In a five-dimensional pseudo-Euclidean space of index two, the geometry on its sphere is studied. The equivalence of the geometry on a sphere of imaginary radius on de Sitter space is shown. The interpretation of the geometry on a sphere of imaginary radius, inside the sphere of imaginary radius of the Minkowski four-dimensional space, is implemented. We study a curve in a five-dimensional pseudo-Euclidean space of index two and determine the membership condition of the curve to a sphere of imaginary radius.
1. Aminov Yu. A. Geometry of Submanifolds, Naukova Dumka, Kiev, (2002). [in Russian].
2. Aminov Yu. A., Nasedkina Ya.S. Membership conditions of a two-dimensional surface from E5to a hypersphere or hyperplane. Mathematical notes, Vol. 94, Issue 2, 163-174 (2013). [in Russian]
3. Artikbaev A., Mamadaliyev B.M. Geometry on subspaces of space 2R5. Bulletin of the Institute of Mathematics, Vol. 3, 42-46 (2021). [in Russian]
4. Artykbaev A., Sokolov D. D. Geometry as a Whole in Flat space-time. Tashkent, Fan, 1-179 (1991). [in Russian]
5. Bang Yen Chen. Marginally trapped surfaces and surfaces with parallel mean curvature vector in lorentzian space forms. Conference, Differential Geometry, Michigan State University, East Lansing, MI, USA, September (2010).
6. Dubrov B.A., Novikov S.P., Fomenko A.T. Modern geometry. Methods and Applications. Vol. 1, Moscow, Nauka, 920 pp (2013).
7. Hasan Altinbas, Bulent Altunkaya, Levent Kula. Polynomial Helices in the n-Dimensional Semi-Euclidean Space with Index Two. Published by Faculty of Sciences and Mathematics, University of Nis, Serbia, 4861-4872 (2010).
8. Hawking J., Ellis. Large-scale structure of space-time. Mir Publishing House, Moscow, (1977). [in Russian]
9. Mamadaliyev B.M. Full two-dimensional surfaces in 2R5. Scientific Journal of Samarkand University, Vol. 199, Issue 1, 35-38 (2020). [in Russian]
10. Mamadaliyev B. M. Problems of modern mathematics and its applications. Properties of a spherical curve in space. June 16-19, Bishkek, 27-28 ( 2021).
11. Narmanov A. Ya. Geometry of orbits of vector fields and singular foliations. Modern mathematics, Fundamental directions. Vol. 65, Issue 1, 54-71 (2019). [in Russian]
12. Rosenfeld B.A. Non-Euclidean spaces. Moscow, Nauka, 548 pp (1969). [in Russian]
Artikbaev, Abdulaziz and Mamadaliyev, Botirjon
"Interpretation of de Sitter space of second kind,"
Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences: Vol. 4:
4, Article 8.