Yıl 2019,
Cilt: 12 Sayı: 1, 9 - 19, 27.03.2019
Muhittin Evren Aydın
Mihriban Alyamaç Külahçı
,
Alper Osman Öğrenmiş
Kaynakça
- [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 H^2 × 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.
Constant Curvature Translation Surfaces in Galilean 3-Space
Yıl 2019,
Cilt: 12 Sayı: 1, 9 - 19, 27.03.2019
Muhittin Evren Aydın
Mihriban Alyamaç Külahçı
,
Alper Osman Öğrenmiş
Öz
There are five different types of translation surfaces in a Galilean 3-space based upon planarity of
generating curves and absolute figure. We obtain these surfaces with arbitrary constant Gaussian
and mean curvature, except the type that both of generating curves are non-planar .
Kaynakça
- [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 H^2 × 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.