Classifications of translation surfaces in isotropic geometry with constant curvature

Abstract

UDC 515.12

We classify translation surfaces in isotropic geometry with arbitrary constant isotropic Gaussian and mean curvatures underthe condition that at least one of translating curves lies in a plane.

References

Aydin, Muhittin Evren. A generalization of translation surfaces with constant curvature in the isotropic space. J. Geom. 107 (2016), no. 3, 603--615. doi: 10.1007/s00022-015-0292-0

Aydin, Muhittin Evren; Ergut, Mahmut. Affine translation surfaces in the isotropic 3-space. Int. Electron. J. Geom. 10 (2017), no. 1, 21--30. MR3636642

Aydin, Muhittin Evren; Mihai, Ion. On certain surfaces in the isotropic 4-space. Math. Commun. 22 (2017), no. 1, 41--51. MR3622033

Chen, Bang-Yen; Decu, Simona; Verstraelen, Leopold. Notes on isotropic geometry of production models. Kragujevac J. Math. 38 (2014), no. 1, 23--33. doi: 10.5937/KgJMath1401023C

J. G. Darboux,Theorie generale des surfaces, Livre I, Gauthier-Villars, Paris (1914).

da Silva, Luiz C. B. The geometry of Gauss map and shape operator in simply isotropic and pseudo-isotropic spaces. J. Geom. 110 (2019), no. 2, Art. 31, 18 pp. doi: 10.1007/s00022-019-0488-9

da Silva, Luiz C. B. Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spaces, Tamkang J. Math., 51 (2020), no 1, 31 – 52. doi: 10.5556/j.tkjm.51.2020.2960

Dillen, Franki; Goemans, Wendy; Van de Woestyne, Ignace. Translation surfaces of Weingarten type in 3-space. Bull. Transilv. Univ. Braşov Ser. III 1(50) (2008), 109--122. MR2478011

F. Dillen, L. Verstraelen, G. Zafindratafa, A generalization of the translation surfaces of Scherk, Different. Geom. in Honor of Radu Rosca: Meeting on Pure and Appl. Different. Geom. (Leuven, Belgium, 1989), KU Leuven, DepartmentWiskunde (1991), p. 107 – 109.

Dillen, F.; Van de Woestyne, I.; Verstraelen, L.; Walrave, J. The surface of Scherk in ${bf E}^3$: a special case in the class of minimal surfaces defined as the sum of two curves. Bull. Inst. Math. Acad. Sinica 26 (1998), no. 4, 257--267. MR1662279

Erjavec, Zlatko; Divjak, Blaženka; Horvat, Damir. The general solution 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. 17-20, 837--856. MR2796105

Goemans, Wendy; Van de Woestyne, Ignace. Translation and homothetical lightlike hypersurfaces of a semi-Euclidean space. Kuwait J. Sci. Engrg. 38 (2011), no. 2A, 35--42. MR2934057

Gray, Alfred. Modern differential geometry of curves and surfaces with Mathematica. Second edition. CRC Press, Boca Raton, FL, 1998. xxiv+1053 pp. ISBN: 0-8493-7164-3 MR1688379

T. Hasanis, R. Lopez, Translation surfaces in Euclidean space with constant Gaussian curvature, Arxiv 8 Sept 2018: https://arxiv.org/abs/1809.02758v1.

T. Hasanis, R. Lopez, Classification and construction of minimal translation surfaces in Euclidean space, Arxiv 8 Sept 2018: https://arxiv.org/abs/1809.02759v1. DOI: https://doi.org/10.1007/s00025-019-1128-2

Hasanis, Thomas. Translation surfaces with non-zero constant mean curvature in Euclidean space. J. Geom. 110 (2019), no. 2, Art. 20, 8 pp. doi: 10.1007/s00022-019-0476-0

Inoguchi, Jun-ichi; López, Rafael; Munteanu, Marian-Ioan. Minimal translation surfaces in the Heisenberg group ${rm Nil}_3$. Geom. Dedicata 161 (2012), 221--231. doi: 10.1007/s10711-012-9702-8

Jung, Seoung Dal; Liu, Huili; Liu, Yixuan. Weingarten affine translation surfaces in Euclidean 3-space. Results Math. 72 (2017), no. 4, 1839--1848. doi: 10.1007/s00025-017-0737-x

Lima, Barnabé P.; Santos, Newton L.; Sousa, Paulo A. Generalized translation hypersurfaces in Euclidean space. J. Math. Anal. Appl. 470 (2019), no. 2, 1129--1135. doi: 10.1016/j.jmaa.2018.10.053

Liu, Huili. Translation surfaces with constant mean curvature in $3$-dimensional spaces. J. Geom. 64 (1999), no. 1-2, 141--149. doi: 10.1007/BF01229219

Liu, Huili; Jung, Seoung Dal. Affine translation surfaces with constant mean curvature in Euclidean 3-space. J. Geom. 108 (2017), no. 2, 423--428. doi: 10.1007/s00022-016-0348-9

Liu, Huili; Yu, Yanhua. Affine translation surfaces in Euclidean 3-space. Proc. Japan Acad. Ser. A Math. Sci. 89 (2013), no. 9, 111--113. doi: 10.3792/pjaa.89.111

López, Rafael. Minimal translation surfaces in hyperbolic space. Beitr. Algebra Geom. 52 (2011), no. 1, 105--112. doi: 10.1007/s13366-011-0008-z

López, Rafael; Munteanu, Marian Ioan. Minimal translation surfaces in ${rm Sol}_3$. J. Math. Soc. Japan 64 (2012), no. 3, 985--1003. doi: 10.2969/jmsj/06430985

López, Rafael; Moruz, Marilena. Translation and homothetical surfaces in Euclidean space with constant curvature. J. Korean Math. Soc. 52 (2015), no. 3, 523--535. doi: 10.4134/JKMS.2015.52.3.523

López, Rafael; Perdomo, Óscar. Minimal translation surfaces in Euclidean space. J. Geom. Anal. 27 (2017), no. 4, 2926--2937. doi: 10.1007/s12220-017-9788-1

Milin Šipuš, Željka. Translation surfaces of constant curvatures in a simply isotropic space. Period. Math. Hungar. 68 (2014), no. 2, 160--175. doi: 10.1007/s10998-014-0027-2

Moruz, Marilena; Munteanu, Marian Ioan. Minimal translation hypersurfaces in $Bbb{E}^4$. J. Math. Anal. Appl. 439 (2016), no. 2, 798--812. doi: 10.1016/j.jmaa.2016.02.077

Munteanu, Marian Ioan; Palmas, Oscar; Ruiz-Hernández, Gabriel. Minimal translation hypersurfaces in Euclidean space. Mediterr. J. Math. 13 (2016), no. 5, 2659--2676. doi: 10.1007/s00009-015-0645-9

Ogrenmis A. O. Rotational surfaces in isotropic spaces satisfying Weingarten conditions, Open Phys. 14 (2016), no 9, 221 – 225. doi: 10.1515/phys-2016-0030

Pottmann, Helmut; Grohs, Philipp; Mitra, Niloy J. Laguerre minimal surfaces, isotropic geometry and linear elasticity. Adv. Comput. Math. 31 (2009), no. 4, 391--419. doi: 10.1007/s10444-008-9076-5

Pottmann, Helmut; Opitz, Karsten. Curvature analysis and visualization for functions defined on Euclidean spaces or surfaces. Comput. Aided Geom. Design 11 (1994), no. 6, 655--674. doi: 10.1016/0167-8396(94)90057-4

Sachs, Hans. Isotrope Geometrie des Raumes. (German) [[Isotropic geometry of space]] Friedr. Vieweg & Sohn, Braunschweig, 1990. {rm viii}+323 pp. ISBN: 3-528-06332-7 doi: 10.1007/978-3-322-83785-1

Scherk, H. F. Bemerkungen über die kleinste Fläche innerhalb gegebener Grenzen. (German) J. Reine Angew. Math. 13 (1835), 185--208. doi: 10.1515/crll.1835.13.185

Seo, Keomkyo. Translation hypersurfaces with constant curvature in space forms. Osaka J. Math. 50 (2013), no. 3, 631--641. MR3128996

Strubecker, Karl. Über die isotropen Gegenstücke der Minimalfläche von Scherk. (German) J. Reine Angew. Math. 293(294) (1977), 22--51. doi: 10.1515/crll.1977.293-294.22

Sun, Huafei. On affine translation surfaces of constant mean curvature. Kumamoto J. Math. 13 (2000), 49--57. MR1759217

Verstraelen, L.; Walrave, J.; Yaprak, S. The minimal translation surfaces in Euclidean space. Soochow J. Math. 20 (1994), no. 1, 77--82. MR1265251

Yang, Dan; Fu, Yu. On affine translation surfaces in affine space. J. Math. Anal. Appl. 440 (2016), no. 2, 437--450. doi: 10.1016/j.jmaa.2016.03.066

D. Yang, J. Zhang, Y. Fu, A note on minimal translation graphs in Euclidean space, Mathematics, 7, No 10 (2019) doi: 10.3390/math7100889

Published
28.03.2020
How to Cite
Aydin , M. E. “Classifications of Translation Surfaces in Isotropic Geometry With Constant Curvature”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 3, Mar. 2020, pp. 291-06, doi:10.37863/umzh.v72i3.505.
Section
Research articles