Diffeomorphisms of foliated manifold 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 in analysis. It is known that the group $Diff(M)$ is topological group in compact open topology. In this paper we investigate the group DiffF(M) of diffeomorphisms foliated manifold $(M,F)$ with foliated compact open topology. It is proven in {Narmanov2021} that foliated compact open topology of the group DiffF(M) has a countable base and the group DiffF(M) is topological group 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 DiffF(M) coincides with compact open topology. In addition it is studied the question on the dimension of the group of isometries of foliated manifold when foliation generated by riemannian submersion.