On the group of isometries of foliated manifold

The question of the group of isometries of a Riemannian manifold is the main problem of the classical Riemannian geometry. Let $G$ denote the group of isometries of a Riemannian manifold $M$ of dimension $n$ with a Riemannian metric $g$. It is known that the group $G$ with the compact-open topology is a Lie group. This paper discusses the question of the existence of isometric maps of the foliated manifold $(M,F)$. We denote the group of all isometries of the foliated Riemannian manifold $(M,F)$ by $G_F$. Studying the structure of the group $G_F$ of the foliated manifold $(M,F)$ is a new and interesting problem. First, this problem is considered in the paper of A. Y. Narmanov and the author, where it was shown that the group $G_F$ with a compact-open topology is a topological group. We consider the question of the structure of the group $G_F$, where $M=R^n$ and $F$ is foliation generated by the connected components of the level surfaces of the smooth function $ f\colon R^n\to R$. It is proved that the group of isometries of foliated Euclidean space is a subgroup of the isometry group of Euclidean space, if the foliation is generated by the level surfaces of a smooth function, which is not a metric.