  A nematic membrane is a sheet with embedded orientational order, which can
occur in biological cells, liquid crystal films, manufactured materials, and
other soft matter systems. By formulating the free energy of nematic films
using tensor contractions from differential geometry, we elucidate the elastic
terms allowed by symmetry, and indicate differences from hexatic membranes. We
find that topological defects in the orientation field can cause the membrane
to buckle over a size set by the competition between surface tension and
in-plane elasticity. In the absence of bending rigidity the resulting shape is
universal, known as a parabolic pseudosphere or a revolved tractrix. Bending
costs oppose such buckling and modify the shape in a predictable manner. In
particular, the anisotropic rigidities of nematic membranes lead to different
shapes for aster and vortex defects, in principle enabling measurement of
couplings specific to nematic membranes.
