On the geometry of Riemannian isometric embeddings

This note pertains to isometric embeddings endowed with certain geometric properties. We study two embedding problems for a Riemannian manifold $M$ which is diffeomorphic to ${\text{RR}}^n$ and admits a Bieberbach group $\mathrm{\Gamma }$ acting by isometries. The first problem concerns the existence of an isometric embedding of $M$ into a bounded subset of some Euclidean space ${\text{RR}}^{D_1}$. The second problem seeks a $\mathrm{\Gamma }$-equivariant isometric embdding of $M$ into ${\text{RR}}^{D_2}$. By using a known trick in a novel way, our idea yields results with $D_1=N+2n$ and $D_2=N+n$, where $N$ is the Nash dimension of $M\mathrm{/}\mathrm{\Gamma }$. Moreover, we also show that an $n$-dimensional smooth manifold, of Nash dimension $N$, can be isometrically embedded into a bounded subset of ${\text{RR}}^{2N}$.