## Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

#### Article Title

### Research of parabolic surface points in Galilean space

#### Abstract

The paper studies the surfaces of the Galilean space $R_3^1$. First, we consider the geometry of the surface in a small neighborhood of a point on the surface. Basically, we studied the points of the surface where at least one of the principal curvature appeals to zero. Two classes of points are defined where at least one of the principal curvature is zero. These points are divided into two types, parabolic and especially parabolic. It is proved that these neighborhoods using the movement of space is impossible to move each other. A sweep of surfaces with parabolic and especially parabolic points is constructed. A geometric image of the cone sweep in Galilean space is given. In Galilean space, we consider surfaces that do not have special planes. A class of surfaces with no special tangent planes is defined. A geometric image of a cone sweep in Galilean space is given. In Galilean space, surfaces that do not have special planes are considered. A class of surfaces is defined that do not have special tangent planes. At the end of the article, a classification of surface points in Galilean space is given.

#### First Page

231

#### Last Page

245

#### References

1. Artykbaev A. Classification of surface points in Galilean space. Research on the theory of surfaces in manifolds of sign-constant curvature. L., 1–15 (1987).

2. Pogorelov A.V. Differential geometry. Nauka, Moscow, (1974). – 560 pp.

3. Kurbonov E.K. Cyclic surfaces of Galilean space. Uzbek Mathematical Journal, No. 2, 51–57 (2001).

4. Kurbonov E.K. About the surface of Galilean space. Uzbek Mathematical Journal, No. 1, 46–52 (2005).

5. Sultanov B.M. Existence of a cyclic surface by a given function of total curvature. Acta NUUz, No. 2/2, 201–204 (2017).

6. Artykbaev A., Sokolov D.D. Geometry as a whole in space-time. Fan, Tashkent, (1991). – 179 pp.

7. Tomter P. Constant mean curvature surface in the Heisenberg group. Proceedings of Symposia in Pure Mathematics, Vol. 54.1, 485–495 (1993).

8. Dairbekov N.S. Mappings with bounded distortion on Heisenberg groups. Siberian Math. J., Vol. 41, Issue 3, 465–486 (2000).

9. Aleksandrov A.D. Convex polyhedra. Gostekhizdat, Moscow, Leningrad, (1950). – 282 pp.

10. Artykbaev A. Total angle about the vertex of a cone in Galilean space. Math. Notes, Vol. 43, Issue 5, 657–661 (1988).

11. Dolgarev A.I. Elements of differential Galilean geometry and odules of Galilean transformations. Saransk: the middle Volga mathematical society, Preprint 63, (2003). – 116 p.

12. Dolgarev A.I. Classical methods in the differential geometry of odular spaces: a monograph. Information and Publishing center of PSU, Penza, (2005). – 306 pp.

13. Dolgarev I.A. Movement of a material point on the surface of Galilee space. Actual problems of mathematics and methods of teaching mathematics: Intercollegiate collection of scientific works Penza: publishing house of PGTA, 3–6 (2007).

14. Divjak B., Milin Šipuš Ž. Some special surface in the pseudo-Galilean space. Acta Mathematica Hungarica, Vol. 118, Issue 3, 209–226 (2008).

15. Dede M., Ekici C., Goemans W. Surface of revolution with vanishing curvature in Galilean 3-space. Journal of Mathematical Physics, Analysis, Geometry, Vol. 14, Issue 2, 141–152 (2018).

16. Dea Won Yoon. Some classification of translation surface in Galilean 3-space. Int. Journal of Math. Analysis, Vol. 6, No. 25–28, 1355–1361 (2012).

17. Jaglom I.M. The principle of relativity of Galilean and non-Euclidean geometry. Nauka, Moscow, (1969). – 394 pp.

18. Chilin V.I, Muminov K.K. A transcendence basis in the differential field of invariants of pseudo-Galilean group. Russian Mathematics, Vol. 63, Issue 3, 15–24 (2019).

19. Aleksandrov A.D. Internal geometry of convex surfaces. OGIZ, M.-L., (1948). – 388 pp.

20. Pogorelov A.V. External geometry of convex surfaces. Nauka, Moscow, (1969). – 759 pp.

#### Recommended Citation

Artykbaev, Abdullaaziz and Sultanov, Bekzod
(2019)
"Research of parabolic surface points in Galilean space,"
*Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences*: Vol. 2:
Iss.
4, Article 2.

DOI: https://doi.org/10.56017/2181-1318.1034