Research Article
BibTex RIS Cite

Year 2019, , 9 - 19, 27.03.2019

Muhittin Evren Aydın Mihriban Alyamaç Külahçı , Alper Osman Öğrenmiş

https://doi.org/10.36890/iejg.545741
Cited By: 7

Abstract

References

  • [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

Year 2019, , 9 - 19, 27.03.2019

Muhittin Evren Aydın Mihriban Alyamaç Külahçı , Alper Osman Öğrenmiş

https://doi.org/10.36890/iejg.545741
Cited By: 7

Abstract

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 .

Keywords

Galilean space translation surface Gaussian curvature mean curvature

References

  • [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.
There are 43 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

Muhittin Evren Aydın This is me

Mihriban Alyamaç Külahçı

Alper Osman Öğrenmiş

Publication Date March 27, 2019
Published in Issue Year 2019

Cite

APA Aydın, M. E., Külahçı, M. A., & Öğrenmiş, A. O. (2019). Constant Curvature Translation Surfaces in Galilean 3-Space. International Electronic Journal of Geometry, 12(1), 9-19. https://doi.org/10.36890/iejg.545741
AMA Aydın ME, Külahçı MA, Öğrenmiş AO. Constant Curvature Translation Surfaces in Galilean 3-Space. Int. Electron. J. Geom. March 2019;12(1):9-19. doi:10.36890/iejg.545741
Chicago Aydın, Muhittin Evren, Mihriban Alyamaç Külahçı, and Alper Osman Öğrenmiş. “Constant Curvature Translation Surfaces in Galilean 3-Space”. International Electronic Journal of Geometry 12, no. 1 (March 2019): 9-19. https://doi.org/10.36890/iejg.545741.
EndNote Aydın ME, Külahçı MA, Öğrenmiş AO (March 1, 2019) Constant Curvature Translation Surfaces in Galilean 3-Space. International Electronic Journal of Geometry 12 1 9–19.
IEEE M. E. Aydın, M. A. Külahçı, and A. O. Öğrenmiş, “Constant Curvature Translation Surfaces in Galilean 3-Space”, Int. Electron. J. Geom., vol. 12, no. 1, pp. 9–19, 2019, doi: 10.36890/iejg.545741.
ISNAD Aydın, Muhittin Evren et al. “Constant Curvature Translation Surfaces in Galilean 3-Space”. International Electronic Journal of Geometry 12/1 (March 2019), 9-19. https://doi.org/10.36890/iejg.545741.
JAMA Aydın ME, Külahçı MA, Öğrenmiş AO. Constant Curvature Translation Surfaces in Galilean 3-Space. Int. Electron. J. Geom. 2019;12:9–19.
MLA Aydın, Muhittin Evren et al. “Constant Curvature Translation Surfaces in Galilean 3-Space”. International Electronic Journal of Geometry, vol. 12, no. 1, 2019, pp. 9-19, doi:10.36890/iejg.545741.
Vancouver Aydın ME, Külahçı MA, Öğrenmiş AO. Constant Curvature Translation Surfaces in Galilean 3-Space. Int. Electron. J. Geom. 2019;12(1):9-19.

Cited By

Translation surfaces generating with some partner curve
Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics
https://doi.org/10.31801/cfsuasmas.1299533
Factorable Surfaces in Pseudo-Galilean Space with Prescribed Mean and Gaussian Curvatures
Journal of New Theory
https://doi.org/10.53570/jnt.1137525
Existence and Uniqueness of Polyhedra with Given Values of the Conditional Curvature
International Electronic Journal of Geometry
https://doi.org/10.36890/iejg.1246589
Special Mean and Total Curvature of a Dual Surface in Isotropic Spaces
International Electronic Journal of Geometry
https://doi.org/10.36890/iejg.972370
RULED SURFACE WITH CONSTANT SLOPE ACCORDING TO OSCULATING PLANE OF BASE CURVE IN GALILEAN 3- SPACE
Journal of Universal Mathematics
https://doi.org/10.33773/jum.952868
Surfaces in pseudo-Galilean space with prescribed mean curvature
Adıyaman University Journal of Science
https://doi.org/10.37094/adyujsci.907765
A new approach to revolution surface with its focal surface in the Galilean 3-space $\mathbb{G}_{3}$
Hacettepe Journal of Mathematics and Statistics
https://doi.org/10.15672/hujms.905636