The Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   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 4:16 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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.If it would be possible, then there is
no need to develop non-euclidean geometry later on. So yes, with
later
non- euclidean metrics in R^3, after more advanced hyperbolic space
models were proposed and definition of corresponding geodesics
developed in H^2.The advent was some 18 centuries after Euclid's
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
Poincaré disk model, and the Poincaré half-plane model, in his paper
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_med.gif

Regards,
Narasimham



Date Subject Author
3/5/07
Read Can the Lobachevsky plane be embedded into R^3 ?
ksp4@msiu.ru
3/6/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
Hero
3/11/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
Thomas Mautsch
3/12/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
Hero
3/12/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
Denis Feldmann
3/12/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
Hero
3/15/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
Hero
3/15/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
Rainer Rosenthal
3/15/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
Hero
3/15/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
Denis Feldmann
3/15/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
Rainer Rosenthal
3/15/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
Hero
3/15/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
David Bernier
3/16/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
Hero
3/15/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
Denis Feldmann
3/15/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
Rainer Rosenthal
3/15/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
Hero
3/15/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
Denis Feldmann
3/15/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
David Bernier
3/16/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
narasimham
3/17/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
narasimham
3/17/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
narasimham
3/17/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
narasimham
3/18/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
Thomas Mautsch
3/28/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
narasimham
3/28/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?-
JEMebius
3/28/07
Read Re: Can the Lobachevsky plane be embedded into R^3 ?
Chan-Ho Suh

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2017. All Rights Reserved.