Explore geometric and stability aspects of quasi-geostrophic flows on the sphere in this 48-minute mathematics lecture from the ESI's Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications." Delve into the curvature of configuration space and analyze long-term behavior of exact solutions generated by spherical harmonics. Examine the intricate relationships between curvature, conjugate points, and the Coriolis force, understanding how these geometric features influence flow stability and evolution. Learn about advanced mathematical concepts at the intersection of geometry and fluid dynamics through detailed analysis of quasi-geostrophic motion on spherical surfaces.