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 Diff₅ (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 Diff₅ (M) has a countable base and the group Diff₅ (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 Diff₅ (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.