In this work we study equilibrium and non-equilibrium structures of smectic block copolymer thin films deposited on a topographically patterned substrate. A Brazovskii free energy model is employed to analyze the coupling between the smectic texture and the local mean curvature of the substrate. The substrate's curvature produces out-of-plane deformations of the block copolymer such that equilibrium textures are modified and dictated by the underlying geometry. For weak curvatures it is shown that the free energy of the block copolymer film follows a Helfrich form, scaling with the square of the mean curvature, with a bending constant dependent on the local pattern orientation. On substrates of varying mean curvature simulations show that topological defects are rapidly expelled from regions with large curvature. These results compare well with available experimental data of poly(styrene)-co-poly(ethylene-alt-propylene) smectic thin films.