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Topic: Can the Lobachevsky plane be embedded into R^3 ?
Replies: 26   Last Post: Mar 28, 2007 4:03 PM

 Messages: [ Previous | Next ]
 narasimham Posts: 2,537 Registered: 12/6/04
Re: Can the Lobachevsky plane be embedded into R^3 ?
Posted: Mar 17, 2007 3:47 AM

On Mar 6, 2:07 am, k...@msiu.ru wrote:

> Can the Lobachevsky plane be embedded into R^3 ?
> (The Lobachevsky metric must be induced by
> the standard eucledean metric in R^3).
> I have heard that the answer is "no". If so, anybody knows what are
> the main ideas of the proof?

No, with the euclidean metric, but yes, with later non-euclidean
metrics in R^3, after more advanced/general hyperbolic space models
were developed and proposed with definition of corresponding geodesics
after Euclid's euclidean metric ds^2 = dx^2 + dy^2.

In 1868, (in Essay on an interpretation of non-Euclidean geometry)
Beltrami gave the first model of hyperbolic geometry. In Beltrami's
model, lines of hyperbolic geometry are represented by geodesics on
the pseudosphere. Thus, Beltrami attempted to prove that Euclid's
parallel postulate could not be derived from the other axioms of
Euclidean geometry; this proof fails however since the pseudosphere
is only a small portion of the hyperbolic plane.

In the same year, however, Beltrami went much farther and gave a
correct proof of the equiconsistency of hyperbolic and Euclidean
geometry, by defining what are now known as the Klein model, the
Teoria fondamentale degli spazii di curvatura costante. For the half-
plane model, Beltrami cited a note by Liouville in a book by Monge on
differential geometry.

The above is from:
http://en.wikipedia.org/wiki/Eugenio_Beltrami

Asymptotic lines which can be defined in hyperbolic geometry models
as geodesics on a pseudosphere are seen in Konrad Polthier's
beautiful 1986 calender pictures:

http://page.mi.fu-berlin.de/polthier/Calendar/Kalender86/December86_m...

Regards,
Narasimham

Date Subject Author
3/5/07 ksp4@msiu.ru
3/6/07 Hero
3/11/07 Thomas Mautsch
3/12/07 Hero
3/12/07 Denis Feldmann
3/12/07 Hero
3/15/07 Hero
3/15/07 Rainer Rosenthal
3/15/07 Hero
3/15/07 Denis Feldmann
3/15/07 Rainer Rosenthal
3/15/07 Hero
3/15/07 David Bernier
3/16/07 Hero
3/15/07 Denis Feldmann
3/15/07 Rainer Rosenthal
3/15/07 Hero
3/15/07 Denis Feldmann
3/15/07 David Bernier
3/16/07 narasimham
3/17/07 narasimham
3/17/07 narasimham
3/17/07 narasimham
3/18/07 Thomas Mautsch
3/28/07 narasimham
3/28/07 JEMebius
3/28/07 Chan-Ho Suh