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Individual cells and multicellular systems respond to cell-scale curvatures in their environments, guiding migration, orientation, and tissue formation. However, it remains largely unclear how cells collectively explore and pattern complex landscapes with curvature gradients across the Euclidean and non-Euclidean spectra. Here, we show that mathematically designed substrates with controlled curvature variations induce multicellular spatiotemporal organization of preosteoblasts. We quantify curvature-induced patterning and find that cells generally prefer regions with at least one negative principal curvature. However, we also show that the developing tissue can eventually cover unfavorably curved territories, can bridge large portions of the substrates, and is often characterized by collectively aligned stress fibers. We demonstrate that this is partly regulated by cellular contractility and extracellular matrix development, underscoring the mechanical nature of curvature guidance. Our findings offer a geometric perspective on cell-environment interactions that could be harnessed in tissue engineering and regenerative medicine applications., (© 2023. The Author(s).)