Date: Apr 29, 2013 4:48 PM
Author: Mike Terry
Subject: Re: closed universe, flat space?

"RichD" <> wrote in message
> On Apr 23, Nicolas Bonneel <>
> wrote:

> > > Supposedly, our universe is closed and finite, although the
> > > geometry and topology isn't precisely known.
> > > This means a straight line (geodesic) traveler
> > > must return to his starting
> > > poiint, yes/no? Hence, curved space.

> >
> > no - a flat torus... is flat, closed and finite,
> > and a traveler returns to his starting point.

> ?
> I'm a lifetime donut connoisseur, but have
> yet to see a flat one.

There are different uses of the term "flat". E.g. for a 2-d surface flat
could mean:

a) Flat when viewed in some higher dimensional space within which the
surface is embedded.


b) Intrinsically flat, i.e. flat in terms of measurements purely within the

The surface of a cylinder illustrates the difference: it is flat in the
sense of (b), but not (a). Local measurements purely within the surface
would not show any curvature - in fact we can imagine flattening out
individual areas to look at (like flattening out a rolled up map), and it is
only in the large-scale topology of the space that we see it is
distinguishable from an infinite plane.

The surface of a sphere, on the other hand is not flat in either sense. The
non-flatness can be determined intrinsically, e.g. by looking at circles
measured within the sphere surface, and comparing the circumference/radius
ratio for the circles. As the circles get bigger we see growing deviations
from the value of Pi which we would expect if the space were flat. (In the
cylinder surface example, no such deviations emerge.)

Geodesics are an intrinsic property of the surface, and when people talk of
"space being flat" they are using the sense (b) above. There is generally
no suggestion (or any requirement) of the space being embedded in any higher
dimension space...

Within a "cylinder" space, we may follow a geodesic path and return to the
starting point, and yet the space is flat everywhere, which is what others
have been explaining. A "donut" space is a "cylinder" space with a finite
axis length, but with the ends of the cylinder "joined up" (identified), and
as with an infinite cylinder space, the donut space is everywhere flat and
unbounded. We could also call the donut space a closed space, although
"compact" is a more accurate term.

Just to be clear, we are not considering these cylinder and donut spaces as
being embedded in any higher dimension space here. We are considering just
the surfaces themselves, along with their intrinsic measurement structures.
Just as with 2d surfaces, 3d space and 4d space-time have higher dimensional
analogues, which are flat, and closed but unbounded. So the fact that
geodesics "loop back" upon themselves does not imply the space is curved.