I'm researching the Hilbert's Theorem.- There is no isometric immersion of the hyperbolic plane $\mathbb{H}^2$ in $\mathbb{R}^3$. But on the way I have come across several results, such as:

Efimov's theorem.- There is no isometric immersion of a complete surface with Gaussian curvature bounded superiorly by a negative constant, at $\mathbb{R}^3$.

Then the question arose, where is it possible to have an isometric immersion of $\mathbb{H}^2$? I read that there is an isometric immersion in $\mathbb{R}^5$, but I only got this result,

Blanusa's theorem.- There is an isometric embedding proper to the hyperbolic plane $\mathbb{H}^2$ in $\mathbb{R}^6$.

From Efimov's theorem I have managed to find references to read, but from Blanusa's Theorem or other similar ones I have not obtained much information. If someone knows the topics and could give me some references or perhaps other theorems that can enrich my research, I would appreciate it very much.

Some documents i have found:

  • D. Brander - Isometric Embeddings between Space Forms
  • T. Milnor - Efimov's theorem about complete immersed surface of negative curvature
  • Do Carmo - Differential geometry
  • And others Here
  • $\begingroup$ Have you seen already this thesis? $\endgroup$ Jun 10, 2021 at 18:52
  • $\begingroup$ Yes, I mentioned it on the list. $\endgroup$
    – Zaragosa
    Jun 10, 2021 at 19:20

1 Answer 1


It is proven by Rozendorn that ${\mathbb H}^2$ admits an isometric immersion in ${\mathbb E}^5$. See Prop. 2.2.5 in

Han, Qing; Hong, Jia-Xing, Isometric embedding of Riemannian manifolds in Euclidean spaces, Mathematical Surveys and Monographs 130. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-4071-1/hbk). xiv, 260 p. (2006). ZBL1113.53002.

This book will contain proofs of most of the known results, assuming that your target space is Euclidean. For non-Euclidean targets, you have to ask a more focused question.


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