  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 formulation 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.
