On equivariant isometric embeddings of Riemannian manifolds with symmetries

Let $(M,g)$ be a $C^{\mathrm{\infty }}$-smooth, $n$-dimensional Riemannian manifold which is diffeomorphic to ${\text{RR}}^n$ and admit an action of a properly discontinuous and cocompact group. This work proves the existence of a $C^{\mathrm{\infty }}$ equivariant isometric embedding of $M$ in some Euclidean space ${\text{RR}}^q$ where $q=\max \{s_n+2n,s_n+n+5\}$ is the same as the dimension of Matthias Günther’s results.