Abstract
Keywords
vector field orbit of vector fields orbit of vector fields orbit of vector fields orbit of vector fields foliation foliation foliation foliation infinitesimal generator infinitesimal generator infinitesimal generator infinitesimal generator heat equation heat equation heat equation heat equation symmetry group.}
References
- [1] Azamov, A, Narmanov, A. : On the Limit Sets of Orbits of Systems of Vector Fields. Differential Equations. 40, (2), 257-260(2004). https://doi.org/10.1023/B:DIEQ.0000033716.96100.06
- [2] Narmanov, A, Saitova, S.: On the geometry of orbits of Killing vector fields. Differential Equations. 50,(12), 1584-1591(2014). https://doi.org/10.1134/S0374064114120024
- [3] Narmanov, A, Saitova, S.: On the geometry of the reachability set of vector fields. Differential Equations. 53, 311-316(2017). https://doi.org/10.1134/S001226611703003X
- [4] Narmanov, A, Rajabov, E. : On the Geometry of Orbits of Conformal Vector Fields. J. Geom. Symmetry Phys. 51, 29–39 (2019). https://doi.org/10.7546/jgsp-51-2019-29-39
- [5] Sussman, H.: Orbits of families of vector fields and integrability of distributions Transactions of the AMS. 180, 171–188 (1973).https://doi.org/10.1090/S0002-9947-1973-0321133-2
- [6] Narmanov, O.: Invariant Solutions of Two Dimensional Heat Equation. Journal of Applied Mathematics and Physics. 7, 1488–1497 (2019). https://doi.org/10.4236/jamp.2019.77100
- [7] Narmanov, O. : Invariant solutions of the two-dimensional heat equation. Bulletin of Udmurt University. Mathematics, Mechanics, ComputerScience. 29, (1), 52–60 (2019).https://doi.org/10.20537/vm190105
- [8] Dorodnitsyn, A, Knyazeva, I, Svirshchevskii,S. : Group properties of the heat equation with source in the two-dimensional and three-dimensionalcases. Differ. Uravn. 19 (7), 1215–1223 (1983).
- [9] Fomenko, V.: Classification of Two-Dimensional Surfaces with Zero Normal Torsion in Four-Dimensional Spaces of Constant Curvature. Math. Notes. 75,(5), 690 –701 (2004). https://doi.org/10.4213/mzm60
- [10] Fomenko, V. : Some properties of two-dimensional surfaces with zero normal torsion in E4, Sb. Math., 35,(2), 251–265 (1979). https://doi.org/10.1070/SM1979v035n02ABEH001473
- [11] Kadomcev, S. : A study of certain properties of normal torsion of a two-dimensional surface in four-dimensional space (Russian). Problems in geometry.Akad. Nauk SSSR Vsesojuz. Inst. Nauch. i Tehn. Informacii, Moscow, 7, pp. 267–278, (1975).
- [12] Olver, P : Applications of Lie Groups to Differential Equations. Springer. (1993).
- [13] Ramazanova, K. : The theory of curvature of X2 in E4. Izv. Vyssh. Uchebn. Zaved. Mat.6, 137–143 (1966).
Abstract
The geometry of orbits of families of smooth vector fields was studied by many mathematicians due
to its importance in applications in the theory of control systems, in dynamic systems, in geometry
and in the theory of foliations.
In this paper it is studied geometry of orbits of vector fields in four dimensional Euclidean space. It is shown that orbits generate
singular foliation every regular leaf of which is a surface of negative Gauss curvature and zero normal torsion.
In addition, the invariant functions of the considered vector fields are used to find solutions of the two-dimensional heat equation that are invariant under the groups of transformations generated by these vector fields.
Keywords
Vector field orbit of vector fields foliation infinitesimal generator heat equation symmetry group
References
- [1] Azamov, A, Narmanov, A. : On the Limit Sets of Orbits of Systems of Vector Fields. Differential Equations. 40, (2), 257-260(2004). https://doi.org/10.1023/B:DIEQ.0000033716.96100.06
- [2] Narmanov, A, Saitova, S.: On the geometry of orbits of Killing vector fields. Differential Equations. 50,(12), 1584-1591(2014). https://doi.org/10.1134/S0374064114120024
- [3] Narmanov, A, Saitova, S.: On the geometry of the reachability set of vector fields. Differential Equations. 53, 311-316(2017). https://doi.org/10.1134/S001226611703003X
- [4] Narmanov, A, Rajabov, E. : On the Geometry of Orbits of Conformal Vector Fields. J. Geom. Symmetry Phys. 51, 29–39 (2019). https://doi.org/10.7546/jgsp-51-2019-29-39
- [5] Sussman, H.: Orbits of families of vector fields and integrability of distributions Transactions of the AMS. 180, 171–188 (1973).https://doi.org/10.1090/S0002-9947-1973-0321133-2
- [6] Narmanov, O.: Invariant Solutions of Two Dimensional Heat Equation. Journal of Applied Mathematics and Physics. 7, 1488–1497 (2019). https://doi.org/10.4236/jamp.2019.77100
- [7] Narmanov, O. : Invariant solutions of the two-dimensional heat equation. Bulletin of Udmurt University. Mathematics, Mechanics, ComputerScience. 29, (1), 52–60 (2019).https://doi.org/10.20537/vm190105
- [8] Dorodnitsyn, A, Knyazeva, I, Svirshchevskii,S. : Group properties of the heat equation with source in the two-dimensional and three-dimensionalcases. Differ. Uravn. 19 (7), 1215–1223 (1983).
- [9] Fomenko, V.: Classification of Two-Dimensional Surfaces with Zero Normal Torsion in Four-Dimensional Spaces of Constant Curvature. Math. Notes. 75,(5), 690 –701 (2004). https://doi.org/10.4213/mzm60
- [10] Fomenko, V. : Some properties of two-dimensional surfaces with zero normal torsion in E4, Sb. Math., 35,(2), 251–265 (1979). https://doi.org/10.1070/SM1979v035n02ABEH001473
- [11] Kadomcev, S. : A study of certain properties of normal torsion of a two-dimensional surface in four-dimensional space (Russian). Problems in geometry.Akad. Nauk SSSR Vsesojuz. Inst. Nauch. i Tehn. Informacii, Moscow, 7, pp. 267–278, (1975).
- [12] Olver, P : Applications of Lie Groups to Differential Equations. Springer. (1993).
- [13] Ramazanova, K. : The theory of curvature of X2 in E4. Izv. Vyssh. Uchebn. Zaved. Mat.6, 137–143 (1966).
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Research Article |
| Authors | |
| Publication Date | April 30, 2023 |
| Acceptance Date | April 2, 2023 |
| Published in Issue | Year 2023 |