Special Mean and Total Curvature of a Dual Surface in Isotropic Spaces

Yıl 2022, Cilt: 15 Sayı: 1, 1 - 10, 30.04.2022

Abdullaaziz Artykbaev Sherzodbek Ismoilov

https://doi.org/10.36890/iejg.972370

Cited By: 1

Öz

The study of the geometry surfaces in spaces with a degenerate metric is one of the urgent problems of modern geometry since its results find numerous applications in problems of mechanics and quantum mechanics.

In this paper, we study the properties of the total and mean curvatures of a surface and its dual image in an isotropic space. We prove the equality of the mean curvature and the second quadratic forms. The relation of the mean curvature of a surface to its dual surface is found. The superimposed space method is used to investigate the geometric characteristics of a surface relative to the normal and special normal.

Anahtar Kelimeler

Isotropic space, mean curvature, total curvature, dual surface, special mean curvature, special total curvature

Destekleyen Kurum

Tashkent State Transport University

Teşekkür

Thanks

Kaynakça

• [1] Artykbaev, A., Ismoilov, Sh. Sh.: The dual surfaces of an isotropic space R^2_3. Bulletin of the Institute of Mathematics. 4, 1-8 (2021).
• [2] Artykbaev, A., Sokolov, D.D.: Geometry as a whole in space-time. Tashkent Fan., (1991).
• [3] Artykbaev, A.: Recovering Convex Surfaces from the Extrinsic Curvature in Galilean Space. Mathematics of the USSR Sbornik. 47(1), 195-214 (1984). https://doi.org/10.1070/SM1984v047n01ABEH002637
• [4] Artikbayev, A., Ismoilov, Sh.: O secheniya ploskosti so izotropnogo prostranstva. Scientific Journal of Samarkand University. 5(123), 84-89 (2020).
• [5] Aleksandrov, A.D.: Internal geometry of convex surfaces. M. L. OGIZ, (1948).
• [6] Aydin, E.M., Mihai I.: On certain surfaces in the isotropic 4- space. Mathematical Communications. 22, 41-51 (2017).
• [7] Aydin, M.E., Mihai, A.: Ruled surfaces generated byelliptic cylindrical curves in the isotropic space, Georgian Math. J. (2017).
• [8] Aydin, M.E.: Classification results on surfaces in the isotropic 3-space. AKU J. Sci. Eng. 16, 239-246 (2016).
• [9] Aydin, M.E., Kulahci, M.A., Ogrenmis, A.O.: Constant curvature translation surfaces in Galilean 3-space. International Electronic Journal of Geometry. 12(1), 9-19 (2019).
• [10] Chen, B.Y.: Solutions to homogeneous Monge - Ampere equations of homothetic functions and their applications to production models in economics. J. Math.Anal.Appl. 411, 223-229 (2014).
• [11] Dede, C., Ekici, Goemans,W.: Surfaces of Revolution with Vanishing Curvature in Galilean 3-Space. Journal of Mathematical Physics, Analysis, Geometry. 14(2), 141-152 (2018). https://doi.org/10.15407/mag14.02.141
• [12] Yoon, D.W.: Some classification of translation surface in Galilean 3-space. Int. Journal of Math. Analysis, 6, 1355-1361 (2012).
• [13] Rosenfelt, B.A.: Non-Euclidean spaces. Moscow. (1969).
• [14] Sachs, H.: Isotrop Geometri des Raumes. (1990).
• [15] Ismoilov, Sh., Sultonov, B.: Cyclic surfaces in pseudo-euclidean space. International Journal of Statistics and Applied Mathematics. 3, 28-31 (2020).
• [16] Ismoilov, Sh.: Dual image in isotropic space. NamSU konf. 1, 36-40 (2016).
• [17] Jaglom, I. M.: The principle of relativity of Galilean and non-Euclidean geometry. M. Nauka., (1969).
• [18] Lone, M.S., Karacan M.K.: Dual translation surfaces in the three dimensional simply isotropic space I^1_3 . Tamkang Journal of mathematics. 49(1), 67-77 (2018). https://doi.org/10.5556/j.tkjm.49.2018.2476
• [19] Pogorelov, A.V.: Differential geometry. Publishing House Nauka, Moscow, (1974).
• [20] Polyanin, A.D., Valentin, Z.: Nonlinear Equations in Mathematical Physics and Mechanics: Solution Methods. Urait Publishing House. 451-461 (2017).
• [21] Strubecker, K.: Differentialgeometrie des isotropen Raumes II. Math.Z.47, 743-777 (1942).
• [22] Strubecker, K.: Differentialgeometrie des isotropen Raumes III. Math.Z.48, 369-427 (1943).
• [23] Yoon, D.W., Lee, J.W.: Linear Weingarten helicoidal surfaces in isotropic space. Symmetry. 8(11), 1-7 (2016). https://doi.org/10.3390/sym8110126