  In two dimensional nematics, topological defects are point like singularities
with both a charge and a phase. We study topological defects within curved
nematic textures on the surface of a cylinder. This allows us to isolate the
effect of extrinsic curvature on the structure of the topological defect. By
minimizing the energy associated with distortions in the nematic director
around the core of a defect we show that the phase of the topological defect is
coupled to the orientation of the cylinder. This coupling depends on the
relative energetic cost associated with splay, bend and twist distortions of
the nematic director. We identify a bistability in the phase of the defects
when twist deformations dominate. Finally, we show a similar effect for integer
charge topological defects.
