In this thesis we study the ground state of ordered phases grown as thin layers on substrates with smooth spatially varying Gaussian curvature. The Gaussian curvature acts as a source for a one body potential of purely geometrical origin that controls the equilibrium distribution of the defects in liquid crystal layers, thin films of He4 and two dimensional crystals on a frozen curved surface. For superfluids, all defects are repelled (attracted) by regions of positive (negative) Gaussian curvature. For liquid crystals, charges between 0 and 4pi are attracted by regions of positive curvature while all other charges are repelled. As the thickness of the liquid crystal film increases, transitions between two and three dimensional defect structures are triggered in the ground state of the system. Thin spherical shells of nematic molecules with planar anchoring possess four short 12 disclination lines but, as the thickness increases, a three dimensional escaped configuration composed of two pairs of half-hedgehogs becomes energetically favorable. Finally, we examine the static and dynamical properties that distinguish two dimensional crystals constrained to lie on a curved substrate from their flat space counterparts. A generic mechanism of dislocation unbinding in the presence of varying Gaussian curvature is presented. We explore how the geometric potential affects the energetics and dynamics of dislocations and point defects such as vacancies and interstitials.
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