Successful cytokinesis requires proper assembly of the contractile actomyosin ring, its stable positioning on the cell surface and proper constriction. Over the years, many of the key molecular components and regulators of the assembly and positioning of the actomyosin ring have been elucidated. Here we show that cell geometry and mechanics play a crucial role in the stable positioning and uniform constriction of the contractile ring. Contractile rings that assemble in locally spherical regions of cells are unstable and slip towards the poles. By contrast, actomyosin rings that assemble on locally cylindrical portions of the cell under the same conditions do not slip, but uniformly constrict the cell surface. The stability of the rings and the dynamics of ring slippage can be described by a simple mechanical model. Using fluorescence imaging, we verify some of the quantitative predictions of the model. Our study reveals an intimate interplay between geometry and actomyosin dynamics, which are likely to apply in a variety of cellular contexts.