The hyperbolic n-space as a graph in Euclidean (6n−6)-space

Nettekoven, Wolfgang and Henke, Wolfgang (1987) The hyperbolic n-space as a graph in Euclidean (6n−6)-space.
Published in: Manuscripta mathematica Vol. 59 (1). pp. 13-20.


Let Hsp n denote the n-dimensional hyperbolic space of constant curvature (-1) and Esp N the N-dimensional Euclidean space. {it D. Blanusa} [Monatsh. Math. 59, 217-229 (1955; Zbl. 67, 144)] constructed an isometric Csp{infty}-imbedding Hsp 2 o Esp 6 whose image is the graph of a Csp{infty}-map {bfR}sp 2 o {bfR}sp 4. For nge 3, in the same article, Blanusa was only able to construct a 1-1 isometric Csp{infty}-immersion Hsp n o Esp{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 nge 2, there exists an isometric Csp{infty}-imbedding Hsp n o Esp{6n-6} whose image is the graph of a C-map {bfR}sp n o {bfR}sp{5n-6}. Moreover explicit formulas are given which apply to the isometric imbedding problem for a larger class of Riemannian manifolds.

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