5
$\begingroup$

I'm researching the isometric dips of the hyperbolic plane and in particular I'm interested in reading the results of Rozendorn who proved that the hyperbolic plane is isometrically immersed in $\mathbb R^5$. The demonstrations that I have been able to find in this regard, for example in Gromov's book, do not go with Rozendorn's initial idea, I say it because he demonstrates it in: enter image description here but I have not been able to locate that article anywhere. Does anyone know where I can find it? or maybe someone knows other results in the same line of the original demonstration of him?

From what I have been able to read, the proof that Rozendorn gives is based entirely on the functions that Blanusa gives in this Article in which he proved that the hyperbolic plane is isometrically embedded in $\mathbb R^6$.

$\endgroup$
4
  • 1
    $\begingroup$ You might wish to write to MSU. Rozendorn (who passed away in 2012) worked there for nearly 50 years, so if anyone has his papers, it'll be them. $\endgroup$ Sep 1, 2021 at 8:30
  • $\begingroup$ The MathSciNet review for the article has an explicit embedding written up, which might give you some idea. Also, the article is only 2 pages long, and in Dokladu, so there might not be a proof in the article. $\endgroup$ Sep 1, 2021 at 8:54
  • $\begingroup$ Crossposted on MSE. $\endgroup$ Sep 1, 2021 at 10:42
  • 2
    $\begingroup$ This journal is not available on Internet. If you have an access to a university library, you can request it by Interlibrary Loan. It will be in Russian. $\endgroup$ Sep 1, 2021 at 12:26

2 Answers 2

3
$\begingroup$

If you don't mind looking at the original Russian version, then the scanned article itself is available at https://arar.sci.am//Content/23775/file_0.pdf

I found it by searching https://www.google.com/search?q=розендорн+реализация+метрики which shows also some Russian citations of that work.

Update. Perhaps even better, you could read more about this in his later review "Surfaces of Negative Curvature" https://link.springer.com/chapter/10.1007%2F978-3-662-02751-6_2

(translated from Э. Р. Розендорн, Поверхности отрицательной кривизны, Итоги науки и техн. Сер. Соврем. пробл. мат. Фундам. направления, 1989, том 48, 98–195, http://www.mathnet.ru/links/ac4e964b7708b97721d5f6525b93dc36/intf145.pdf section 5.2, p. 180~)

$\endgroup$
5
$\begingroup$

If you don't read Russian: Rozendorn's construction is presented by Aminov in Extrinsic geometric properties of the Rozendorn surface, which is an isometric immersion of the Lobachevskiĭ plane into $E^5$. Now Aminov's paper was also in Russian, but the article does have an English translation available, which you can find at https://iopscience.iop.org/article/10.1070/SM2009v200n11ABEH004051. On pages 1576-1578 there's a pretty complete reproduction of Rozendorn's arguments.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy