  It is shown that curved and flat helical double twisted liquid crystal (DTLC)
in blue phase, can be unstable (stable) depending of the sign, negative
(positive) of sectional curvature, depending on the pitch of the helix of the
nematic crystal. In both cases Cartan torsion is presented. It is also shown
that the instability or stability depends on the value of the pitch of the
helix in nematic crystals. Frank energy stability A similar result using the
method of Frank energy stability in the twist of cholesteric liquid crystal was
given by Kiselev and Sluckin [PRE 71(2005)], where the free twist number
determines the equilibrium value of the cholesteric liquid crystals (CLC) pitch
of the helix. As a final example we solve the geodesic equations in twisted
nematics with variable pitch helix and non-constant torsion. This
non-Riemannian geometrical approach, also seems to unify two recent analysis of
cylindrical columns given by Santangelo et al [PRL 99,(2007)] and the curved
crystal endowed with torsion, given by Vitelli et al [Proc Nat Acad Sci
(2006)]. Stability of toroidal curved surfaces were also previously considered
by Bowick et al [PRE 69,(2004)], as an example of curvature-induced defect.
Investigation of the Lagrangean instability may be useful in the investigation
of HIV viruses and proteins.
