  We study, analytically and theoretically, defects in a nematically-ordered
surface that couple to the extrinsic geometry of a surface. Though the
intrinsic geometry tends to confine topological defects to regions of large
Gaussian curvature, extrinsic couplings tend to orient the nematic in the local
direction of maximum or minimum bending. This additional frustration is
unavoidable and most important on surfaces of negative Gaussian curvature,
where it leads to a complex ground state thermodynamics. We show, in
contradistinction to the well-known effects of intrinsic geometry, that
extrinsic curvature expels disclinations from the region of maximum curvature
above a critical coupling threshold. On catenoids lacking an "inside-outside"
symmetry, defects are expelled altogether.
