From: benzvi@noneuclid.geom.umn.edu (David Ben-zvi)
Subject: Re: Geometry - Euclidean/elliptic/hyperbolic.
Organization: Geometry Center, University of Minnesota
Date: Thu, 30 Jul 1992 19:03:14 GMT

In article <1992Jul30.172517.6127@csc.canterbury.ac.nz> wft@math.canterbury.ac.nz (Bill Taylor) writes:
>I have a small list here of simple characteristic differences between the
>three types of homogeneous geometry.
>
>If anyone can add further items of interest, I would be most grateful.
>---------------
>      Euclidean                               Elliptic            Hyperbolic
>      =========                               ========            ==========
>One parallel to a line through a given pt.   No parallels.      Many parallels. 
> 
>Triangle angles add to 180.                  More than 180.     Less than 180. 
> 
>Triangles have arbitrary area.               Upper limt to area.   <-- ditto
> 
>Whole space has infinite area.               Finite.            Infinite. 
> 
>Minimum distance between parallel lines      <inapplicable>     At a particular
> achieved by every point on either line.                        point on each.

The comment about Euclidean and Hyperbolic spaces having
infinite area is not true for all spaces having these kinds
of geometry. It is true of the model spaces for these geometries-
the simply connected (in these cases, contractible) homogeneous spaces
E^n and H^n (Euclidean and Hyperbolic n-space)-this is probably what you meant. 
Every euclidean/hyperbolic
manifold has E^n/H^n as its universal covering space. But for many purposes
euclidean/hyperbolic manifolds of finite volume such as the torus/genus g
surface (g>1) are the most interesting. (All of these can be represented
as the actions of some nice groups -Kleinian groups in the hyperbolic case,
pretty boring groups in the euclidean - acting on the universal cover.)

An interesting note about parallels in hyperbolic space - we can talk about lines
parallel to a given line, these being required to meet the other line at 
infinity (i.e. on the unit disc in the Poincare or Klein models). This corresponds
to our regular notion of parallels, meeting at the horizon. There are also 
lines that don't intersect a given line either in the finite or infinite part
of the hyperbolic space (i.e. not even asymptotically). These are called 
ultraparallels. Given two ultraparallel lines, there is a unique line perpendicular 
to both. If we think of our hyperbolic space as lying in projective space
(projective = Klein model), then every two ultraparallels meet at a point 
OUTSIDE infinity, and these points correpond to the perpendiculars we constructed.
Since projective space=disc+Mobius strip, we can say that there is a Mobius strip
outside infinity correponding to the perpendiculars (and thus intrinsically 
defined!) For higher dimensional hyperbolic spaces there is a simillarly
defined perpendicular hyperplane given two ultraparallels, and the set of these
hyperplanes form the higher dimensional analogues of Mobius strips
(= tautological or canonical line bundle over projective space P^(n-1)).
Thus we can truly say that there is a Mobius strip outside of infinity!

Final note - in your enumeration of homogeneous geometries you might want to include
Thurston's list of eight homogeneous geometries for three-manifolds.
The three you mentioned are the isotropic ones. Besides these
there are:
 H^2 x E geometry - it looks like the hyperbolic plane in one
	direction and like the Euclidean line in another.

S^2 x E : spherical in some directions, 1-D Euclidean looking "up".

Twisted H^2 x E, or PSL2(R) geometry: this looks like H^2xE except
	that it's twisted - in moving along the H^2 direction you slide
	down the E direction. (This is a natural geometry on the 
	universal cover of PSL2(R), the group of hyperbolic isometries)

Twisted Euclidean (or Nilgeometry) - same idea. there's a good picture of this
	as a "junglejim" in Jeff Weeks' book.

Solvegeometry - this is the really wierd one. Essentially no two
	directions look the same. It is made out of Euclidean planes, 
	but when you move up you get squished in one direction and expanded in
	another, viceversa for moving down. Geodesics look very strange-
	to get from one point to another in the same Euclidean plane, it's
	better to move up (say) so that lengths shrink, and then move back down.

(For a complete description of these geometries, see Thurston's preprint
"Three Dimensional Geometry and Topology". There's also an article
by Peter Scott, "The Geometries of 3-manifolds"  in the Bulletin of the London
Mathematical Society" 15, 1983, - I haven't read this).   


Hope this helps,
David Ben-Zvi (benzvi@geom.umn.edu)

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"The surface was invented by the devil" - Wolfgang Pauli