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Nil-manifolds cannot be immersed as hypersurfaces in Euclidean spaces. (English. Russian original) Zbl 1114.53051

Math. Notes 76, No. 6, 810-815 (2004); translation from Mat. Zametki 76, No. 6, 868-873 (2004).
Summary: We prove that the \(2n + 1\)-dimensional Heisenberg group \(H_n\) and the 4-manifolds \(\text{Nil}^4\) and \(\text{Nil}^3 \times \mathbb R\) 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.

MSC:

53C40 Global submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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