Diffeomorphisms of Foliated Manifolds I

Year 2024, Volume: 17 Issue: 2, 531 - 537

Narmanov Abdugappar Yakubovich , Guzal Abdishukurova

https://doi.org/10.36890/iejg.1441930

Abstract

The set $Diff(M)$ of all diffeomorphisms of 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 analysis. It is known that the group $Diff(M)$ is topological group in compact open topology.In this paper we investigate the group $Diff_{F}(M)$ of diffeomorphisms foliated manifold $(M,F)$ with foliated compact open topology.

In this paper we prove that if all leaves of the the foliation $F$ are closed subsets of $M$ then the foliated compact open topology of the group $Diff_{F}(M)$ coincides with compact open topology. In addition it is studied the question on the dimension of the group of isometries of foliated manifold is studied when foliation generated by riemannian submersion.

Keywords

Foliated manifold, submersion, foliated compact open topology, diffeomorphisms of foliated manifold, Lie group

References

• [1] Abdishukurova G., Narmanov.A.: Diffeomorphisms of Foliated Manifolds, Methods Funct. Anal. Topology, 27(1),1–9 (2021).
• [2] Azamov A., Narmanov A.: On the Limit Sets of Orbits of Systems of Vector Fields, Differential Equations, 40 (2), 271-275 (2004).
• [3] Hermann R., A Sufficient Condition That a Mapping of Riemannian Manifolds To Be a Fiber bundle, Proc. Amer. Math. Soc., 11(4), 236–242 (1960).
• [4] Kobayashi Sh. and Nomizu K., Foundations of Differential Geometry,New York- London, Interscience 1963. [5] Molino P., Riemannian Foliations, Burkhauser, Boston 1988.
• [6] Narmanov A., Sharipov A., On the Group of Foliation Isometries, Methods Funct. Anal. Topology, 15,(2),195–200 (2009).
• [7] Narmanov A., Zoyidov A., On the Group of Diffeomorphisms of Foliated Manifolds, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Kompyuternye Nauki, 30(1),49–58 (2020).
• [8] Narmanov A., Tursunov B., Geometry of Submersions on Manifolds of Nonnegative Curvature, Mathematica Aeterna, 5 (1),169 – 174 (2015).
• [9] Narmanov A. and Abdushukurova G., On the Geometry of Riemannian Submersions, Uzbek Mathematical journal, 2,3–8 (2016).
• [10] Reinhart B., Foliated Manifolds With Bundle-like Metrics, Annals of Mathematics, Second Series 69, 119–132 (1959).
• [11] O’Neill B., The Fundamental Equations of Submersions, Michigan Mathematical Journal 13 459–469 (1966).