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Can a complete hyperbolic surface be isometrically embedded into flat $\mathbb R^4$?

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It is known that the hyperbolic plane can be isometrically immersed in $E^5$ (and isometrically embedded in $E^6$). Existence of an isometric immersion into $E^4$ is an open problem. Thus, the answer to your question is "unknown".

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  • $\begingroup$ Thanks you studiosus. Can you tell me where to find some information about this open problem? And about the embedding into 6-space? $\endgroup$ – Hyperbolic Asker Nov 14 '15 at 4:39
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    $\begingroup$ @HyperbolicAsker: davidbrander.org/penn.pdf as well as "Isometric Embedding of Riemannian Manifolds in Euclidean Spaces", By Qing Han, Jia-Xing Hong $\endgroup$ – Moishe Kohan Nov 14 '15 at 4:48

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