Rigidity transitions in simple models of confluent cells have been a powerful organizing principle in understanding the dynamics and mechanics of dense biological tissue. In this work we explore the interplay between geometry and rigidity in two-dimensional vertex models confined to the surface of a sphere. By considering shapes of cells defined by perimeters whose magnitude depends on geodesic distances and areas determined by spherical polygons, the critical shape index in such models is affected by the size of the cell relative to the radius of the sphere on which it is embedded. This implies that cells can collectively rigidify by growing the size of the sphere, i.e. by tuning the curvature of their domain. Finite-temperature studies indicate that cell motility is affected well away from the zero-temperature transition point.
On the existence of a finite-temperature transition in the two-dimensional gauge glass