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## Archive of Issues

Uzbekistan Tashkent
Year
2020
Volume
30
Issue
1
Pages
49-58
 Section Mathematics Title On the group of diffeomorphisms of foliated manifolds Author(-s) Narmanov A.Ya.a, Zoyidov A.N.a Affiliations National University of Uzbekistana Abstract Now the foliations theory is intensively developing branch of modern differential geometry, there are numerous researches on the foliation theory. The purpose of our paper is study the structure of the group $Diff_{F}(M)$ of diffeomorphisms and the group $Iso_{F}(M)$ of isometries of foliated manifold $(M,F)$. It is shown the group $Diff_{F}(M)$ is closed subgroup of the group $Diff(M)$ of diffeomorphisms of the manifold $M$ in compact-open topology and also it is proven the group $Iso_{F}(M)$ is Lie group. It is introduced new topology on $Diff_{F}(M)$ which depends on foliation $F$ and called $F$- compact open topology. It's proven that some subgroups of the group $Diff_F(M)$ are topological groups with $F$-compact open topology. Keywords manifold, foliation, group of diffeomorphisms, compact open topology UDC 517.977 MSC 22A05, 54H15, 57R50 DOI 10.35634/vm200104 Received 1 February 2020 Language English Citation Narmanov A.Ya., Zoyidov A.N. On the group of diffeomorphisms of foliated manifolds, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 1, pp. 49-58. References Arnold V. Sur la geometrie differentielle des groupes de Lie de dimension infinite et ses applications a l'hidrodynamique des fluides parfaits, Annales de l'Institut Fourier, 1966, vol. 16, pp. 318-361. https://doi.org/10.5802/aif.233 Chevalley C. Theory of Lie groups. I, Princeton: Princeton University press, 1966. https://archive.org/details/in.ernet.dli.2015.86469/page/n7/mode/2up Gromoll D., Klingenberg W., Meyer W. Riemannsche Geometrie im Grossen, Berlin: Springer, 1968. http://dx.doi.org/10.1007/978-3-540-35901-2 Helgason S. Differential geometry, Lie groups and symmetric spaces, Toronto: Academic Press, 1978. Hirsch M.W. Differential topology, New York: Springer, 1976. Lukatskii A.M. Finite generation of groups of diffeomorphisms, Russian Mathematical Surveys, 1978, vol. 33, no. 1, pp. 207-261. https://doi.org/10.1070/RM1978v033n01ABEH002248 Lukatsky A.M. Investigation of the geodesic flow on an infinite-dimensional Lie group by means of the coadjoint action operator, Proceedings of the Steklov Institute of Mathematics, 2009, vol. 267, pp. 195-204. https://doi.org/10.1134/S0081543809040166 Molino P. Riemannian foliations, Boston-Basel: Burkhauser, 1988. Myers S.B., Steenrod N.E. The group of isometries of a Riemannian manifold, Annals of Mathematics. Second Series, 1939, vol. 40, pp. 400-416. https://doi.org/https://doi.org/10.2307/1968928 Narmanov A.Ya., Saitova S.S. On geometry of vector fields, Journal of Mathematical Sciences, 2020, vol. 245, pp. 375-381. https://doi.org/10.1007/s10958-020-04699-z Narmanov A.Y. Geometry of orbits of vector fields and singular foliations, Contemporary Mathematics. Fundamental Directions, 2019, vol. 65, issue 1, pp. 54-71 (in Russian). https://doi.org/10.22363/2413-3639-2019-65-1-54-71 Narmanov A., Rajabov E. On the geometry of orbits of conformal vector fields, Journal of Geometry and Symmetry in Physics, 2019, vol. 51, pp. 29-39. https://doi.org/10.7546/jgsp-51-2019-29-39 Narmanov A., Sharipov A. On the group of foliation isometries, Methods of Functional Analysis and Topology, 2009, vol. 15, pp. 195-209. http://mfat.imath.kiev.ua/article/?id=435 Narmanov A.Ya., Saitova S.S. On the geometry of orbits of Killing vector fields, Differential Equations, 2014, vol. 50, pp. 1584-1591. https://doi.org/10.1134/S0012266114120027 Omori H. On the group of diffeomorphisms on a compact manifold, Proc. Symp. Pure Math., 1970, vol. 15, pp. 167-183. https://mathscinet.ams.org/mathscinet-getitem?mr=0271983 Omori H. Group of diffeomorphisms and thier subgroups, Trans. Amer. Math. Soc., 1973, vol. 179, pp. 85-122. https://doi.org/10.1090/S0002-9947-1973-0377975-0 Postnikov M.M. Lectures on geometry. Semester V. Lie groups and Lie algebras, Moscow: Nauka, 1982. https://archive.org/details/postnikovliegroups/page/n5/mode/2up Rokhlin V.A., Fuks D.B. Initial course of topology. Geometrical chapters, Moscow: Mir, 1977. Tamura I. Topology of foliations: An introduction, Providence, Rhode Island: American Mathematical Society, 1992. Tondeur Ph. Foliations on Riemannian manifolds, New York: Springer, 1988. Full text