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

"RichD" <r_delaney2001@yahoo.com> wrote in messagenews:8b41d76d-051f-4cc0-9c23-4acd51aea047@gh3g2000pbd.googlegroups.com...> On Apr 23, Nicolas Bonneel <nicolas.bonn...@wanadooooo.fr.invalid>> 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 flatcould mean:a)  Flat when viewed in some higher dimensional space within which thesurface is embedded.orb)  Intrinsically flat, i.e. flat in terms of measurements purely within thesurface.The surface of a cylinder illustrates the difference: it is flat in thesense of (b), but not (a).  Local measurements purely within the surfacewould not show any curvature - in fact we can imagine flattening outindividual areas to look at (like flattening out a rolled up map), and it isonly in the large-scale topology of the space that we see it isdistinguishable from an infinite plane.The surface of a sphere, on the other hand is not flat in either sense.  Thenon-flatness can be determined intrinsically, e.g. by looking at circlesmeasured within the sphere surface, and comparing the circumference/radiusratio for the circles.  As the circles get bigger we see growing deviationsfrom the value of Pi which we would expect if the space were flat.  (In thecylinder 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 generallyno suggestion (or any requirement) of the space being embedded in any higherdimension space...Within a "cylinder" space, we may follow a geodesic path and return to thestarting point, and yet the space is flat everywhere, which is what othershave been explaining.  A "donut" space is a "cylinder" space with a finiteaxis length, but with the ends of the cylinder "joined up" (identified), andas with an infinite cylinder space, the donut space is everywhere flat andunbounded.  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 asbeing embedded in any higher dimension space here.  We are considering justthe surfaces themselves, along with their intrinsic measurement structures.Just as with 2d surfaces, 3d space and 4d space-time have higher dimensionalanalogues, which are flat, and closed but unbounded.  So the fact thatgeodesics "loop back" upon themselves does not imply the space is curved.Regards,Mike.
```