Fluid membranes endowed with tangent-plane order (TPO) such as tilt- and hexatic order afford unique soft matter systems for investigating the interplay between elasticity, shape, topology, and thermal fluctuations. Using the spin-connection formulat
ion of membrane energy we obtain equa- tions of equilibrium together with free boundary conditions for ground states of such membranes. We extend the spin-connection formulation to smectic liquid crystals with TPO and show that for chiral smectics-C* this generalization leads to experimentally verifiable consequences for dispirations having topological indices (helicities) of the same magnitude but opposite signs.
