Geodesic
Nil-Manifolds Cannot be Immersed as Hypersurfaces in Euclidean Spaces
Matematičeskie zametki, Tome 76 (2004) no. 6, pp. 868-873 
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/
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     author = {L. A. Masal'tsev},
     title = {Nil-Manifolds {Cannot} be {Immersed} as {Hypersurfaces} in {Euclidean} {Spaces}},
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     pages = {868--873},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_76_6_a6/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Rivertz H. J., An obstruction to isometric immersion of the threedimensional Heisenberg group into $\mathbb R^4$, Preprint series in Pure Mathematics No. 22, Matematick institut Univ. Oslo, Oslo, 1999

[2] Gromov M., Differentsialnye sootnosheniya s chastnymi proizvodnymi, Mir, M., 1990

[3] Hillman J. A., Four-manifolds, Geometries and Knots, Geometry and Topology Monographs, 5, University of Sydney, 2002 | MR

[4] Kobayasi Sh., Nomidzu K., Osnovy differentsialnoi geometrii, T. 2, Nauka, M., 1981

[5] Gordon C., “Riemannian manifolds isospectral on functions but not on $1$-forms”, J. Differential Geometry, 24 (1986), 79–96 | MR | Zbl

[6] Besse A., Mnogoobraziya Einshteina, Mir, M., 1990 | Zbl

[7] Aminov Yu. A., Geometriya podmnogoobrazii, Naukova dumka, Kiev, 2002 | Zbl