Physical and Mathematical Sciences | Article | Published 2021

The dual surfaces of an isotropic space R32

Authors:

Абдуллаазиз Артикбаев

Publisher: Bulletin of the Institute of Mathematics
Collection: Bulletin of the Institute of Mathematics
Keywords: Paraboloid; plane; isotropic space; dual mapping; dual surfaces.

Abstract

In the paper, we study some properties of convex surfaces in an isotropic space $R_3^2.$ The definitions of $R_3^2,$ the first and second quadratic forms of the surface are given, and an analogue of the sphere of the Euclidean space is found. The sphere of an isotropic space is a paraboloid of revolution. It is proved that the intersection of the sphere of an isotropic space is always an ellipse. The definition of the dual image of a plane with respect to the sphere of an isotropic space is given. Using the dual image, the generalized surface mapping is defined. Some properties of the dual surface are proved.

References

  1. Артыкбаев А., Восстановление выпуклых поверхностей по внешней кривизне в галилеевом пространстве, Математический сборник, 119(2), 1982, pp.204.
  2. Artykbaev, A., & Sokolov, D. D. (1991). Geometry in the large in a flat space-time. Fan, Tashkent.
  3. Ismoilov, S., & Sultonov, B. (2020). Cyclic surfaces in pseudo-euclidean space. International Journal of Statistics and Applied Mathematics, 3, 28-31.
  4. Bakelman I.Ya., Verner A.L., Kontor Vvedenie v differensialnuyu geometriyu, v selom, Москва, Наука, 1991.
  5. Rezanfeld B. A. Neyevklidovi pronstranstva, New York & Basel : Dekker, 1969. [In russian]
  6. Manfredo P. Do Carmo. Differential geometry of Curves and surface, Dover publications. Inc. Mineola, New York, 2016.
  7. Masalsev L.A. Constant mean curvature surface in the Heisenberg group. Proe. of Symp. pure math. 54(1), 1993, pp. 485-495.
  8. Artikbayev, A., & Ismoilov, S. (2020). O secheniya ploskosti so izotropnogo prostranstva. Scientific Journal of Samarkand University, 5(123), 84-89.
  9. Исмоилов Ш., Тиллаев Д., Юсупова З. ГЕОМЕТРИЯ ПРЕОБРАЗОВАНИЙ, СОХРАНЯЮЩИХ ПЛОЩАДЬ ИЛИ ОБЪЕМ //CONTEMPORARY PROBLEMS IN MATHEMATICS AND PHYSICS. – 2017. – С. 175.
  10. Исмоилов, Ш. Ш. (2021). СУЩЕСТВОВАНИЕ ПОВЕРХНОСТИ В ИЗОТРОПНОМ ПРОСТРАНСТВЕ С ЗАДАННОЙ СРЕДНЕЙ КРИВИЗНОЙ ДВОЙСТВЕННОГО ОБРАЗА. ББК 22.151 я431 К476, 1, 78.
  11. Strubecker K, Differentialgeometrie des isotropen Raumes II, Mathematische Zeitschrift, 47, 743-777.
  12. Strubecker K, Differentialgeometrie des isotropen Raumes II, Mathematische Zeitschrift 1943; 48: 372-417.
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