Equivariant embeddings of manifolds into Euclidean spaces

Suppose a finite group $G$ acts on a manifold $M$. By a theorem of Mostow, also Palais, there is a $G$-equivariant embedding of $M$ into the $m$-dimensional Euclidean space ${\text{RR}}^m$ for some $m$. We are interested in some explicit bounds of such $m$. First we provide an upper bound: there exists a $G$-equivariant embedding of $M$ into ${\text{RR}}^{d\mathrm{\mid }G\mathrm{\mid }+1}$, where $\mathrm{\mid }G\mathrm{\mid }$ is the order of $G$ and $M$ embeds into ${\text{RR}}^d$. Next we provide a lower bound for finite cyclic group action $G$: If there are $l$ points having pairwise co-prime lengths of $G$-orbits greater than $1$ and there is a $G$-equivariant embedding of $M$ into ${\text{RR}}^m$, then $m\ge 2l$. Some applications to surfaces are given.