Geodesic
Nil-Manifolds Cannot be Immersed as Hypersurfaces in Euclidean Spaces
Matematičeskie zametki, Tome 76 (2004) no. 6, pp. 868-873

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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. 
@article{MZM_2004_76_6_a6,
     author = {L. A. Masal'tsev},
     title = {Nil-Manifolds {Cannot} be {Immersed} as {Hypersurfaces} in {Euclidean} {Spaces}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {868--873},
     publisher = {mathdoc},
     volume = {76},
     number = {6},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_76_6_a6/}
}
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L. A. Masal'tsev. Nil-Manifolds Cannot be Immersed as Hypersurfaces in Euclidean Spaces. Matematičeskie zametki, Tome 76 (2004) no. 6, pp. 868-873. http://geodesic.mathdoc.fr/item/MZM_2004_76_6_a6/