Curvature of a torus

Torus from Wikipedia

Carroll seems to say that a three-torus can have curvature zero everywhere. A two-torus does not. 

It has a curvature ##S## $$S=\frac{2\cos{\theta}}{r\left(R+r\cos{\theta}\right)}$$where ##r,R,\theta## are shown on my wire diagram. 

So the curvature is not constant. So a torus is not maximally symmetric. The curvature vanishes at the top and bottom of the torus, is negative on the inside and positive on the outside which is exactly what you would expect. Note that it gets more negative than it gets positive - probably because the inside is closer to the centre and therefore has to curve harder. If ##R=20## the minima and maxima are almost equally far from zero. 

Wrong!

I asked for help on Physics Forums here and you can have a flat torus as shown in the diagram on the right. It is is like a rectangle size ##A\times B## with coordinates ##x,y## but if you go off an edge, you come back to the opposite edge, so adding multiples of ##A## to the ##x## coordinate of ##B## to the ##y## coordinate also keep you in the same place. Pretty similar to the regular torus where adding ##2\pi## to ##\theta## or ##\phi## does the same. The only trick is that you have to start with two circles in four dimensions not three! So Carroll was messing about in six dimensions.

Here's why Commentary 8.2 Tori.pdf. (4 pages).