On isometric immersion of nilmanifolds in Euclidean space
Let \(M\) be an \(n\)-dimensional Riemannian manifold and denote by \(K\) the sectional curvature function on \(M\). Assume that there exists a direct sum decomposition \(TM = {\mathcal Z} \oplus {\mathcal V}\) of the tangent bundle \(TM\) of \(M\) such that (i) \(K(X,Y) < 0\) for mutually orthogonal unit vectors \(X,Y \in {\mathcal V}\), (ii) \(K(X,Z) > 0\) for all unit vectors \(X \in {\mathcal V}\) and \(Z \in {\mathcal Z}\), (iii) \(K(Z_1,Z_2) \leq 0\) for orthogonal unit vectors \(Z_1,Z_2 \in {\mathcal Z}\). Denote by \(n_1\) and \(n_2\) the rank of \({\mathcal Z}\) and \({\mathcal V}\) respectively. The author proves that there exists no regular isometric \(C^2\) immersion from \(M\) into the Euclidean space \({\mathbb E}^{n+p}\) with \(p < \max\{n_1,n_2\}\). As a corollary this implies that there exists no regular isometric immersion from the \((2n+1)\)-dimensional Heisenberg group with a left-invariant Riemanian metric into \({\mathbb E}^{2n+1 + p}\) with \(p < 2n\).
scientific article; zbMATH DE number 5791027