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Diffeomorphisms of foliated manifolds


Abstract

The set $\rm{Diff}(M)$ of all diffeomorphisms of a manifold $M$ onto itself is the group related to composition and inverse mapping. The group of diffeomorphisms of smooth manifolds is of great importance in differential geometry and in analysis. It is known that the group $\rm{Diff}(M)$ is a topological group in compact open topology. In this paper we investigate the group $\rm{Diff}_{F}(M)$ of diffeomorphisms foliated manifold $(M,F)$ with foliated compact open topology. It is proven that foliated compact open topology of the group $\rm{Diff}_{F}(M)$ has a countable base. It is also proven that the group $\rm{Diff}_{F}(M)$ is a topological group with foliated compact open topology. Also some one-parameter subgroups of the group $\rm{Diff}_{F}(M)$ are found and studied for the foliations generated by special submersions.

Множина $\rm{Diff}(M )$ всіх дифеоморфізмів многовиду $M$ є групою відносно композиції та взяття оберненого і топологічною групою в компактно-відкритій топології. Групи дифеоморфізмів гладких многовидів мають велике значення в диференціальній геометрії та аналізі. У цій роботі досліджується група дифеоморфізмів шаруватого многовиду з розшарованою компактно-відкритою топологією. Показано, що ця топологія має зліченну базу. Знайдені деякі однопараметричні підгрупи групи $\rm{Diff}(M )$ і досліджені для шарувань, породжених спеціальними субмерсіями.

Key words: Foliated manifold, submersion, foliated compact open topology, diffeomorphisms of foliated manifold, Lie group.


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Article Information

TitleDiffeomorphisms of foliated manifolds
SourceMethods Funct. Anal. Topology, Vol. 27 (2021), no. 1, 1-9
DOI10.31392/MFAT-npu26_1.2021.01
MathSciNet   MR4252992
Milestones  Received 17/08/2020; Revised 02/09/2020
CopyrightThe Author(s) 2021 (CC BY-SA)

Authors Information

G. M. Abdishukurova
Faculty of Mathematics, National University of Uzbekistan, Tashkent, 100174, Tashkent, Uzbekistan

A. Ya. Narmanov
Faculty of Mathematics, National University of Uzbekistan, Tashkent, 100174, Tashkent, Uzbekistan


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Citation Example

G. M. Abdishukurova and A. Ya. Narmanov, Diffeomorphisms of foliated manifolds, Methods Funct. Anal. Topology 27 (2021), no. 1, 1-9.


BibTex

@article {MFAT1500,
    AUTHOR = {G. M. Abdishukurova and A. Ya. Narmanov},
     TITLE = {Diffeomorphisms of foliated manifolds},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {27},
      YEAR = {2021},
    NUMBER = {1},
     PAGES = {1-9},
      ISSN = {1029-3531},
  MRNUMBER = {MR4252992},
       DOI = {10.31392/MFAT-npu26_1.2021.01},
       URL = {http://mfat.imath.kiev.ua/article/?id=1500},
}


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