Diffeomorphisms of Foliated Manifolds I

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.