The hyperbolic n-space as a graph in Euclidean (6n-6)-space
Let \(H^ n\) denote the n-dimensional hyperbolic space of constant curvature (-1) and \(E^ N\) the N-dimensional Euclidean space. \textit{D. Blanusa} [Monatsh. Math. 59, 217-229 (1955; Zbl 0067.144)] constructed an isometric \(C^{\infty}\)-imbedding \(H^ 2\to E^ 6\) whose image is the graph of a \(C^{\infty}\)-map \({\mathbb{R}}^ 2\to {\mathbb{R}}^ 4\). For \(n\geq 3\), in the same article, Blanusa was only able to construct a 1-1 isometric \(C^{\infty}\)-immersion \(H^ n\to E^{6n-5}\) which is not an imbedding in the strong sense (i.e. not a homeomorphism onto a topological subspace). The present paper generalizes the stronger 2- dimensional result of Blanusa. Theorem: For each \(n\geq 2\), there exists an isometric \(C^{\infty}\)-imbedding \(H^ n\to E^{6n-6}\) whose image is the graph of a C-map \({\mathbb{R}}^ n\to {\mathbb{R}}^{5n-6}\). Moreover explicit formulas are given which apply to the isometric imbedding problem for a larger class of Riemannian manifolds.
scientific article; zbMATH DE number 3990294