I think I've missed this point somehow. I've just started with Compactification and so far, I don't really see why it is restricted to the above mentioned types of manifolds?

I have to admit, when studying T-Duality, I simply took Toroidal compactification as a kind of "why not"-thing.

Could someone point me in the right direction? Been skipping through my books to look for some explanation of this, but couldn't find anything. Thanks!