Nil-Manifolds Cannot be Immersed as Hypersurfaces in Euclidean Spaces

We prove that the $2n+1$-dimensional Heisenberg group $H_n$ and the $4$-manifolds $\operatorname{Nil}^4$ and $\operatorname{Nil}^3\times\mathbb R$ endowed with an arbitrary left-invariant metric admit no $C^3$-regular immersions into Euclidean spaces $\mathbb R^{2n+2}$ and $\mathbb R^5$, respectively.