  We study the effect of thermal fluctuations in the XY-model on a surface with
non vanishing mean curvature and zero Gaussian curvature. Unlike Gaussian
curvature that typically frustrates orientational order, the extrinsic
curvature of the surface can act as a local field that promotes long range
order at low temperature. We find numerically that the transition from the high
temperature isotropic phase to the true long range ordered phase is
characterized by critical exponents consistent with those of the flat space
Ising model in two dimensions, up to finite size effects. Our results suggest a
versatile strategy to achieve geometric control of liquid crystal order by
suitable design of the underlying curvature of a substrate or bounding surface.
