Compatible Director Fields in $\mathbb{R}^{3}$

Relieving nematic geometric frustration in the plane

“…where (fS, f T, f b 1 , f b 2 , fq), with (S, T, b 1 , b 2 , q) all constants, are the corresponding distortion characteristics. These equations are equivalent versions, fit to the present purpose, of the compatibility conditions considered in [19]. As for using them as sufficient conditions to determine quasi-uniform distortions, the same caveats issued in [19] also apply here, as witnessed by the limited use we make of them.…”

“…11)-(B 19[4], which in a Cartesian frame (e x , e y , e z ) can be described as n = sin α cos ge x ± sin α sin ge y + cos αe z , ± sin g)e x + (cos α sin g ∓ cos g)e y − sin αe z ],…”

“…The method of connectors is in a way intermediate between Cartan's method of moving frames[18] employed in[3,4] and classical vector calculus employed in[19], being perhaps closer in spirit to the latter.…”