and Yoon D.W., Translation surfaces in the 3-dimensional Galilean space satisfying MII xi = xi; Bull. Korean Math. Soc. 54 (2017), no. 4, 1241-1254.
[5] Darboux, J. G., Theorie Generale des Surfaces. Livre I, Gauthier-Villars, Paris, 1914.
[6] Dede, M., Tubular surfaces in Galilean space. Math. Commun. 18 (2013), 209–217.
[7] Dede, M., Tube surfaces in pseudo-Galilean space. Int. J. Geom. Methods Mod. Phys. 13 (2016), no. 5, 1650056, 10 pp.
[8] Dede, M., Ekici, C., Goemans, W. and Unluturk, Y., Twisted surfaces with vanishing curvature in Galilean 3-space. Int. J. Geom. Methods Mod. Phys. 15(1) (2018), 1850001, 13pp.
[9] Dede, M., Ekici, C. and Goemans, W., Surfaces of revolution with vanishing curvature in Galilean 3-space. J. Math. Phys. Anal. Geom. 14 (2018), no. 2, 141-152.
[10] Dillen, F., Goemans, W. and Woestyne, Van De I., Translation surfaces of Weigarten type in 3-space. Bull. Transilv. Univ. Brasov Ser. III, Math. Inform. Phys. 1 (2008), no. 50, 109-122.
[11] Dillen, F., Verstraelen, L. and G. Zafindratafa, A generalization of the translation surfaces of Scherk. Differential Geometry in Honor of Radu Rosca: Meeting on Pure and Applied Differential Geometry, Leuven, Belgium, 1989, KU Leuven, Departement Wiskunde (1991), pp. 107–109.
[12] Divjak, B. and Milin-Sipus, Z., Special curves on ruled surfaces in Galilean and pseudo-Galilean spaces. Acta Math. Hungar. 98 (2003), 175–187.
[13] Erjavec, Z., Divjak, B. and Horvat, D., The general solutions of Frenet’s system in the equiform geometry of the Galilean, pseudo-Galilean, simple isotropic and double isotropic space. Int. Math. Forum. 6 (2011), no. 1, 837-856.
[14] Erjavec, Z., On generalization of helices in the Galilean and the pseudo-Galilean space. J. Math. Research 6 (2014), no. 3, 39-50.
[15] Giering O., Vorlesungen uber hohere Geometrie, Friedr. Vieweg & Sohn, Braunschweig, Germany, 1982.
[16] Gray A., Modern Differential Geometry of Curves and Surfaces with Mathematica. CRC Press LLC, 1998.
[17] Inoguchi, J., Lopez, R. and Munteanu M.I., Minimal translation surfaces in the Heisenberg group Nil3. Geom. Dedicata 161 (2012), 221-231.
[18] Kazan, A. and Karadag, H.B., Twisted Surfaces in the Pseudo-Galilean Space. NTMSCI 5 (2017), no.4, 72-79.
[19] Liu, H., Translation surfaces with constant mean curvature in 3-dimensional spaces. J. Geom. 64 (1999), no. 1-2, 141-149.
[20] Liu, H. and Yu, Y., Affine translation surfaces in Euclidean 3-space. In: Proceedings of the Japan Academy, Ser. A, Mathematical Sciences, vol. 89, pp. 111–113, Ser. A (2013).
[21] Liu, H. and Jung, S.D., Affine translation surfaces with constant mean curvature in Euclidean 3-space. J. Geom. 108 (2017), no. 2, 423-428.
[22] Lopez, R. and Munteanu M.I., Minimal translation surfaces in Sol 3. J. Math. Soc. Japan 64 (2012), no. 3, 985-1003.
[23] Lopez, R., Moruz, M., Translation and homothetical surfaces in Euclidean space with constant curvature. J. Korean Math. Soc. 52 (2015), no. 3, 523-535.
[24] Lopez, R., Minimal translation surfaces in hyperbolic space. Beitr. Algebra Geom. 52 (2011), no. 1, 105-112.
[25] Milin-Sipus, Z., Ruled Weingarten surfaces in the Galilean space.Period. Math. Hungar. 56 (2008), 213–225.
[26] Milin-Sipus, Z. and Divjak, B., Some special surfaces in the pseudo-Galilean Space. Acta Math. Hungar. 118 (2008), 209–226.
[27] Milin-Sipus, Z. and Divjak, B., Translation surface in the Galilean space. Glas. Mat. Ser. III 46 (2011), no. 2, 455–469.
[28] Milin-Sipus, Z., and Divjak, B., Surfaces of constant curvature in the pseudo-Galilean space. Int. J. Math. Sci., 2012, Art ID375264, 28pp.
[29] Milin-Sipus, Z., On a certain class of translation surfaces in a pseudo-Galilean space. Int. Mat. Forum 6 (2012), no. 23, 1113-1125.
[30] Milin-Sipus, Z., Translation surfaces of constant curvatures in a simply isotropic space. Period. Math. Hung. 68 (2014), 160–175.
[31] Moruz, M. and Munteanu M.I., Minimal translation hypersurfaces in E4: J. Math. Anal. Appl. 439 (2016), no. 2, 798-812.
[32] Munteanu M.I., Palmas, O. and Ruiz-Hernandez, G., Minimal translation hypersurfaces in Euclidean spaces. Mediterranean J. Math. 13 (2016), 2659–2676.
[33] Onishchick, A. and Sulanke, R., Projective and Cayley-Klein Geometries. Springer, 2006.
[34] Pavkovic, B.J. and Kamenarovic, I., The equiform differential geometry of curves in the Galilean space G3. Glasnik Math. 22 (1987), no. 42, 449-457.
[35] Roschel, O., Die Geometrie des Galileischen Raumes. Forschungszentrum Graz, Mathematisch-Statistische Sektion, Graz, 1985.
[36] Roschel, O., Torusflachen des Galileischen Raumes. Studia Sci. Math. Hungarica, 23 (1988), no. 3-4, 401–410.
[37] Scherk, H.F., Bemerkungen uber die kleinste Flache innerhalb gegebener Grenzen. J. Reine Angew. Math. 13 (1835), 185-208.
[38] Seo, K., Translation Hypersurfaces with constant curvature in space forms. Osaka J. Math. 50 (2013), 631-641.
[39] Sun, H., On affine translation surfaces of constant mean curvature. Kumamoto J. Math. 13 (2000), 49-57.
[40] Verstraelen, L., Walrave, J. and Yaprak, S., The minimal translation surfaces in Euclidean space. Soochow J. Math. 20 (1994), 77–82.
[41] Yang, D. and Fu, Y., On affine translation surfaces in affine space. J. Math. Anal. Appl. 440 (2016), no. 2, 437–450.
[42] Yoon, D.W., Minimal Translation Surfaces in H2 R. Taiwanese J. Math. 17(5) (2013), 1545-1556.
[43] Yoon, D.W., Classification of rotational surfaces in pseudo-Galilean space. Glas. Mat. Ser. III 50 (2015), no. 2, 453-465.

APA	Aydin M, ALYAMAÇ KÜLAHCI M, ÖĞRENMİŞ A (2019). Constant Curvature Translation Surfaces in Galilean 3-Space. , 9 - 19.
Chicago	Aydin Muhittin Evren,ALYAMAÇ KÜLAHCI MİHRİBAN,ÖĞRENMİŞ Alper Osman Constant Curvature Translation Surfaces in Galilean 3-Space. (2019): 9 - 19.
MLA	Aydin Muhittin Evren,ALYAMAÇ KÜLAHCI MİHRİBAN,ÖĞRENMİŞ Alper Osman Constant Curvature Translation Surfaces in Galilean 3-Space. , 2019, ss.9 - 19.
AMA	Aydin M,ALYAMAÇ KÜLAHCI M,ÖĞRENMİŞ A Constant Curvature Translation Surfaces in Galilean 3-Space. . 2019; 9 - 19.
Vancouver	Aydin M,ALYAMAÇ KÜLAHCI M,ÖĞRENMİŞ A Constant Curvature Translation Surfaces in Galilean 3-Space. . 2019; 9 - 19.
IEEE	Aydin M,ALYAMAÇ KÜLAHCI M,ÖĞRENMİŞ A "Constant Curvature Translation Surfaces in Galilean 3-Space." , ss.9 - 19, 2019.
ISNAD	Aydin, Muhittin Evren vd. "Constant Curvature Translation Surfaces in Galilean 3-Space". (2019), 9-19.

APA	Aydin M, ALYAMAÇ KÜLAHCI M, ÖĞRENMİŞ A (2019). Constant Curvature Translation Surfaces in Galilean 3-Space. International Electronic Journal of Geometry, 12(1), 9 - 19.
Chicago	Aydin Muhittin Evren,ALYAMAÇ KÜLAHCI MİHRİBAN,ÖĞRENMİŞ Alper Osman Constant Curvature Translation Surfaces in Galilean 3-Space. International Electronic Journal of Geometry 12, no.1 (2019): 9 - 19.
MLA	Aydin Muhittin Evren,ALYAMAÇ KÜLAHCI MİHRİBAN,ÖĞRENMİŞ Alper Osman Constant Curvature Translation Surfaces in Galilean 3-Space. International Electronic Journal of Geometry, vol.12, no.1, 2019, ss.9 - 19.
AMA	Aydin M,ALYAMAÇ KÜLAHCI M,ÖĞRENMİŞ A Constant Curvature Translation Surfaces in Galilean 3-Space. International Electronic Journal of Geometry. 2019; 12(1): 9 - 19.
Vancouver	Aydin M,ALYAMAÇ KÜLAHCI M,ÖĞRENMİŞ A Constant Curvature Translation Surfaces in Galilean 3-Space. International Electronic Journal of Geometry. 2019; 12(1): 9 - 19.
IEEE	Aydin M,ALYAMAÇ KÜLAHCI M,ÖĞRENMİŞ A "Constant Curvature Translation Surfaces in Galilean 3-Space." International Electronic Journal of Geometry, 12, ss.9 - 19, 2019.
ISNAD	Aydin, Muhittin Evren vd. "Constant Curvature Translation Surfaces in Galilean 3-Space". International Electronic Journal of Geometry 12/1 (2019), 9-19.