Geodesics on a sphere are great circles — the shortest paths between two points on the surface.
Parallel transport: moving a vector along a closed path on a curved surface returns it rotated by the holonomy angle γ = solid angle Ω enclosed = 2π(1 − cos θ) for a circle at colatitude θ.
This is the geometric (Berry) phase on the sphere. At the equator (θ=90°), γ=2π (full rotation). At the pole (θ=0°), γ=0. The holonomy equals the integral of curvature (Gauss-Bonnet theorem).