Soft and biological matter come in a variety of shapes and geometries. When soft surfaces that do not fit into each other due to a mismatch in Gaussian curvatures form an interface, beautiful geometry-induced patterns are known to emerge. In this paper, we study the effect of geometry on the dynamical response of soft surfaces moving relative to each other. Using a simple experimental scheme, we measure friction between a highly bendable thin polymer sheet and a hydrogel substrate. At this soft and low-friction interface, we find a strong dependence of friction on the relative geometry of the two surfaces—a flat sheet experiences significantly larger friction on a spherical substrate than on flat or cylindrical substrate. We show that the stress developed in the sheet due to its geometrically incompatible confinement is responsible for the enhanced friction. This mechanism also leads to a transition in the nature of friction as the sheet radius is increased beyond a critical value. Our finding reveals a hitherto unnoticed mechanism based on an interplay between geometry and elasticity that may influence friction significantly in soft, biological, and nanoscale systems. In particular, it provokes us to reexamine our understanding of phenomena such as the curvature dependence of biological cell mobility. The coming together of soft objects with incompatible geometries often leads to a rich glossary of patterns and phenomena, for example, the plethora of phases seen in bent-core liquid crystals ( 1), size-selection in the self-assembly of systems with incompatible building blocks ( 2, 3) and the emergence of beautiful wrinkle patterns when thin sheets are confined to substrates with incompatible geometries ( 4– 7). In the last case above, the incompatibility lies in the Gaussian curvature mismatch between the thin sheet and the substrate and is a consequence of the Gauss’s Theorema Egregium ( 8). This problem has been studied in various settings, including a flat sheet on spherical liquid drop ( 6), a flat sheet on a spherical solid substrate ( 9– 11), and a thin spherical shell on a flat liquid surface ( 7). The ground state obtained in these problems usually involves a nontrivial stress distribution arising due to geometrical incompatibility. While many recent papers have studied the effect of such geometrical incompatibility-induced stress distribution on static wrinkle patterns, its effect on the dynamics of such systems remains relatively unexplored.