  Here we show that coupling to curvature has profound effects on collective
motion in active systems, leading to patterns not observed in flat space.
Biological examples of such active motion in curved environments are numerous:
curvature and tissue folding are crucial during gastrulation, epithelial and
endothelial cells move on constantly growing, curved crypts and vili in the
gut, and the mammalian corneal epithelium grows in a steady-state vortex
pattern. On the physics side, droplets coated with actively driven microtubule
bundles show active nematic patterns. We study a model of self-propelled
particles with polar alignment on a sphere. Hallmarks of these motion patterns
are a polar vortex and a circulating band arising due to the incompatibility
between spherical topology and uniform motion - a consequence of the hairy ball
theorem. We present analytical results showing that frustration due to
curvature leads to stable elastic distortions storing energy in the band.
