On equivariant isometric embeddings of Riemannian manifolds with symmetries

Abstract

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