On the isometry groups of foliated manifolds

In this paper, we study the isometry group $\mathrm{Iso}_{F}(M)$ of a foliated manifold with an $F$-compact-open topology. This topology depends on the foliation $F$ and coincides with the compact-open topology if $F$ is an $n$-dimensional foliation. If the codimension of the foliation is equal to $n$, then the convergence in this topology coincides with the pointwise convergence. Some properties of the group $\mathrm{Iso}_F(M)$ are proved.