 It is diffeomorphic to a regular torus but not isometric.It can not be analytically embedded (smooth of class ) into Euclidean 3-space.Mapping it into "3"-space requires you to stretch it, in which case it looks like a regular torus, for example, the following map

If "R" and "P" in the above flat torus form a unit vector then "u", "v", and "η" can be used to parameterize the unit 3-sphere in a parameterization associated with the Hopf map.