  In many biological systems, the curvature of the surfaces cells live on
influence their collective properties. Curvature should likewise influence the
behavior of active colloidal particles. We show using molecular simulation of
self-propelled active particles on surfaces of Gaussian curvature (both
positive and negative) that curvature sign and magnitude can alter the system's
collective behavior. Curvature acts as a geometrical lens and shifts the
critical density of motility-induced phase separation (MIPS) to lower values
for positive curvature and higher values for negative curvature, which we
explain theoretically by the nature of parallel lines in spherical and
hyperbolic space. Curvature also fluidizes dense MIPS clusters due to the
emergence of defect patterns disrupting the crystalline order inside the
clusters. Using our findings, we engineer three confining surfaces that
strategically combine regions of different curvature to produce a host of novel
dynamic phases, including cyclic MIPS on sphercylinders, wave-like MIPS on
spherocones, and cluster fluctuations on metaballs.
