Suppose that $M$ is a smooth manifold of dimension $k$ embedded in $\mathbb{R}^n$. Are there any results about when one can find another smooth manifold $M'$ of dimension $k'$ embedded in $\mathbb{R}^{n'}$ such that there is a global diffeomorphism between an open subset of $\mathbb{R}^{k+k'}$ and $M\times M'$?
For instance, there is no global diffeomorphism between the sphere $\mathbb{S}^2$ and $\mathbb{R}^3$ (they aren't even the same dimension). However, if we consider the product $\mathbb{S}^2\times\mathbb{R}_+$ then we have a global diffeomorphism with $\mathbb{R}^3\setminus \left\{0\right\}$ in the form spherical coordinates.