Re: Can the Lobachevsky plane be embedded into R^3 ?-

ksp4@xxxxxxx 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?


"Can the Lobachevsky plane be embedded into R^3 ?" Locally: yes; globally: no.

The hyperbolic plane has constant negative total curvature, i.e. it has the same negative total curvature at each point (i.e. the reciprocal of the principal radii of curvature).

The pseudosphere ( http://en.wikipedia.org/wiki/Pseudosphere ) is a surface in R^3 with constant negative curvature. Therefore it is locally isometric with the hyperbolic plane.
The proof is a standard subject in 19th-century differential geometry.
However, the pseudosphere has a singularity consisting of a cuspoidal circle, and furthermore it is mapped one-to-one onto part of hyperbolic plane only.

See also

http://en.wikipedia.org/wiki/Hyperbolic_geometry
http://en.wikipedia.org/wiki/Tractrix
http://en.wikipedia.org/wiki/William_Thurston
http://www.math.princeton.edu/Thurston60th/index.html
(Conference "Geometry and the Imagination" June 7th - 11th 2007 Princeton University)


Johan E. Mebius
.