On the group of diffeomorphisms of foliated manifolds

Now the foliations theory is intensively developing branch of modern differential geometry, there are numerous researches on the foliation theory. The purpose of our paper is study the structure of the group $Diff_{F}(M)$ of diffeomorphisms and the group $Iso_{F}(M)$ of isometries of foliated manifold $(M,F)$. It is shown the group $Diff_{F}(M)$ is closed subgroup of the group $Diff(M)$ of diffeomorphisms of the manifold $M$ in compact-open topology and also it is proven the group $Iso_{F}(M)$ is Lie group. It is introduced new topology on $Diff_{F}(M)$ which depends on foliation $F$ and called $F$-compact open topology. It's proven that some subgroups of the group $Diff_F(M)$ are topological groups with $F$-compact open topology.