
Re: Can the Lobachevsky plane be embedded into R^3 ?
Posted:
Mar 17, 2007 4:16 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.If it would be possible, then there is no need to develop noneuclidean 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 nonEuclidean 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 PoincarĂ© disk model, and the PoincarĂ© halfplane 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.fuberlin.de/~polthier/Calendar/Kalender86/December86_med.gif
Regards, Narasimham

