Parallel Transport along Closed Loop Doesn't Meet!  
Blue Phase Parallel Transport has
Negative Curvature in Flat Space
Rijij = 6 q^2 
On sphere of radius 1/q in four dimensions, the curvature vanishes!
There is an ideal template for doubletwist there:

The Blue Phase is like a carpet made of orange peel (figure copyright Pamela Davis Kivelson): the edges of the cuts are the defect lines. 
James P. Sethna, sethna@lassp.cornell.edu
Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).