My question is if a torus admits coordinates which guarantuee isotropy. Background: In cosmology one usually assumes homogenity and isotropy of the universe. These principles are respected by most prominent cosmological models. Now what about the torus universe? Is there a choice of coordinates for which isotropy is guarantueed? Using standard torus coordinates via embedding obviously violates isotropy. Using a square with opposite edges identified and cartesian coordinates on this square violates isotropy as well. My feeling is that isotropy is always violated, i.e. that the torus topology does not allow for a geometry which respects isotropy. My reasoning goes as follows: using the square (cube, ...) one immediately sees that for a straight curve parallel to the edges of the square the curve always closes with winding number 1. A curve not parallel to the edges will close with winding number >1 (in rational cases) or it will never close (in irrational cases). But of course this is only one counter example for one specific geometry, not a general proof. So my question is if there is a geometry on a torus which respects isotropy.