Three-torus model of the universe

I want to ask a question about the Three-torus model of the universe, as described in wikipedia:

https://en.wikipedia.org/wiki/Three-torus_model_of_the_universe

"is a proposed model describing the shape of the universe as a three-dimensional torus."

I'm not sure i understand the maths of three-dimensional tori, but I want to ask:

Does this model posit that the shape of the universe is that of a ring doughnut? If so, is the universe contained inside the ring doughnut, or is it supposed to exist only on the surface?

Or does the model say that the universe has the shape of a higher dimensional version of a ring doughnut? If so, does it exist inside the higher dimensional ring doughnut, or on its surface?

• Note that this model is inconsistent with the Cosmological Principle, since it's negatively curved "on the inside", and positively curved "on the outside". – pela Apr 17 '16 at 7:52
• @pela - The only notion of curvature that's physically relevant in general relativity is intrinsic curvature, not the extrinsic curvature of a surface relative to some higher-dimensional embedding space, and the intrinsic curvature of a torus is zero everywhere, just like the intrinsic curvature of a cylinder (see the discussion on pages 2-3 here). – Hypnosifl Apr 17 '16 at 13:55
• @Hypnosifl: Hmm, I can't visualize this, but I believe you. Thanks! – pela Apr 17 '16 at 18:00